Finite Group Databasebeta

← #10844622

Subgroups — #10844622

Order 768.

964 conjugacy classes of subgroups.

The subgroup lattice has 736 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2370C211yesmaximal, characteristic, complemented
2C2×#5066C211yesmaximal, characteristic, complemented
3C2×#5067C211yesmaximal, characteristic, complemented
4#73799C211yesmaximal, characteristic
5#69630C211yesmaximal, characteristic, complemented
6#69629C211yesmaximal, characteristic, complemented
7#69627C211yesmaximal, characteristic, complemented
8#69627C211yesmaximal, characteristic, complemented
9#69487C211yesmaximal, characteristic, complemented
10#69487C211yesmaximal, characteristic, complemented
11#69487C211yesmaximal, characteristic, complemented
12#69485C211yesmaximal, characteristic, complemented
13#69474C211yesmaximal, characteristic, complemented
14#69463C211yesmaximal, characteristic, complemented
15#69462C211yesmaximal, characteristic, complemented
16#25288—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C2×#895—21no—
19C2×#895—21no—
20C2×#874C2211yescharacteristic
21C2×#874C2211yescharacteristic
22C2×#874C2211yescharacteristic
23C2×C4×D12C2211yescharacteristic
24C3×#389—21no—
25C3×#389C2211yescharacteristic
26C3×#389C2211yescharacteristic
27C3×#387—21no—
28C3×#387—21no—
29C3×C4×C4⋊C4—21no—
30C3×C4×C4⋊C4C2211yescharacteristic
31C3×C4×C4⋊C4C2211yescharacteristic
32C3×C4×C4⋊C4C2211yescharacteristic
33C3×C4×C4⋊C4C2211yescharacteristic
34#5067—24no—
35#5066—24no—
36C2×#809C2211yescharacteristic
37C2×#809C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#809C2211yescharacteristic
46C2×#809C2211yescharacteristic
47C2×#809C2211yescharacteristic
48C2×#808C2211yescharacteristic
49C2×#807C2211yescharacteristic
50#4939C2211yescharacteristic
51#4939C2211yescharacteristic
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54C2×C4⋊(C3⋊C8)C2211yescharacteristic
55C2×#803C2211yescharacteristic
56#4671—21no—
57#4671—21no—
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#490—31no—
63C2×#478—31no—
64#2541—31no—
65#2370D311yescharacteristic, complemented
66#2067—31no—
67#2065—31no—
68#2041—31no—
69#2041—31no—
70#1982—31no—
71#1982—31no—
72#1982—31no—
73#1940—31no—
74#1926—31no—
75#1925—31no—
76#1762—31no—
77C22×D12—21no—
78C22×D12—21no—
79C22×D12—21no—
80C22×D12—21no—
81C22×D12—21no—
82C22×D12—21no—
83C22×D12C2311yescharacteristic
84C22×D12C2311yescharacteristic
85C22×D12C2311yescharacteristic
86C22×C4×D3—21no—
87C3×#125—22no—
88C3×#125D414yescomplemented
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4D411yescharacteristic
94C2×C3×C4⋊C4D411yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C42—21no—
100C2×C3×C42—21no—
101C2×C3×C42—21no—
102C2×C3×C42C2311yescharacteristic
103C2×C3×C42C2311yescharacteristic
104C2×C3×C42C2311yescharacteristic
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×C4⋊Dic3C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115#895—44no—
116#895—44no—
117#874—22no—
118#874—22no—
119#874—22no—
120#874—22no—
121#874—41no—
122#874—41no—
123C4×D12—22no—
124C4×D12—22no—
125C3×#95—21no—
126C3×#95—21no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—21no—
130C3×#95D411yescharacteristic
131C3×#95D411yescharacteristic
132#809—22no—
133#809—22no—
134#809—22no—
135#809—22no—
136#809—22no—
137#809—22no—
138#809—22no—
139#809—22no—
140#809—22no—
141#809—22no—
