Finite Group Databasebeta

← #10698529

Subgroups — #10698529

Order 768.

646 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82679C211yesmaximal, characteristic
2C2×C3×#327C211yesmaximal, characteristic
3#71677C211yesmaximal, characteristic
4#69930C211yesmaximal, characteristic, complemented
5#69419C211yesmaximal, characteristic, complemented
6C2×#4452C211yesmaximal, characteristic, complemented
7#67674C211yesmaximal, characteristic, complemented
8#10405—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C22×C4⋊Dic3—21no—
11C22×C4⋊Dic3C2211yescharacteristic
12C2×#924C2211yescharacteristic
13C2×C3×C22⋊C8C2211yescharacteristic
14C2×C3×#95C2211yescharacteristic
15#5222—21no—
16#5220—21no—
17#5220—21no—
18C2×#832—21no—
19C2×C3⋊#95—21no—
20C2×C3⋊#95C2211yescharacteristic
21C2×#830C2211yescharacteristic
22C2×#816—21no—
23#4985—21no—
24#4953—21no—
25#4947—21no—
26C3×#327—22no—
27#4551—21no—
28#4529—21no—
29#4452—22no—
30#2639—31no—
31#1874—31no—
32#1872—31no—
33C2×#328—31no—
34C2×#327D311yescharacteristic
35#1444—31no—
36#1280—31no—
37C22×C3×D4—21no—
38C22×C3×D4D411yescharacteristic
39C23×C3×C4—21no—
40C23×C3×C4C2311yescharacteristic
41C23×Dic3—21no—
42C23×Dic3D411yescharacteristic
43C23×Dic3D411yescharacteristic
44C22×Dic6D411yescharacteristic
45C22×C3×C8—21no—
46C3×#121—21no—
47C3×#121—21no—
48C3×#121—21no—
49C3×#121—21no—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C22⋊C4—21no—
52C2×#217—21no—
53C2×#217—21no—
54C2×#217—21no—
55C2×#217—21no—
56C2×#217—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3—42no—
60C2×C4⋊Dic3—42no—
61C2×C4⋊Dic3—42no—
62C2×C4⋊Dic3—42no—
63C2×C4⋊Dic3D411yescharacteristic
64#924—21no—
65#924—21no—
66#924—21no—
67#924—21no—
68C2×Dic3⋊C4—21no—
69C22×C3⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×#95—21no—
73C3×#95D412yes—
74#832—42no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81C3⋊#95—42no—
82#830—21no—
83#830—21no—
84#816—42no—
85C2×C22⋊Q8—31no—
86C2×#121C2×D311yescharacteristic
87C22×C4⋊C4—31no—
88C22×C4⋊C4—61no—
89C2×#103—61no—
90C2×#102—61no—
91C2×C22⋊C8—31no—
92C2×C22⋊C8C2×D311yescharacteristic
93#400—61no—
94#397—61no—
95#394—61no—
96#393—61no—
97#386—61no—
98#386—61no—
99C2×#95—31no—
100C2×#95—61no—
101C2×#95C2×D311yescharacteristic
102#340—61no—
103#340—61no—
104#328—62no—
105#327—22no—
106C24×C3—21no—
107C24×C3C2×D411yescharacteristic
108C2×C3×D4—21no—
109C2×C3×D4—21no—
110C2×C3×D4—41no—
111C2×C3×D4—41no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—21no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4#4111yescharacteristic
132C22×C3×C4#4111yescharacteristic
133C22×C3×C4#4111yescharacteristic
134C22×Dic3—22no—
135C22×Dic3—21no—
136C22×Dic3—21no—
137C22×Dic3—21no—
138C22×Dic3—21no—
139C22×Dic3—21no—
140C22×Dic3—21no—
141C22×Dic3—21no—
142C22×Dic3—21no—
143C22×Dic3—41no—
144C22×Dic3—41no—
145C22×Dic3—41no—
146C22×Dic3—41no—
147C22×Dic3—41no—
148C22×Dic3—41no—
149C22×Dic3—41no—
150C22×Dic3—41no—
151C22×Dic3—41no—
152C22×Dic3—41no—
153C2×Dic6—21no—
154C2×Dic6—21no—
155C2×Dic6—41no—
156C2×Dic6—41no—
157C2×C3×C8—41no—
158C2×C3×C8—41no—
159C2×C3×C8—41no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—21no—
162C3×C22⋊C4—41no—
163C3×C22⋊C4—41no—
164#217—41no—
165#217—41no—
166#217—81no—
167#217—81no—
168#217—81no—
169#217—81no—
170C4⋊Dic3—22no—
171C4⋊Dic3—22no—
172C4⋊Dic3—21no—
173C4⋊Dic3—21no—
