Finite Group Databasebeta

← #10695521

Subgroups — #10695521

Order 768.

808 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 808 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3193C211yesmaximal, characteristic, complemented
2#82735C211yesmaximal, characteristic, complemented
3#82709C211yesmaximal, characteristic
4#71701C211yesmaximal, characteristic, complemented
5#71673C211yesmaximal, characteristic, complemented
6#69991C211yesmaximal, characteristic, complemented
7#67684C211yesmaximal, characteristic, complemented
8#9934—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C2×C3×#121C2211yescharacteristic
12C2×#942—21no—
13C2×#938—21no—
14C3×#484—21no—
15C3×#480—21no—
16C3×#468—22no—
17C3×#450—21no—
18C3×#448—21no—
19C3×#407C2211yescharacteristic
20#5223—22no—
21#5212C2211yescharacteristic
22C4×#217C2211yescharacteristic
23#5207C2211yescharacteristic
24C2×#825—22no—
25#5050—21no—
26#5047—22no—
27#5046—21no—
28#4987—21no—
29#4966—21no—
30#4960—21no—
31#4947—21no—
32#4537—22no—
33#4526—22no—
34#3193D311yescharacteristic, complemented
35#3007—31no—
36#2615—31no—
37#1996—31no—
38#1789—31no—
39#1507—31no—
40#1305—31no—
41C22×C3×Q8D411yescharacteristic, complemented
42C22×C3×D4—21no—
43C22×C3×D4D411yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C22×#67—21no—
46C22×#67—22no—
47C2×#235—21no—
48C2×#235—21no—
49C2×#235—21no—
50C2×#235—21no—
51C2×#235—44no—
52C22×D12D411yescharacteristic
53C22×C4×D3—21no—
54C22×Dic6D411yescharacteristic
55C2×C3×SD16—22no—
56C2×C3×M16D412yescomplemented
57C3×C22⋊Q8—21no—
58C3×C22⋊Q8—21no—
59C3×C22⋊Q8—21no—
60C3×C22⋊Q8—21no—
61C3×#121—21no—
62C3×#121—21no—
63C3×#121—21no—
64C3×#121—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4D411yescharacteristic
67C2×C3×C4⋊C4D411yescharacteristic, complemented
68C2×C3×C22⋊C4—21no—
69C2×C3×C22⋊C4—21no—
70C2×#217—21no—
71C2×#217—22no—
72C2×#217D412yes—
73C2×#216—22no—
74#942—44no—
75C2×#215—22no—
76#938—44no—
77C2×(C2×D3)⋊C4—22no—
78C2×C4⋊Dic3—21no—
79C2×C4×Dic3—21no—
80C2×#208—21no—
81C3×#107—22no—
82C3×#103—22no—
83C3×#102—22no—
84C3×C22⋊C8—21no—
85C3×C22⋊C8—21no—
86C3⋊#95—22no—
87C3⋊#95—21no—
88C3⋊#95—21no—
89#830—21no—
90#830D412yescomplemented
91#825—44no—
92#810—22no—
93#809—22no—
94C22×#41—31no—
95C2×#137—61no—
96C2×C22⋊Q8C2×D311yescharacteristic
97C2×#121—61no—
98C2×#121C2×D311yescharacteristic
99#484—21no—
100#480—21no—
101#468—22no—
102#452—62no—
103#451—61no—
104#450—21no—
105#450—61no—
106#448—21no—
107C2×#104—62no—
108#417—61no—
109#407—31no—
110#407C2×D311yescharacteristic
111#393—61no—
112#392—61no—
113#392—62no—
114#380—31no—
115#380—61no—
116C4×C22⋊C4—31no—
117#342—62no—
118#327—62no—
119C24×C3—21no—
120C24×C3C2×D411yescharacteristic
121C23×D3—21no—
122C2×C3×Q8—21no—
123C2×C3×Q8—21no—
124C2×C3×Q8—42no—
125C2×C3×D4—21no—
126C2×C3×D4—21no—
127C2×C3×D4—42no—
128C2×C3×D4—41no—
129C2×C3×D4—41no—
130C2×C3×D4—82no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—42no—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4#4111yescharacteristic
141C2×#67—42no—
142C2×#67—42no—
143C2×#67—41no—
144C2×#67—41no—
145C2×#67—44no—
146C2×#67—82no—
147C2×#67—82no—
148C22×Dic3—21no—
149C22×Dic3—22no—
150C22×Dic3—41no—
151#235—24no—
152#235—24no—
153#235—42no—
154#235—42no—
155#235—42no—
156#235—42no—
157#235—42no—
158#235—42no—
159C2×D12—21no—
160C2×D12—21no—
161C2×D12—42no—
162C2×C4×D3—24no—
163C2×C4×D3—42no—
164C2×C4×D3—41no—
165C2×C4×D3—41no—
166C2×Dic6—21no—
