Finite Group Databasebeta

← #10876103

Subgroups — #10876103

Order 768.

772 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2349C211yesmaximal, characteristic, complemented
2#73694C211yesmaximal, characteristic, complemented
3#71613C211yesmaximal, characteristic
4#71613C211yesmaximal, characteristic
5#71613C211yesmaximal, characteristic
6#71609C211yesmaximal, characteristic, complemented
7#71606C211yesmaximal, characteristic, complemented
8#71605C211yesmaximal, characteristic, complemented
9#69630C211yesmaximal, characteristic, complemented
10#69617C211yesmaximal, characteristic, complemented
11#69613C211yesmaximal, characteristic, complemented
12#69600C211yesmaximal, characteristic, complemented
13#69557C211yesmaximal, characteristic, complemented
14#69542C211yesmaximal, characteristic, complemented
15#69483C211yesmaximal, characteristic, complemented
16#27581—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C22×#213C2211yescharacteristic
21C3×#389C2211yescharacteristic
22C3×#389C2211yescharacteristic
23C3×#385—21no—
24C3×#385—21no—
25C3×#382C2211yescharacteristic
26C3×#380—21no—
27C3×C4×C4⋊C4C2211yescharacteristic
28C2×C3×#95—21no—
29C2×#832C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C2×#832C2211yescharacteristic
32#4992—21no—
33#4990—21no—
34C2×#809C2211yescharacteristic
35C2×#809C2211yescharacteristic
36C2×#809C2211yescharacteristic
37C2×#807C2211yescharacteristic
38#4939C2211yescharacteristic
39#4939C2211yescharacteristic
40#4939C2211yescharacteristic
41#4936C2211yescharacteristic
42C4×C4⋊Dic3C2211yescharacteristic
43#4928C2211yescharacteristic
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45C2×#803C2211yescharacteristic
46#4675—21no—
47#4669—21no—
48#4529C2211yescharacteristic
49#4529C2211yescharacteristic
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4524C2211yescharacteristic
57#4524C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2513—31no—
61#2349D311yescharacteristic, complemented
62#2101—31no—
63#2101—31no—
64#2073—31no—
65#2067—31no—
66#1936—31no—
67#1918—31no—
68#1917—31no—
69#1879—31no—
70#1879—31no—
71#1879—31no—
72#1862—31no—
73#1804—31no—
74#1762—31no—
75C22×C3×D4C2311yescharacteristic
76C23×C3×C4—21no—
77C23×C3×C4—21no—
78C23×C3×C4—21no—
79C22×D12C2311yescharacteristic
80C3×C4×D4—22no—
81C3×C4×D4—22no—
82C3×C4×D4—22no—
83C3×C4×D4—22no—
84C3×C4×D4—41no—
85C3×C4×D4—41no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C4⋊C4D411yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C42C2311yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C2×C3×C42C2311yescharacteristic
99C2×#213—22no—
100C2×#213—22no—
101C2×#213—22no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×Dic3⋊C4—21no—
112C2×C4×Dic3—21no—
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C3×#95—21no—
118C3×#95—21no—
119C3×#95—21no—
120C3×#95—21no—
121C3×#95D411yescharacteristic
122C3×#95D411yescharacteristic
123#832—22no—
124#832—22no—
125#832—22no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130#809—22no—
131#809—41no—
132#809—41no—
133#807—22no—
134C4⋊(C3⋊C8)—41no—
135#803—41no—
136C22×D8—31no—
137C2×C4×D4C2×D311yescharacteristic
138C2×C4×D4C2×D311yescharacteristic
139C2×C4×D4C2×D311yescharacteristic
140C2×#107—31no—
141C2×C4⋊C8—31no—
142C2×#102—31no—
143C2×#102—31no—
144C2×#102—31no—
145C2×#102—31no—
146C2×#102—31no—
147C2×#102—31no—
148C2×#97—31no—
149#397—61no—
150#397—61no—
151#390—31no—
152#390—31no—
153#390—31no—
154#389—31no—
155#389C2×D311yescharacteristic
156#389C2×D311yescharacteristic
157#388—61no—
158#388—61no—
159#385—21no—