174C4⋊Dic3—41no—
175C4⋊Dic3—41no—
176C4⋊Dic3—48no—
177Dic3⋊C4—41no—
178Dic3⋊C4—41no—
179C2×C3⋊C8—41no—
180C2×C3⋊C8—41no—
181C2×C3⋊C8—41no—
182C22×Q8—31no—
183C22×D4—21no—
184C22×D4D1211yescharacteristic
185C23×C4—21no—
186C23×C4—31no—
187C23×C4—31no—
188C23×C4—61no—
189C23×C4C22×D311yescharacteristic
190C22×C8—21no—
191C22×C8—61no—
192C22⋊Q8—61no—
193C22⋊Q8—61no—
194C22⋊Q8—61no—
195C22⋊Q8—61no—
196#121—21no—
197#121—21no—
198#121—21no—
199#121—21no—
200C2×C4⋊C4—31no—
201C2×C4⋊C4—61no—
202C2×C4⋊C4—61no—
203C2×C4⋊C4—61no—
204C2×C4⋊C4—122no—
205C2×C4⋊C4—122no—
206C2×C4⋊C4—122no—
207C2×C4⋊C4—122no—
208C2×C4⋊C4D1211yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—61no—
211C2×C22⋊C4—61no—
212C2×C22⋊C4—61no—
213C2×C22⋊C4—61no—
214C2×C22⋊C4—61no—
215#103—122no—
216#102—122no—
217C22⋊C8—21no—
218C22⋊C8—21no—
219C22⋊C8—61no—
220C22⋊C8—61no—
221#95—21no—
222#95—61no—
223#95—61no—
224#95—61no—
225#95—61no—
226#95—61no—
227#95—61no—
228#95—122no—
229#95#6712yes—
230C23×C3—21no—
231C23×C3—21no—
232C23×C3—21no—
233C23×C3—21no—
234C23×C3—41no—
235C23×C3—41no—
236C23×C3—41no—
237C23×C3—41no—
238C23×C3—41no—
239C23×C3—41no—
240C23×C3—41no—
241C23×C3#12011yescharacteristic
242C23×C3#12111yescharacteristic
243C23×C3#12411yescharacteristic
244C23×C3#13511yescharacteristic
245C23×C3#13511yescharacteristic
246C23×C3AGL(1,ℤ/8)11yescharacteristic
247C23×C3#13711yescharacteristic
248C3×D4—42no—
249C3×D4—81no—
250C3×D4—81no—
251C3×D4—81no—
252C3×D4—81no—
253C3×D4—81no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—24no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4#12111yescharacteristic
273C2×C3×C4#12111yescharacteristic
274C2×C3×C4#12311yescharacteristic
275C2×C3×C4#12311yescharacteristic
276C2×Dic3—42no—
277C2×Dic3—42no—
278C2×Dic3—42no—
279C2×Dic3—42no—
280C2×Dic3—41no—
281C2×Dic3—41no—
282C2×Dic3—41no—
283C2×Dic3—41no—
284C2×Dic3—41no—
285C2×Dic3—41no—
286C2×Dic3—41no—
287C2×Dic3—41no—
288C2×Dic3—42no—
289C2×Dic3—41no—
290C2×Dic3—41no—
291C2×Dic3—41no—
292C2×Dic3—41no—
293C2×Dic3—81no—
294C2×Dic3—81no—
295C2×Dic3—81no—
296C2×Dic3—81no—
297C2×Dic3—81no—
298C2×Dic3—81no—
299Dic6—42no—
300Dic6—81no—
301C3×C8—81no—
302C3⋊C8—81no—
303C24—21no—
304C24D3×D411yescharacteristic
305C2×Q8—61no—
306C2×Q8—61no—
307C2×Q8—121no—
308C2×Q8—121no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—62no—
330C22×C4—61no—
331C22×C4—61no—
332C22×C4—61no—
333C22×C4—61no—
334C22×C4—61no—
335C22×C4—61no—
336C22×C4—61no—
337C22×C4—61no—
338C22×C4—121no—
339C22×C4—121no—
340C22×C4—121no—
341C22×C4—121no—
342C22×C4—121no—
343C22×C4—121no—
344C22×C4—121no—
345C22×C4—121no—
346C22×C4—121no—
347C22×C4—121no—
348C22×C4C2×D1211yescharacteristic
349C22×C4#23511yescharacteristic
350C22×C4D3×D411yescharacteristic
351C22×C4#23711yescharacteristic
352C22×C4#23711yescharacteristic
353C22×C4C2×#6711yescharacteristic
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—121no—
358C2×C8—121no—
359C2×C8—121no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—62no—
363C4⋊C4—62no—
364C4⋊C4—61no—
365C4⋊C4—61no—
366C4⋊C4—121no—
367C4⋊C4—121no—
368C4⋊C4—128no—
369C4⋊C4—121no—
370C4⋊C4—121no—
371C22⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—121no—
374C22⋊C4—121no—
375C22⋊C4—241no—
376C22⋊C4—241no—
377C22⋊C4—241no—
378C22⋊C4—241no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—81no—
392C22×C3—81no—