142#809—41no—
143#809—41no—
144#808—22no—
145#807—22no—
146C4⋊(C3⋊C8)—22no—
147C4⋊(C3⋊C8)—41no—
148C4⋊(C3⋊C8)—41no—
149#803—22no—
150C2×#127—31no—
151C2×#127—31no—
152C2×#127—31no—
153C2×#125C2×D311yescharacteristic
154C2×#121—61no—
155C2×#121—61no—
156C2×C4×D4—31no—
157#490—64no—
158#478—64no—
159C2×#107—31no—
160C2×#106—31no—
161C2×C4⋊C8—31no—
162C2×C4⋊C8—31no—
163C2×C4⋊C8—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×#102—31no—
171C2×#102—31no—
172C2×#102—31no—
173C2×#102—31no—
174C2×#102—31no—
175C2×#102—31no—
176C2×#97—31no—
177#392—61no—
178#392—61no—
179#390—31no—
180#390—31no—
181#389—21no—
182#389C2×D311yescharacteristic
183#389C2×D311yescharacteristic
184#387—21no—
185#387—21no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4C2×D311yescharacteristic
188C4×C4⋊C4C2×D311yescharacteristic
189C4×C4⋊C4C2×D311yescharacteristic
190C4×C4⋊C4C2×D311yescharacteristic
191#340—31no—
192#340—31no—
193#336—31no—
194#336—31no—
195C23×D3—21no—
196C23×D3—21no—
197C23×D3—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2411yescharacteristic
209C22×Dic3—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12—42no—
228C2×D12—42no—
229C2×D12—42no—
230C2×D12—42no—
231C2×D12—42no—
232C2×D12—42no—
233C2×D12—42no—
234C2×D12—42no—
235C2×D12—42no—
236C2×D12—42no—
237C2×D12—42no—
238C2×D12—42no—
239C2×D12—81no—
240C2×D12—81no—
241C2×D12—81no—
242C2×D12—81no—
243C2×D12—81no—
244C2×D12—81no—
245C2×D12#4112yes—
246C2×C4×D3—42no—
247C2×C4×D3—42no—
248C2×C4×D3—42no—
249C3×C4⋊C4—24no—
250C3×C4⋊C4—24no—
251C3×C4⋊C4—22no—
252C3×C4⋊C4—22no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—42no—
259C3×C4⋊C4—81no—
260C3×C4⋊C4C2×D412yes—
261C3×C4⋊C4C2×D412yes—
262C3×C4⋊C4#4112yes—
263C3×C42—21no—
264C3×C42—21no—
265C3×C42—21no—
266C3×C42—21no—
267C3×C42—42no—
268C3×C42—81no—
269C3×C42—81no—
270C3×C42C2×D412yes—
271C3×C42#4112yes—
272(C2×D3)⋊C4—42no—
273(C2×D3)⋊C4—42no—
274(C2×D3)⋊C4—42no—
275(C2×D3)⋊C4—42no—
276(C2×D3)⋊C4—42no—
277C4⋊Dic3—21no—
278C4⋊Dic3—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C22×D4—31no—
292C22×D4—31no—
293C22×D4—31no—
294C22×D4—61no—
295C22×D4—61no—
296C22×D4—61no—
297C22×D4—61no—
298C22×D4—61no—
299C22×D4—61no—
300C23×C4—61no—
301C22×C8—31no—
302C22×C8—31no—
303C22×C8—31no—
304C22×C8—31no—
305#127—62no—
306#127—62no—
307#127—62no—
308#127—62no—
309#127—121no—
310#127—121no—
311#125—22no—
312#125D1214yescomplemented
313#121—124no—
314#121—124no—
315C4×D4—62no—
316C4×D4—62no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—31no—
322C2×C4⋊C4#6711yescharacteristic
323C2×C4⋊C4#6711yescharacteristic
324C2×C4⋊C4C22×D311yescharacteristic
325C2×C4⋊C4C22×D311yescharacteristic
326C2×C4⋊C4C22×D311yescharacteristic
327C2×C4⋊C4C22×D311yescharacteristic
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42C22×D311yescharacteristic
337C2×C42C22×D311yescharacteristic
338C2×C42C22×D311yescharacteristic
339#107—62no—
340#106—62no—
341C4⋊C8—62no—
342C4⋊C8—121no—
343C4⋊C8—121no—
344#102—62no—
345#102—62no—
346#102—62no—
347#102—62no—
348#102—62no—
349#102—62no—
350#102—62no—
351#102—62no—
352#102—62no—
353#102—62no—
354#102—121no—
355#102—121no—
356#97—62no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95#6711yescharacteristic
363#95#6711yescharacteristic
364C23×C3C22×D411yescharacteristic
365C22×D3—22no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C22×D3—41no—
370C22×D3—41no—