160#385—21no—
161#382—31no—
162#382C2×D311yescharacteristic
163#380—21no—
164C4×C4⋊C4—31no—
165C4×C4⋊C4C2×D311yescharacteristic
166C2×#95—21no—
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#336—31no—
176#336—31no—
177#336—31no—
178#336—31no—
179C24×C3—21no—
180C23×D3—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4#4112yes—
185C2×C3×D4#4112yes—
186C2×C3×D4#4112yes—
187C22×C3×C4—21no—
188C22×C3×C4—24no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—42no—
192C22×C3×C4—42no—
193C22×C3×C4—42no—
194C22×C3×C4—42no—
195C22×C3×C4—81no—
196C22×C3×C4—81no—
197C22×C3×C4—81no—
198C22×C3×C4—81no—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2411yescharacteristic
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—41no—
221C3×C4⋊C4—41no—
222C3×C4⋊C4—81no—
223C3×C4⋊C4—81no—
224C3×C4⋊C4#4112yes—
225C3×C22⋊C4—24no—
226C3×C22⋊C4—42no—
227C3×C22⋊C4—42no—
228C3×C42—21no—
229C3×C42—21no—
230C3×C42—21no—
231C3×C42—21no—
232C3×C42—21no—
233C3×C42—21no—
234#213—44no—
235(C2×D3)⋊C4—42no—
236(C2×D3)⋊C4—42no—
237(C2×D3)⋊C4—81no—
238(C2×D3)⋊C4—81no—
239(C2×D3)⋊C4—81no—
240(C2×D3)⋊C4—81no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247Dic3⋊C4—81no—
248C4×Dic3—81no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C22×D4—31no—
262C22×D4C22×D311yescharacteristic
263C23×C4—21no—
264C23×C4—21no—
265C23×C4—21no—
266C2×D8—62no—
267C2×D8—62no—
268C2×D8—62no—
269C22×C8—31no—
270C22×C8—31no—
271C22×C8—31no—
272C22×C8—31no—
273C4×D4—22no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—41no—
278C4×D4—41no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4#6711yescharacteristic
286C2×C4⋊C4#6711yescharacteristic
287C2×C4⋊C4C22×D311yescharacteristic
288C2×C4⋊C4C22×D311yescharacteristic
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C42—61no—
300C2×C42C22×D311yescharacteristic
301C2×C42C22×D311yescharacteristic
302C2×C42C22×D311yescharacteristic
303#107—62no—
304C4⋊C8—121no—
305#102—62no—
306#102—62no—
307#102—62no—
308#102—62no—
309#102—121no—
310#102—121no—
311#97—121no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320#95#6711yescharacteristic
321#95#6711yescharacteristic
322C23×C3—22no—
323C23×C3—22no—
324C23×C3—22no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3C22×D411yescharacteristic
330C22×D3—41no—
331C22×D3—41no—
332C22×D3—41no—
333C22×D3—41no—
334C22×D3—41no—
335C22×D3—41no—
336C22×D3—41no—
337C3×D4—22no—
338C3×D4—22no—
339C3×D4—22no—
340C3×D4#12614yes—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—22no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—84no—
369C2×C3×C4—84no—
370C2×C3×C4—84no—
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4H(5,𝔽2)11yescharacteristic
376C2×C3×C42−1+411yescharacteristic
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386D12—41no—
387D12—41no—
388D12—41no—
389D12—41no—
390C3⋊C8—41no—
391C3⋊C8—41no—
392C3⋊C8—41no—
393C3⋊C8—41no—
394C24—21no—
395C24—61no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—61no—
400C2×D4—61no—
401C2×D4—61no—
402C2×D4—61no—
403C2×D4—61no—
404C2×D4—61no—
405C2×D4#23512yes—
406C2×D4#23712yes—
407C2×D4#23912yes—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4—81no—
422C22×C4—81no—
423C22×C4C2×#6711yescharacteristic
424C22×C4C2×#6711yescharacteristic
425C22×C4C2×#6711yescharacteristic
426C22×C4C2×#6711yescharacteristic
427C22×C4C2×#6711yescharacteristic
428C22×C4C2×#6711yescharacteristic
429C22×C4C23×D311yescharacteristic
430D8—124no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—61no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—81no—
457C4⋊C4—81no—
