Finite Group Databasebeta

← #10844667

Subgroups — #10844667

Order 768.

828 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2370C211yesmaximal, characteristic, complemented
2C2×#5063C211yesmaximal, characteristic, complemented
3C2×#5067C211yesmaximal, characteristic
4#73712C211yesmaximal, characteristic
5#69630C211yesmaximal, characteristic, complemented
6#69629C211yesmaximal, characteristic, complemented
7#69628C211yesmaximal, characteristic, complemented
8#69628C211yesmaximal, characteristic, complemented
9#69488C211yesmaximal, characteristic, complemented
10#69488C211yesmaximal, characteristic, complemented
11#69487C211yesmaximal, characteristic
12#69485C211yesmaximal, characteristic, complemented
13#69475C211yesmaximal, characteristic, complemented
14#69463C211yesmaximal, characteristic, complemented
15#69462C211yesmaximal, characteristic, complemented
16#25291—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C2×#875C2211yescharacteristic
19C2×#875C2211yescharacteristic
20C2×#874C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C3×#389—21no—
23C3×#389C2211yescharacteristic
24C3×#389C2211yescharacteristic
25C3×#387—21no—
26C3×#387—21no—
27C3×C4×C4⋊C4—21no—
28C3×C4×C4⋊C4C2211yescharacteristic
29C3×C4×C4⋊C4C2211yescharacteristic
30C3×C4×C4⋊C4C2211yescharacteristic
31C3×C4×C4⋊C4C2211yescharacteristic
32#5067—24no—
33#5063—24no—
34#4988—21no—
35#4988—21no—
36C2×#810C2211yescharacteristic
37C2×#810C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#810C2211yescharacteristic
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#4672—21no—
57#4672—21no—
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#490—31no—
63C2×#479—31no—
64#2548—31no—
65#2370D311yescharacteristic, complemented
66#2067—31no—
67#2065—31no—
68#2042—31no—
69#2042—31no—
70#1983—31no—
71#1983—31no—
72#1982—31no—
73#1940—31no—
74#1926—31no—
75#1925—31no—
76#1763—31no—
77C22×D12—21no—
78C22×D12—21no—
79C22×D12C2311yescharacteristic
80C22×D12C2311yescharacteristic
81C22×Dic6C2311yescharacteristic
82C3×#125—22no—
83C3×#125D414yes—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
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×C4⋊C4C2311yescharacteristic
94C2×C3×C42—21no—
95C2×C3×C42—21no—
96C2×C3×C42—21no—
97C2×C3×C42C2311yescharacteristic
98C2×C3×C42C2311yescharacteristic
99C2×C3×C42C2311yescharacteristic
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
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×Dic3⋊C4—21no—
110C2×C4×Dic3—21no—
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115#875—22no—
116#875—22no—
117#875—41no—
118#875—41no—
119#874—22no—
120#874—22no—
121C4×Dic6—22no—
122C4×Dic6—22no—
123C3×#95—21no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95—21no—
127C3×#95—21no—
128C3×#95D411yescharacteristic
129C3×#95D411yescharacteristic
130#810—22no—
131#810—22no—
132#810—22no—
133#810—22no—
134#809—22no—
135#809—22no—
136#809—22no—
137#809—22no—
138#809—22no—
139#809—22no—
140#809—41no—
141#809—41no—
142#808—22no—
143#807—22no—
144C4⋊(C3⋊C8)—22no—
145C4⋊(C3⋊C8)—41no—
146C4⋊(C3⋊C8)—41no—
147#803—22no—
148C2×#127—31no—
149C2×#125C2×D311yescharacteristic
150C2×#124—31no—
151C2×#124—31no—
152C2×C4×Q8—31no—
153#490—64no—
154#479—64no—
155C2×#107—31no—
156C2×#106—31no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—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×#97—31no—
173#394—61no—
174#394—61no—
175#390—31no—
176#390—31no—
177#390—61no—
178#390—61no—
179#389—21no—
180#389C2×D311yescharacteristic
181#389C2×D311yescharacteristic
182#387—21no—
183#387—21no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4C2×D311yescharacteristic
186C4×C4⋊C4C2×D311yescharacteristic
187C4×C4⋊C4C2×D311yescharacteristic
188C4×C4⋊C4C2×D311yescharacteristic
189#340—31no—
190#340—31no—
191#336—31no—
192#336—31no—
193C23×D3—21no—