167C2×Dic6—21no—
168C2×Dic6—42no—
169C3×SD16—82no—
170C3×M16—42no—
171C2×C3×C8—22no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—41no—
177C3×C4⋊C4—41no—
178C3×C22⋊C4—41no—
179C3×C22⋊C4—41no—
180C3×C22⋊C4—41no—
181C3×C22⋊C4—41no—
182#217—22no—
183#217—44no—
184#217—41no—
185#217—42no—
186#216—44no—
187#215—44no—
188(C2×D3)⋊C4—82no—
189C4⋊Dic3—42no—
190C4×Dic3—42no—
191#208—42no—
192C2×C3⋊C8—22no—
193C2×#41—61no—
194C2×#41—61no—
195C2×#41—61no—
196C2×#41—61no—
197C2×#41—124no—
198C22×Q8—31no—
199C22×Q8#6711yescharacteristic, complemented
200C22×D4—21no—
201C22×D4—31no—
202C22×D4—61no—
203C22×D4—62no—
204C22×D4#6711yescharacteristic
205C23×C4—61no—
206C23×C4C22×D311yescharacteristic
207#137—124no—
208C2×Q16—62no—
209C2×SD16—22no—
210C2×SD16—62no—
211C2×M16—61no—
212C2×M16#6712yescomplemented
213C22⋊Q8—21no—
214C22⋊Q8—21no—
215C22⋊Q8—21no—
216C22⋊Q8—21no—
217#121—21no—
218#121—21no—
219#121—21no—
220#121—21no—
221#121—124no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—61no—
224C2×C4⋊C4#6711yescharacteristic
225C2×C4⋊C4#6711yescharacteristic, complemented
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—32no—
229C2×C22⋊C4—61no—
230C2×C22⋊C4—62no—
231C2×C22⋊C4—62no—
232C2×C42—61no—
233#107—22no—
234#104—124no—
235#103—22no—
236#103—62no—
237#102—22no—
238#102—62no—
239C22⋊C8—21no—
240C22⋊C8—21no—
241C22⋊C8—32no—
242C22⋊C8—61no—
243#95—62no—
244#95—61no—
245#95—61no—
246C23×C3—21no—
247C23×C3—21no—
248C23×C3—41no—
249C23×C3—41no—
250C23×C3—42no—
251C23×C3—42no—
252C23×C3—41no—
253C23×C3—41no—
254C23×C3—41no—
255C23×C3#12011yescharacteristic
256C23×C3#12111yescharacteristic
257C23×C3#12711yescharacteristic
258C22×D3—22no—
259C22×D3—41no—
260C22×D3—41no—
261C22×D3—41no—
262C22×D3—41no—
263C22×D3—41no—
264C22×D3—41no—
265C3×Q8—42no—
266C3×Q8—42no—
267C3×D4—42no—
268C3×D4—81no—
269C3×D4—82no—
270C3×D4—82no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—42no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—81no—
282C2×C3×C4—81no—
283C2×C3×C4#12011yescharacteristic
284C2×C3×C4#12011yescharacteristic
285C2×C3×C4#12111yescharacteristic
286C2×C3×C4#12111yescharacteristic
287#67—44no—
288#67—44no—
289#67—88no—
290C2×Dic3—22no—
291C2×Dic3—41no—
292C2×Dic3—41no—
293C2×Dic3—42no—
294C2×Dic3—42no—
295C2×Dic3—42no—
296C2×Dic3—42no—
297C2×Dic3—81no—
298C2×Dic3—81no—
299D12—22no—
300D12—22no—
301D12—41no—
302D12—41no—
303C4×D3—44no—
304C4×D3—44no—
305Dic6—22no—
306Dic6—22no—
307Dic6—41no—
308Dic6—41no—
309C3×C8—42no—
310C3⋊C8—42no—
311C24—21no—
312C24—61no—
313C24D3×D411yescharacteristic
314#41—64no—
315#41—64no—
316#41—122no—
317#41—122no—
318#41—122no—
319#41—122no—
320#41—122no—
321#41—122no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8—42no—
325C2×Q8—61no—
326C2×Q8—61no—
327C2×Q8—122no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—42no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—61no—
334C2×D4—61no—
335C2×D4—82no—
336C2×D4—122no—
337C2×D4—122no—
338C2×D4—122no—
339C2×D4—121no—
340C2×D4—121no—
341C2×D4—124no—
342C2×D4—242no—
343C2×D4—242no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—42no—
348C22×C4—64no—
349C22×C4—61no—
350C22×C4—62no—
351C22×C4—122no—
352C22×C4—121no—
353C22×C4—121no—
354C22×C4—121no—
355C22×C4D3×D411yescharacteristic
356C22×C4D3×D411yescharacteristic
357C22×C4#23711yescharacteristic
358C22×C4C2×#6711yescharacteristic
359C22×C4C2×#6711yescharacteristic
360C22×C4C2×#6711yescharacteristic