371C22×D3—41no—
372C22×D3—41no—
373C22×D3—41no—
374C22×D3—41no—
375C22×D3—41no—
376C22×D3—41no—
377C22×D3—41no—
378C22×D3—41no—
379C22×D3—41no—
380C22×D3—41no—
381C22×D3—41no—
382C22×D3—41no—
383C22×D3—41no—
384C22×D3—41no—
385C22×D3—41no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—22no—
395C2×C3×C4—22no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—22no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4—41no—
408C2×C3×C4—41no—
409C2×C3×C4—41no—
410C2×C3×C4—41no—
411C2×C3×C4—41no—
412C2×C3×C4#12112yes—
413C2×C3×C4#12112yes—
414C2×C3×C4AGL(1,ℤ/8)12yes—
415C2×C3×C4AGL(1,ℤ/8)12yes—
416C2×C3×C4C22×D411yescharacteristic
417C2×C3×C4C2×#4111yescharacteristic
418C2×C3×C4C2×#4111yescharacteristic
419C2×C3×C4C2×#4111yescharacteristic
420C2×C3×C4H(5,𝔽2)11yescharacteristic
421C2×C3×C42−1+411yescharacteristic
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425D12—22no—
426D12—22no—
427D12—41no—
428D12—41no—
429D12—41no—
430D12—41no—
431D12—41no—
432D12—41no—
433D12—41no—
434D12—41no—
435D12—41no—
436D12—41no—
437D12—84no—
438D12—84no—
439D12—84no—
440D12—84no—
441D12—84no—
442D12—84no—
443C4×D3—84no—
444C3⋊C8—41no—
445C3⋊C8—41no—
446C3⋊C8—41no—
447C3⋊C8—41no—
448C24—61no—
449C24—61no—
450C24—61no—
451C2×D4—32no—
452C2×D4—61no—
453C2×D4—61no—
454C2×D4—61no—
455C2×D4—61no—
456C2×D4—61no—
457C2×D4—61no—
458C2×D4—61no—
459C2×D4—61no—
460C2×D4—61no—
461C2×D4—61no—
462C2×D4—61no—
463C2×D4—61no—
464C2×D4—61no—
465C2×D4—61no—
466C2×D4—61no—
467C2×D4—61no—
468C2×D4—61no—
469C2×D4—122no—
470C2×D4—122no—
471C2×D4—122no—
472C2×D4—122no—
473C2×D4—122no—
474C2×D4—122no—
475C2×D4—122no—
476C2×D4—122no—
477C2×D4—122no—
478C2×D4—122no—
479C2×D4—122no—
480C2×D4—122no—
481C2×D4—241no—
482C2×D4—241no—
483C2×D4—241no—
484C2×D4—241no—
485C2×D4—241no—
486C2×D4—241no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—61no—
492C22×C4—122no—
493C22×C4—122no—
494C22×C4—122no—
495C22×C4C2×#6711yescharacteristic
496C22×C4C2×#6711yescharacteristic
497C22×C4C2×#6711yescharacteristic
498C22×C4C2×#6711yescharacteristic
499C22×C4C2×#6711yescharacteristic
500C22×C4C2×#6711yescharacteristic
501C22×C4C23×D311yescharacteristic
502C2×C8—61no—
503C2×C8—61no—
504C2×C8—61no—
505C2×C8—61no—
506C2×C8—61no—
507C2×C8—61no—
508C2×C8—61no—
509C2×C8—61no—
510C2×C8—61no—
511C2×C8—61no—
512C2×C8—61no—
513C2×C8—61no—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—42no—
524C4⋊C4—61no—
525C4⋊C4—61no—
526C4⋊C4—81no—
527C4⋊C4C2×D1212yes—
528C4⋊C4C2×D1212yes—
529C4⋊C4#23912yes—
530C22⋊C4—122no—
531C22⋊C4—122no—
532C22⋊C4—122no—
533C22⋊C4—122no—
534C22⋊C4—122no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—42no—
540C42—81no—
541C42—81no—
542C42C2×D1212yes—
543C42#23512yes—
544C22×C3C2×#12111yescharacteristic
545C22×C3#54611yescharacteristic
546C22×C3#54811yescharacteristic
547C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
548C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
549C22×C3#57611yescharacteristic
550C22×C3#57711yescharacteristic
551C2×D3—42no—
552C2×D3—42no—
553C2×D3—42no—
554C2×D3—81no—
555C2×D3—81no—
556C2×D3—81no—
557C2×D3—81no—
558C2×D3—81no—
559C2×D3—81no—
560C2×D3—81no—
561C2×D3—81no—
562C2×D3—81no—
563C2×D3—81no—
564C2×D3—81no—
565C2×D3—81no—
566C2×D3—81no—
567C2×D3—81no—
568C2×D3—81no—
569C2×D3—81no—
570C2×D3—81no—
571C2×D3—81no—
572C3×C4—22no—
573C3×C4—22no—
574C3×C4—22no—
575C3×C4—22no—
576C3×C4—42no—