194C23×D3—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
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—
208C2×D12—21no—
209C2×D12—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—42no—
221C2×D12—42no—
222C2×D12—42no—
223C2×D12—42no—
224C2×D12—81no—
225C2×D12—81no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6#4112yes—
232C3×C4⋊C4—24no—
233C3×C4⋊C4—24no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—41no—
240C3×C4⋊C4—41no—
241C3×C4⋊C4—42no—
242C3×C4⋊C4—81no—
243C3×C4⋊C4C2×D412yes—
244C3×C4⋊C4C2×D412yes—
245C3×C4⋊C4#4112yes—
246C3×C42—21no—
247C3×C42—21no—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42—42no—
251C3×C42—81no—
252C3×C42—81no—
253C3×C42C2×D412yes—
254C3×C42#4112yes—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4—42no—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—42no—
259(C2×D3)⋊C4—81no—
260(C2×D3)⋊C4—81no—
261(C2×D3)⋊C4—81no—
262(C2×D3)⋊C4—81no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265Dic3⋊C4—42no—
266C4×Dic3—42no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C22×Q8—31no—
280C22×D4—31no—
281C22×D4—31no—
282C22×D4—61no—
283C22×D4—61no—
284C22×C8—31no—
285C22×C8—31no—
286C22×C8—31no—
287C22×C8—31no—
288#127—62no—
289#127—62no—
290#125—22no—
291#125D1214yes—
292#124—62no—
293#124—62no—
294#124—121no—
295#124—121no—
296C4×Q8—62no—
297C4×Q8—62no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4#6711yescharacteristic
305C2×C4⋊C4#6711yescharacteristic
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C4⋊C4C22×D311yescharacteristic
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—61no—
322C2×C42C22×D311yescharacteristic
323C2×C42C22×D311yescharacteristic
324C2×C42C22×D311yescharacteristic
325#107—62no—
326#106—62no—
327C4⋊C8—62no—
328C4⋊C8—121no—
329C4⋊C8—121no—
330#103—62no—
331#103—62no—
332#103—62no—
333#103—62no—
334#102—62no—
335#102—62no—
336#102—62no—
337#102—62no—
338#102—62no—
339#102—62no—
340#102—121no—
341#102—121no—
342#97—62no—
343#95—21no—
344#95—21no—
345#95—21no—
346#95—21no—
347#95—21no—
348#95#6711yescharacteristic
349#95#6711yescharacteristic
350C23×C3C22×D411yescharacteristic
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—41no—
355C22×D3—41no—
356C22×D3—41no—
357C22×D3—41no—
358C22×D3—41no—
359C22×D3—41no—
360C22×D3—41no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—22no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4#12112yes—
394C2×C3×C4#12112yes—
395C2×C3×C4C22×D411yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×C3×C4C2×#4111yescharacteristic
398C2×C3×C4C2×#4111yescharacteristic
399C2×C3×C4H(5,𝔽2)11yescharacteristic
400C2×C3×C42−1+411yescharacteristic
401C2×Dic3—22no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407D12—41no—
408D12—41no—
409D12—41no—
410D12—41no—
411D12—41no—
412D12—41no—
413D12—41no—
414D12—41no—
415D12—84no—
416D12—84no—
417Dic6—22no—
418Dic6—22no—
419Dic6—41no—
420Dic6—41no—
421C3⋊C8—41no—
422C3⋊C8—41no—
423C3⋊C8—41no—
424C3⋊C8—41no—
425C24—61no—
426C24—61no—
427C2×Q8—32no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×D4—61no—
434C2×D4—61no—
435C2×D4—61no—
436C2×D4—61no—
437C2×D4—61no—
438C2×D4—61no—
439C2×D4—61no—
440C2×D4—61no—
441C2×D4—61no—
442C2×D4—61no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4—122no—
446C2×D4—122no—
447C2×D4—122no—
448C2×D4—122no—
449C2×D4—241no—
450C2×D4—241no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4C2×#6711yescharacteristic
458C22×C4C2×#6711yescharacteristic
459C22×C4C2×#6711yescharacteristic
460C22×C4C2×#6711yescharacteristic
461C22×C4C2×#6711yescharacteristic
462C22×C4C2×#6711yescharacteristic
463C22×C4C23×D311yescharacteristic
464C2×C8—61no—
465C2×C8—61no—
466C2×C8—61no—
467C2×C8—61no—
468C2×C8—61no—
469C2×C8—61no—
470C2×C8—61no—
471C2×C8—61no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—42no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—81no—
