Finite Group Databasebeta

← #10695524

Subgroups — #10695524

Order 768.

808 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 808 conjugacy classes of subgroups — too many to draw. Only its 83 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
1#84098C211yesmaximal, characteristic
2#84084C211yesmaximal, characteristic, complemented
3#78642C211yesmaximal, characteristic, complemented
4C3×#1789C211yesmaximal, characteristic, complemented
5#69457C211yesmaximal, characteristic, complemented
6#69444C211yesmaximal, characteristic, complemented
7#67684C211yesmaximal, characteristic, complemented
8#9934—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×#951—21no—
11C2×#930—21no—
12C2×#930C2211yescharacteristic
13C2×#924C2211yescharacteristic
14C2×C3×#104—22no—
15C3×#407C2211yescharacteristic
16C3×C4×C22⋊C4C2211yescharacteristic
17#5239—22no—
18#5232—21no—
19#5231—21no—
20#5228—21no—
21#5223—21no—
22#5223—22no—
23#5218—21no—
24#5212—21no—
25#5212C2211yescharacteristic
26#5207C2211yescharacteristic
27#5134—22no—
28#4738—21no—
29#4736—21no—
30#4731—21no—
31#4722—21no—
32#4527—22no—
33#4451—22no—
34#3193—31no—
35#3007—31no—
36#2615—31no—
37#1996—31no—
38#1789D311yescharacteristic, complemented
39#1507—31no—
40#1305—31no—
41C2×C3×#41—21no—
42C2×C3×#41—21no—
43C2×C3×#41—21no—
44C2×C3×#41—21no—
45C2×C3×#41—44no—
46C22×C3×Q8D411yescharacteristic
47C22×C3×D4—21no—
48C22×C3×D4D411yescharacteristic
49C23×C3×C4—21no—
50C23×C3×C4C2311yescharacteristic
51C22×#67—22no—
52C22×#67—21no—
53C22×D12D411yescharacteristic
54C22×Dic6D411yescharacteristic, complemented
55C2×C3×M16D412yescomplemented
56C2×C3×C22⋊C4—21no—
57C2×C3×C42—21no—
58C2×#217—21no—
59C2×#217—22no—
60C2×#217D412yes—
61#951—44no—
62C2×#216—22no—
63C2×#214—22no—
64#930—21no—
65#930—21no—
66#930—21no—
67#930—21no—
68#930—44no—
69C2×(C2×D3)⋊C4—22no—
70C2×(C2×D3)⋊C4—21no—
71C2×C4⋊Dic3—21no—
72C2×C4⋊Dic3D411yescharacteristic
73C2×C4⋊Dic3D411yescharacteristic, complemented
74#924—21no—
75#924—21no—
76#924—21no—
77#924—21no—
78C2×Dic3⋊C4—21no—
79C2×#208—21no—
80C2×#204—22no—
81C3×#104—44no—
82C3×C22⋊C8—21no—
83C3×C22⋊C8—21no—
84C3×#95—21no—
85#835—22no—
86#832—22no—
87C3⋊#95—22no—
88C3⋊#95—21no—
89#830—21no—
90#830D412yescomplemented
91#821—22no—
92#818—22no—
93#816—22no—
94C22×#41C2×D311yescharacteristic
95C2×#137—61no—
96C2×C22⋊Q8—31no—
97C2×#121—31no—
98C2×#121—61no—
99#484—61no—
100#480—61no—
101#468—62no—
102#452—62no—
103#451—61no—
104#450—61no—
105#450—61no—
106#448—61no—
107C2×#104—22no—
108#417—61no—
109#407—31no—
110#407C2×D311yescharacteristic
111#393—61no—
112#392—62no—
113#392—61no—
114#380—31no—
115#380—61no—
116C4×C22⋊C4C2×D311yescharacteristic
117#342—62no—
118#327—62no—
119C24×C3—21no—
120C24×C3C2×D411yescharacteristic
121C23×D3—21no—
122C3×#41—24no—
123C3×#41—24no—
124C3×#41—42no—
125C3×#41—42no—
126C3×#41—42no—
127C3×#41—42no—
128C3×#41—42no—
129C3×#41—42no—
130C2×C3×Q8—21no—
131C2×C3×Q8—21no—
132C2×C3×Q8—42no—
133C2×C3×D4—21no—
134C2×C3×D4—21no—
135C2×C3×D4—42no—
136C2×C3×D4—42no—
137C2×C3×D4—42no—
138C2×C3×D4—41no—
139C2×C3×D4—41no—
140C22×C3×C4—24no—
141C22×C3×C4—21no—
142C22×C3×C4—42no—
143C22×C3×C4—42no—
144C22×C3×C4—41no—
145C22×C3×C4—41no—
146C22×C3×C4—41no—
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4#4111yescharacteristic
153C2×#67—44no—
154C2×#67—41no—
155C2×#67—41no—