577C3×C4—42no—
578C3×C4—42no—
579C3×C4—41no—
580C3×C4—41no—
581C3×C4—81no—
582C3×C4#52911yescharacteristic
583C3×C4#54211yescharacteristic
584C3×C4#54811yescharacteristic
585C3×C4#55011yescharacteristic
586Dic3—81no—
587C23—62no—
588C23—121no—
589C23—121no—
590C23—121no—
591C23—121no—
592C23—121no—
593C23—121no—
594C23—121no—
595C23—121no—
596C23—121no—
597C23—121no—
598C23—121no—
599C23—121no—
600C23—121no—
601C23—121no—
602C23—121no—
603C23—121no—
604C23—121no—
605C23—121no—
606C23—121no—
607C23—121no—
608C23C22×#6711yescharacteristic
609D4—62no—
610D4—62no—
611D4—121no—
612D4—121no—
613D4—121no—
614D4—121no—
615D4—121no—
616D4—121no—
617D4—121no—
618D4—121no—
619D4—121no—
620D4—121no—
621D4—244no—
622D4—244no—
623D4—244no—
624D4—244no—
625D4—244no—
626D4—244no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—121no—
654C2×C4—121no—
655C2×C4—121no—
656C2×C4—244no—
657C2×C4#93012yes—
658C2×C4#93012yes—
659C2×C4#94912yes—
660C2×C4#94912yes—
661C2×C4C22×D1211yescharacteristic
662C2×C4C2×#23511yescharacteristic
663C2×C4C2×#23911yescharacteristic
664C2×C4#100711yescharacteristic
665C2×C4D3×#4111yescharacteristic
666C2×C4#100911yescharacteristic
667C8—121no—
668C8—121no—
669C8—121no—
670C8—121no—
671C2×C3#242611yescharacteristic
672C2×C3#303811yescharacteristic
673C2×C3#304511yescharacteristic
674C2×C3#331011yescharacteristic
675C2×C3#331811yescharacteristic
676C2×C3#336611yescharacteristic
677C2×C3#336811yescharacteristic
678D3—82no—
679D3—161no—
680D3—161no—
681C22—122no—
682C22—122no—
683C22—122no—
684C22—241no—
685C22—241no—
686C22—241no—
687C22—241no—
688C22—241no—
689C22—241no—
690C22—241no—
691C22—241no—
692C22—241no—
693C22—241no—
694C22—241no—
695C22—241no—
696C22—241no—
697C22—241no—
698C22—241no—
699C22—241no—
700C22—241no—
701C22—241no—
702C22C2×#93011yescharacteristic
703C22#581711yescharacteristic
704C22C2×#94911yescharacteristic
705C22C2×#94911yescharacteristic
706C22#582911yescharacteristic
707C22#583511yescharacteristic
708C22#583711yescharacteristic
709C4—22no—
710C4—22no—
711C4—22no—
712C4—22no—
713C4—42no—
714C4—42no—
715C4—41no—
716C4—41no—
717C4—42no—
718C4—81no—
719C4—241no—
720C4#557611yescharacteristic
721C4#557711yescharacteristic
722C4#558011yescharacteristic
723C4#558211yescharacteristic
724C3#2528811yescharacteristic, complemented, Sylow
725C2—242no—
726C2—481no—
727C2—481no—
728C2#8249511yescharacteristic
729C2#8252611yescharacteristic
730C2#8256511yescharacteristic
731C2#8359911yescharacteristic
732C2#8361611yescharacteristic
733C2#8363811yescharacteristic
734C2#8369011yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2370C2
socleC23×C3C22×D4
prime radical#73799C2
hypercenter#2370D3
hypocenterC3#25288
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2370

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×#2370C2semidirect
2. C2×#5066C2semidirect
3. C2×#5067C2semidirect
5. #69630C2semidirect
6. #69629C2semidirect
7. #69627C2semidirect
8. #69627C2semidirect
9. #69487C2semidirect
10. #69487C2semidirect
11. #69487C2semidirect
12. #69485C2semidirect
13. #69474C2semidirect
14. #69463C2semidirect
15. #69462C2semidirect
65. #2370D3semidirect
88. C3×#125D4semidirect
312. #125D12semidirect
724. C3#25288semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×#23702
C2×#50662
C2×#50672
#737992
#696302
#696292
#696272
#694872
#694852
#694742
#694632
#694622
#252883