489C4⋊C4—122no—
490C4⋊C4C2×D1212yes—
491C4⋊C4C2×D1212yes—
492C4⋊C4#23912yes—
493C22⋊C4—122no—
494C22⋊C4—122no—
495C22⋊C4—122no—
496C22⋊C4—122no—
497C22⋊C4—241no—
498C22⋊C4—241no—
499C22⋊C4—241no—
500C22⋊C4—241no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—42no—
506C42—81no—
507C42—81no—
508C42—122no—
509C42C2×D1212yes—
510C42#23512yes—
511C22×C3C2×#12111yescharacteristic
512C22×C3#54611yescharacteristic
513C22×C3#54811yescharacteristic
514C22×C3#57511yescharacteristic
515C22×C3#57511yescharacteristic
516C22×C3#57611yescharacteristic
517C22×C3#57711yescharacteristic
518C2×D3—81no—
519C2×D3—81no—
520C2×D3—81no—
521C2×D3—81no—
522C2×D3—81no—
523C2×D3—81no—
524C2×D3—81no—
525C2×D3—81no—
526C2×D3—81no—
527C2×D3—81no—
528C2×D3—81no—
529C2×D3—81no—
530C2×D3—81no—
531C2×D3—81no—
532C3×C4—22no—
533C3×C4—22no—
534C3×C4—42no—
535C3×C4—42no—
536C3×C4—42no—
537C3×C4—41no—
538C3×C4—41no—
539C3×C4—41no—
540C3×C4—41no—
541C3×C4—81no—
542C3×C4#52911yescharacteristic
543C3×C4#54211yescharacteristic
544C3×C4#54811yescharacteristic
545C3×C4#55011yescharacteristic
546Dic3—42no—
547Dic3—81no—
548C23—121no—
549C23—121no—
550C23—121no—
551C23—121no—
552C23—121no—
553C23—121no—
554C23—121no—
555C23—121no—
556C23—121no—
557C23—121no—
558C23—121no—
559C23—121no—
560C23—121no—
561C23—121no—
562C23C22×#6711yescharacteristic
563Q8—62no—
564Q8—62no—
565Q8—121no—
566Q8—121no—
567D4—121no—
568D4—121no—
569D4—121no—
570D4—121no—
571D4—121no—
572D4—121no—
573D4—121no—
574D4—121no—
575D4—244no—
576D4—244no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—62no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4#93012yes—
612C2×C4#93012yes—
613C2×C4C22×D1211yescharacteristic
614C2×C4C2×#23511yescharacteristic
615C2×C4C2×#23911yescharacteristic
616C2×C4#100711yescharacteristic
617C2×C4D3×#4111yescharacteristic
618C2×C4#100911yescharacteristic
619C8—121no—
620C8—121no—
621C8—121no—
622C8—121no—
623C2×C3#242611yescharacteristic
624C2×C3#304011yescharacteristic
625C2×C3#304511yescharacteristic
626C2×C3#330911yescharacteristic
627C2×C3#331711yescharacteristic
628C2×C3#337011yescharacteristic
629C2×C3#337111yescharacteristic
630D3—161no—
631D3—161no—
632C22—241no—
633C22—241no—
634C22—241no—
635C22—241no—
636C22—241no—
637C22—241no—
638C22—241no—
639C22—241no—
640C22—241no—
641C22—241no—
642C22—241no—
643C22—241no—
644C22—241no—
645C22—241no—
646C22C2×#93011yescharacteristic
647C22#581711yescharacteristic
648C22#582211yescharacteristic
649C22#582211yescharacteristic
650C22#582911yescharacteristic
651C22#583511yescharacteristic
652C22#583711yescharacteristic
653C4—22no—
654C4—22no—
655C4—42no—
656C4—42no—
657C4—41no—
658C4—41no—
659C4—41no—
660C4—41no—
661C4—42no—
662C4—81no—
663C4—122no—
664C4—241no—
665C4#557611yescharacteristic
666C4#557711yescharacteristic
667C4#558011yescharacteristic
668C4#558211yescharacteristic
669C3#2529111yescharacteristic, complemented, Sylow
670C2—481no—
671C2—481no—
672C2#8249511yescharacteristic
673C2#8252711yescharacteristic
674C2#8256511yescharacteristic
675C2#8360411yescharacteristic
676C2#8361911yescharacteristic
677C2#8363911yescharacteristic
678C2#8369111yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2370C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenter#2370D3
hypocenterC3#25291
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×#5063C2semidirect
5. #69630C2semidirect
6. #69629C2semidirect
7. #69628C2semidirect
8. #69628C2semidirect
9. #69488C2semidirect
10. #69488C2semidirect
12. #69485C2semidirect
13. #69475C2semidirect
14. #69463C2semidirect
15. #69462C2semidirect
65. #2370D3semidirect
669. C3#25291semidirect

Maximal subgroups

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

subgroupindex
C3×#23702
C2×#50632
C2×#50672
#737122
#696302
#696292
#696282
#694882
#694872
#694852
#694752
#694632
#694622
#252913