156C2×#67—82no—
157C2×#67—82no—
158C2×#67—82no—
159C22×Dic3—22no—
160C22×Dic3—21no—
161C22×Dic3—21no—
162C22×Dic3—21no—
163C2×D12—21no—
164C2×D12—21no—
165C2×D12—42no—
166C2×Dic6—21no—
167C2×Dic6—21no—
168C2×Dic6—42no—
169C3×M16—42no—
170C2×C3×C8—22no—
171C3×C22⋊C4—22no—
172C3×C22⋊C4—41no—
173C3×C42—42no—
174#217—41no—
175#217—41no—
176#217—42no—
177#217—82no—
178#216—44no—
179#214—44no—
180(C2×D3)⋊C4—44no—
181(C2×D3)⋊C4—41no—
182(C2×D3)⋊C4—41no—
183C4⋊Dic3—21no—
184C4⋊Dic3—21no—
185C4⋊Dic3—21no—
186C4⋊Dic3—21no—
187C4⋊Dic3—42no—
188Dic3⋊C4—41no—
189Dic3⋊C4—41no—
190#208—42no—
191C2×C3⋊C8—22no—
192#204—82no—
193C2×#41—21no—
194C2×#41—21no—
195C2×#41—21no—
196C2×#41—21no—
197C2×#41—44no—
198C22×Q8—31no—
199C22×Q8D1211yescharacteristic
200C22×D4—21no—
201C22×D4—31no—
202C22×D4—62no—
203C22×D4—61no—
204C22×D4D1211yescharacteristic
205C23×C4—21no—
206C23×C4C22×D311yescharacteristic
207#137—124no—
208C2×Q16—62no—
209C2×SD16—62no—
210C2×SD16—62no—
211C2×M16—61no—
212C2×M16#6712yescomplemented
213C22⋊Q8—61no—
214C22⋊Q8—61no—
215C22⋊Q8—61no—
216C22⋊Q8—61no—
217#121—61no—
218#121—61no—
219#121—61no—
220#121—61no—
221#121—124no—
222C2×C4⋊C4—31no—
223C2×C4⋊C4—31no—
224C2×C4⋊C4—61no—
225C2×C4⋊C4—61no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—32no—
228C2×C22⋊C4—62no—
229C2×C22⋊C4—61no—
230C2×C22⋊C4—61no—
231C2×C22⋊C4—62no—
232C2×C42—21no—
233#107—62no—
234#104—44no—
235#103—62no—
236#103—62no—
237#102—62no—
238#102—62no—
239C22⋊C8—21no—
240C22⋊C8—21no—
241C22⋊C8—32no—
242C22⋊C8—61no—
243#95—21no—
244#95—62no—
245#95—61no—
246C23×C3—21no—
247C23×C3—21no—
248C23×C3—22no—
249C23×C3—41no—
250C23×C3—41no—
251C23×C3—41no—
252C23×C3—41no—
253C23×C3—41no—
254C23×C3—41no—
255C23×C3—41no—
256C23×C3—41no—
257C23×C3#12011yescharacteristic
258C23×C3#12111yescharacteristic
259C23×C3#12711yescharacteristic
260C22×D3—42no—
261C22×D3—42no—
262C22×D3—41no—
263C22×D3—41no—
264C22×D3—41no—
265C3×Q8—22no—
266C3×Q8—22no—
267C3×Q8—41no—
268C3×Q8—41no—
269C3×D4—22no—
270C3×D4—22no—
271C3×D4—41no—
272C3×D4—41no—
273C3×D4—44no—
274C3×D4—44no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—44no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—44no—
286C2×C3×C4—42no—
287C2×C3×C4—81no—
288C2×C3×C4—81no—
289C2×C3×C4#12011yescharacteristic
290C2×C3×C4#12011yescharacteristic
291C2×C3×C4#12111yescharacteristic
292C2×C3×C4#12111yescharacteristic
293#67—82no—
294#67—82no—
295#67—88no—
296C2×Dic3—42no—
297C2×Dic3—42no—
298C2×Dic3—42no—
299C2×Dic3—41no—
300C2×Dic3—41no—
301C2×Dic3—41no—
302C2×Dic3—42no—
303C2×Dic3—81no—
304C2×Dic3—81no—
305D12—42no—
306D12—81no—
307Dic6—42no—
308Dic6—42no—
309C3×C8—42no—
310C3⋊C8—42no—
311C24—21no—
312C24—61no—
313C24D3×D411yescharacteristic
314#41—24no—
315#41—24no—
316#41—42no—
317#41—42no—
318#41—42no—
319#41—42no—
320#41—42no—
321#41—42no—
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—42no—
332C2×D4—42no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—61no—
336C2×D4—61no—
337C2×D4—122no—
338C2×D4—124no—
339C2×D4—121no—
340C2×D4—121no—
341C2×D4—242no—
342C2×D4—242no—
343C2×D4—242no—
344C22×C4—24no—
345C22×C4—21no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—62no—
352C22×C4—61no—
353C22×C4—61no—
354C22×C4—61no—
355C22×C4C2×D1211yescharacteristic
356C22×C4#23511yescharacteristic
357C22×C4D3×D411yescharacteristic