458C4⋊C4—241no—
459C4⋊C4#23512yes—
460C22⋊C4—24no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—122no—
464C22⋊C4—122no—
465C22⋊C4—241no—
466C22⋊C4—241no—
467C22⋊C4—241no—
468C22⋊C4—241no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—241no—
476C22×C3—24no—
477C22×C3—42no—
478C22×C3—42no—
479C22×C3—42no—
480C22×C3—42no—
481C22×C3—42no—
482C22×C3—42no—
483C22×C3#52511yescharacteristic
484C22×C3#54611yescharacteristic
485C22×C3#54811yescharacteristic
486C22×C3#57511yescharacteristic
487C22×C3#57511yescharacteristic
488C22×C3#57611yescharacteristic
489C22×C3#57611yescharacteristic
490C2×D3—81no—
491C2×D3—81no—
492C2×D3—81no—
493C2×D3—81no—
494C2×D3—81no—
495C2×D3—81no—
496C2×D3—81no—
497C3×C4—41no—
498C3×C4—41no—
499C3×C4—42no—
500C3×C4—41no—
501C3×C4—41no—
502C3×C4—41no—
503C3×C4—41no—
504C3×C4—81no—
505C3×C4—81no—
506C3×C4C2×#12611yescharacteristic
507C3×C4#53911yescharacteristic
508C3×C4#55211yescharacteristic
509C3×C4#55311yescharacteristic
510Dic3—81no—
511Dic3—81no—
512Dic3—81no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—121no—
521C23—121no—
522C23—121no—
523C23—121no—
524C23—121no—
525C23—121no—
526C23—121no—
527C23C22×#6711yescharacteristic
528D4—22no—
529D4—22no—
530D4—22no—
531D4—121no—
532D4—121no—
533D4—121no—
534D4—121no—
535D4#89814yes—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—84no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4C2×#23511yescharacteristic
576C2×C4C2×#23511yescharacteristic
577C2×C4C2×#23711yescharacteristic
578C2×C4#100411yescharacteristic
579C2×C4C2×#23911yescharacteristic
580C2×C4#100711yescharacteristic
581C8—121no—
582C8—121no—
583C8—121no—
584C8—121no—
585C2×C3—44no—
586C2×C3#248111yescharacteristic
587C2×C3#311111yescharacteristic
588C2×C3#312211yescharacteristic
589C2×C3#331911yescharacteristic
590C2×C3#332211yescharacteristic
591C2×C3#337011yescharacteristic
592C2×C3#337011yescharacteristic
593D3—161no—
594C22—24no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—241no—
602C22—241no—
603C22—241no—
604C22—241no—
605C22—241no—
606C22—241no—
607C22—241no—
608C22#579111yescharacteristic
609C22#580611yescharacteristic
610C22#581711yescharacteristic
611C22#582211yescharacteristic
612C22#582211yescharacteristic
613C22#583511yescharacteristic
614C22#583511yescharacteristic
615C4—41no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—42no—
622C4—81no—
623C4—81no—
624C4—241no—
625C4—241no—
626C4—241no—
627C4C2×#89811yescharacteristic
628C4#551311yescharacteristic
629C4#551411yescharacteristic
630C4#551511yescharacteristic
631C3#2758111yescharacteristic, complemented, Sylow
632C2—44no—
633C2—481no—
634C2#8217511yescharacteristic
635C2#8253011yescharacteristic
636C2#8254211yescharacteristic
637C2#8349911yescharacteristic
638C2#8350011yescharacteristic
639C2#8363911yescharacteristic
640C2#8363911yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2349C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenter#2349D3
hypocenterC3#27581
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
1. C3×#2349C2semidirect
2. #73694C2semidirect
6. #71609C2semidirect
7. #71606C2semidirect
8. #71605C2semidirect
9. #69630C2semidirect
10. #69617C2semidirect
11. #69613C2semidirect
12. #69600C2semidirect
13. #69557C2semidirect
14. #69542C2semidirect
15. #69483C2semidirect
61. #2349D3semidirect
631. C3#27581semidirect

Maximal subgroups

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

subgroupindex
C3×#23492
#736942
#716132
#716092
#716062
#716052
#696302
#696172
#696132
#696002
#695572
#695422
#694832
#275813