393C22×C3—81no—
394C22×C3—81no—
395C22×C3—81no—
396C22×C3—81no—
397C22×C3—81no—
398C22×C3#39411yescharacteristic
399C22×C3#44911yescharacteristic
400C22×C3#45211yescharacteristic
401C22×C3#45311yescharacteristic
402C22×C3#45611yescharacteristic
403C22×C3B(4,𝔽2)11yescharacteristic
404C22×C3#45811yescharacteristic
405C22×C3#46311yescharacteristic
406C22×C3#46411yescharacteristic
407C22×C3#48811yescharacteristic
408C22×C3#48911yescharacteristic
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—44no—
414C3×C4—81no—
415Dic3—81no—
416Dic3—82no—
417Dic3—81no—
418Dic3—81no—
419Dic3—81no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23#88211yescharacteristic
432C23#88511yescharacteristic
433C23#91111yescharacteristic
434C23#91211yescharacteristic
435C23#91511yescharacteristic
436C23#91611yescharacteristic
437C23#93911yescharacteristic
438Q8—122no—
439Q8—241no—
440D4—42no—
441D4—81no—
442D4—81no—
443D4—81no—
444D4—81no—
445D4—81no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—24no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—122no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C4—121no—
469C2×C4—121no—
470C2×C4—121no—
471C2×C4—121no—
472C2×C4—122no—
473C2×C4—121no—
474C2×C4—121no—
475C2×C4—121no—
476C2×C4—121no—
477C2×C4—121no—
478C2×C4—121no—
479C2×C4—121no—
480C2×C4—121no—
481C2×C4—241no—
482C2×C4—241no—
483C2×C4—241no—
484C2×C4—241no—
485C2×C4—241no—
486C2×C4—241no—
487C2×C4#88311yescharacteristic
488C2×C4#88611yescharacteristic
489C2×C4#93011yescharacteristic
490C2×C4#93811yescharacteristic
491C8—81no—
492C8—241no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—161no—
498C2×C3#160011yescharacteristic
499C2×C3#160311yescharacteristic
500C2×C3#160911yescharacteristic
501C2×C3#161811yescharacteristic
502C2×C3#203911yescharacteristic
503C2×C3#204211yescharacteristic
504C2×C3#204411yescharacteristic
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—81no—
518C22—81no—
519C22—81no—
520C22—81no—
521C22—81no—
522C22—81no—
523C22—81no—
524C22#474111yescharacteristic
525C22#474211yescharacteristic
526C22#476511yescharacteristic
527C22#476811yescharacteristic
528C22#478111yescharacteristic
529C22#478311yescharacteristic
530C22#482211yescharacteristic
531C22#482611yescharacteristic
532C22#496111yescharacteristic
533C22#515311yescharacteristic
534C22#516611yescharacteristic
535C4—21no—
536C4—21no—
537C4—21no—
538C4—21no—
539C4—44no—
540C4—81no—
541C4—241no—
542C4—241no—
543C4—242no—
544C4—241no—
545C4—241no—
546C3#1040511yescharacteristic, complemented, Sylow
547C2—21no—
548C2—21no—
549C2—21no—
550C2—21no—
551C2—161no—
552C2#6828511yescharacteristic
553C2#6829111yescharacteristic
554C2#6829411yescharacteristic
555C2#6829611yescharacteristic
556C2#7003211yescharacteristic
557C2#7017811yescharacteristic
558C2#7036111yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#327C2
socleC23×C3#120
prime radicalC2×C3×#121C22
hypercenterC2×#327D3
hypocenterC3#10405
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#327

Complements

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

NH ≅ G/Ntype
4. #69930C2semidirect
5. #69419C2semidirect
6. C2×#4452C2semidirect
7. #67674C2semidirect
546. C3#10405semidirect

Maximal subgroups

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

subgroupindex
#826792
C2×C3×#3272
#716772
#699302
#694192
C2×#44522
#676742
#104053