358C22×C4D3×D411yescharacteristic
359C22×C4D3×D411yescharacteristic
360C22×C4C2×#6711yescharacteristic
361Q16—124no—
362SD16—124no—
363SD16—242no—
364M16—42no—
365M16—122no—
366C2×C8—22no—
367C2×C8—62no—
368C4⋊C4—61no—
369C4⋊C4—61no—
370C4⋊C4—61no—
371C4⋊C4—61no—
372C4⋊C4—122no—
373C4⋊C4—121no—
374C4⋊C4—121no—
375C22⋊C4—22no—
376C22⋊C4—41no—
377C22⋊C4—124no—
378C22⋊C4—121no—
379C22⋊C4—121no—
380C22⋊C4—121no—
381C22⋊C4—121no—
382C22⋊C4—122no—
383C22⋊C4—242no—
384C42—42no—
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—42no—
394C22×C3—42no—
395C22×C3—42no—
396C22×C3—81no—
397C22×C3—81no—
398C22×C3—81no—
399C22×C3—81no—
400C22×C3—81no—
401C22×C3—81no—
402C22×C3#39211yescharacteristic
403C22×C3#45511yescharacteristic
404C22×C3#45611yescharacteristic
405C2×D3—82no—
406C2×D3—82no—
407C2×D3—81no—
408C2×D3—81no—
409C2×D3—81no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—42no—
414C3×C4—82no—
415C3×C4B(4,𝔽2)12yes—
416Dic3—82no—
417Dic3—81no—
418Dic3—81no—
419Dic3—81no—
420C23—21no—
421C23—21no—
422C23—22no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—122no—
432C23—122no—
433C23—121no—
434C23—121no—
435C23—121no—
436C23#88211yescharacteristic
437C23#88411yescharacteristic
438C23#94011yescharacteristic
439Q8—22no—
440Q8—22no—
441Q8—41no—
442Q8—41no—
443Q8—122no—
444Q8—122no—
445D4—22no—
446D4—22no—
447D4—41no—
448D4—41no—
449D4—44no—
450D4—44no—
451D4—122no—
452D4—241no—
453D4—242no—
454D4—242no—
455D4—248no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—44no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—122no—
471C2×C4—122no—
472C2×C4—122no—
473C2×C4—121no—
474C2×C4—121no—
475C2×C4—121no—
476C2×C4—122no—
477C2×C4—241no—
478C2×C4—241no—
479C2×C4#88211yescharacteristic
480C2×C4#88411yescharacteristic
481C2×C4#93011yescharacteristic
482C2×C4#93711yescharacteristic
483C8—42no—
484C8—122no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—41no—
488C2×C3—41no—
489C2×C3—82no—
490C2×C3#200711yescharacteristic
491C2×C3#200811yescharacteristic
492C2×C3#202111yescharacteristic
493D3—161no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—242no—
512C22—242no—
513C22—241no—
514C22—241no—
515C22—241no—
516C22#482511yescharacteristic
517C22#482611yescharacteristic
518C22#496011yescharacteristic
519C4—21no—
520C4—21no—
521C4—21no—
522C4—42no—
523C4—82no—
524C4—242no—
525C4—241no—
526C4—241no—
527C4—241no—
528C4#503212yes—
529C3#993411yescharacteristic, complemented, Sylow
530C2—21no—
531C2—21no—
532C2—41no—
533C2—41no—
534C2—82no—
535C2—481no—
536C2#6950011yescharacteristic
537C2#7039311yescharacteristic
538C2#7040911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1789C2
socleC22×C3#392
prime radical#84098C2
hypercenter#1789D3
hypocenterC3#9934
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1789

Complements

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

NH ≅ G/Ntype
2. #84084C2semidirect
3. #78642C2semidirect
4. C3×#1789C2semidirect
5. #69457C2semidirect
6. #69444C2semidirect
7. #67684C2semidirect
38. #1789D3semidirect
54. C22×Dic6D4semidirect
55. C2×C3×M16D4semidirect
73. C2×C4⋊Dic3D4semidirect
90. #830D4semidirect
212. C2×M16#67semidirect
529. C3#9934semidirect

Maximal subgroups

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

subgroupindex
#840982
#840842
#786422
C3×#17892
#694572
#694442
#676842
#99343