361Q16—124no—
362SD16—82no—
363SD16—124no—
364M16—42no—
365M16—122no—
366C2×C8—22no—
367C2×C8—62no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—122no—
375C22⋊C4—41no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—62no—
380C22⋊C4—124no—
381C22⋊C4—121no—
382C22⋊C4—122no—
383C22⋊C4—242no—
384C42—122no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—81no—
394C22×C3—81no—
395C22×C3—82no—
396C22×C3—82no—
397C22×C3—81no—
398C22×C3—81no—
399C22×C3—81no—
400C22×C3#39211yescharacteristic
401C22×C3#45511yescharacteristic
402C22×C3#45611yescharacteristic
403C2×D3—42no—
404C2×D3—42no—
405C2×D3—42no—
406C2×D3—81no—
407C2×D3—81no—
408C2×D3—81no—
409C2×D3—81no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—81no—
414C3×C4—81no—
415C3×C4—81no—
416C3×C4B(4,𝔽2)12yes—
417Dic3—42no—
418Dic3—82no—
419Dic3—82no—
420C23—21no—
421C23—21no—
422C23—41no—
423C23—41no—
424C23—42no—
425C23—42no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—62no—
430C23—121no—
431C23—121no—
432C23—121no—
433C23—121no—
434C23—121no—
435C23—121no—
436C23#93711yescharacteristic
437C23#93911yescharacteristic
438C23#94011yescharacteristic
439Q8—42no—
440Q8—42no—
441Q8—62no—
442Q8—62no—
443Q8—121no—
444Q8—121no—
445D4—42no—
446D4—62no—
447D4—62no—
448D4—81no—
449D4—82no—
450D4—82no—
451D4—121no—
452D4—121no—
453D4—124no—
454D4—124no—
455D4—248no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—62no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—124no—
470C2×C4—121no—
471C2×C4—121no—
472C2×C4—124no—
473C2×C4—122no—
474C2×C4—122no—
475C2×C4—122no—
476C2×C4—122no—
477C2×C4—241no—
478C2×C4—241no—
479C2×C4#93711yescharacteristic
480C2×C4#93811yescharacteristic
481C2×C4#93911yescharacteristic
482C2×C4#95311yescharacteristic
483C8—42no—
484C8—122no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—41no—
488C2×C3—41no—
489C2×C3—161no—
490C2×C3#200711yescharacteristic
491C2×C3#200811yescharacteristic
492C2×C3#202111yescharacteristic
493D3—82no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—81no—
503C22—81no—
504C22—82no—
505C22—82no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—122no—
510C22—122no—
511C22—122no—
512C22—241no—
513C22—241no—
514C22—241no—
515C22—241no—
516C22#520211yescharacteristic
517C22#520511yescharacteristic
518C22#522311yescharacteristic
519C4—21no—
520C4—21no—
521C4—21no—
522C4—81no—
523C4—81no—
524C4—81no—
525C4—122no—
526C4—242no—
527C4—242no—
528C4#524412yes—
529C3#993411yescharacteristic, complemented, Sylow
530C2—21no—
531C2—21no—
532C2—41no—
533C2—41no—
534C2—161no—
535C2—242no—
536C2#7185111yescharacteristic
537C2#7209911yescharacteristic
538C2#7211011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#3193C2
socleC22×C3#392
prime radical#82709C2
hypercenter#3193D3
hypocenterC3#9934
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3193

Complements

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

NH ≅ G/Ntype
1. C3×#3193C2semidirect
2. #82735C2semidirect
4. #71701C2semidirect
5. #71673C2semidirect
6. #69991C2semidirect
7. #67684C2semidirect
34. #3193D3semidirect
41. C22×C3×Q8D4semidirect
56. C2×C3×M16D4semidirect
67. C2×C3×C4⋊C4D4semidirect
90. #830D4semidirect
199. C22×Q8#67semidirect
212. C2×M16#67semidirect
225. C2×C4⋊C4#67semidirect
529. C3#9934semidirect

Maximal subgroups

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

subgroupindex
C3×#31932
#827352
#827092
#717012
#716732
#699912
#676842
#99343