Finite Group Databasebeta

← #11922948

Subgroups — #11922948

Order 1152.

792 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C2×#64264C211yesmaximal, characteristic, complemented
2C22×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3×D4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#10593655C211yesmaximal, characteristic, complemented
5#10593653C211yesmaximal, characteristic, complemented
6#10593629C211yesmaximal, characteristic, complemented
7#10593628C211yesmaximal, characteristic, complemented
8#85398—33nomaximal
9#83559—31nomaximal
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11C22×C3×SL(2,𝔽3)C2211yescharacteristic, complemented
12#64264—22no—
13C2×C3⋊GL(2,𝔽3)—22no—
14C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
15C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
16C2×C3⋊#226C2211yescharacteristic, complemented
17#64146—41nomaximal
18C3×C4×SL(2,𝔽3)C2211yescharacteristic, complemented
19C3⋊#859C2211yescharacteristic, complemented
20C2×#983—33no—
21C22×GL(2,𝔽3)—33no—
22C3×D4×Q8D311yescharacteristic, complemented
23C2×#942—31no—
24C22×#215—31no—
25#5550—31no—
26D4×SL(2,𝔽3)D313yescomplemented
27#5421—33no—
28#5419—33no—
29#5395—33no—
30#5394—33no—
31#5228—31no—
32#5227—31no—
33#5086—31no—
34#5083—31no—
35#5051—31no—
36#5049—31no—
37#5047—31no—
38#5046—31no—
39#5028—31no—
40#5026—31no—
41#5011—31no—
42C23×C3⋊D3—41no—
43C2×C32×D4—41no—
44C2×#638—41no—
45C2×#638—41no—
46C2×C3×SL(2,𝔽3)—21no—
47C2×C3×SL(2,𝔽3)—21no—
48C2×C3×SL(2,𝔽3)C2311yescharacteristic
49C3⋊GL(2,𝔽3)—21no—
50C3⋊GL(2,𝔽3)D412yescomplemented
51C3⋊#226—21no—
52#3757—41no—
53#3752—41no—
54#3752—41no—
55#3266—91noSylow
56C22×C3×Q8C2×D311yescharacteristic
57C22×C3×Q8C2×D311yescharacteristic, complemented
58C2×#235—31no—
59C22×D12—31no—
60C22×SL(2,𝔽3)C2×D313yes—
61C22×SL(2,𝔽3)C2×D313yescomplemented
62#983—66no—
63C2×GL(2,𝔽3)—33no—
64C2×GL(2,𝔽3)—33no—
65C2×GL(2,𝔽3)—66no—
66C2×#226—33no—
67C3×C4⋊Q8—31no—
68C3×C22⋊Q8—31no—
69C3×C22⋊Q8—31no—
70C3×C4×Q8C2×D311yescharacteristic, complemented
71C3×C4×D4—31no—
72C2×#216—31no—
73C2×#216—31no—
74C2×#216—31no—
75#942—62no—
76#942—62no—
77C2×#215—31no—
78C2×#215—31no—
79C2×#215—31no—
80C2×#215—31no—
81C2×#215—31no—
82C2×#215—31no—
83C2×#215—62no—
84C2×#215—62no—
85#937—61no—
86#937—123no—
87#930—31no—
88#924—31no—
89C2×#208—31no—
90C22×C3⋊C8—31no—
91#894—61no—
92#884—61no—
93D3×C22⋊C4—61no—
94#875—31no—
95C4×D12—31no—
96C4×SL(2,𝔽3)C2×D313yescomplemented
97#859—33no—
98#835—31no—
99#835—31no—
100#835—31no—
101#830—31no—
102#830—31no—
103#810—31no—
104#809—31no—
105#809—31no—
106#807—31no—
107C4⋊(C3⋊C8)—31no—
108#803—31no—
109C23×C32—41no—
110C23×C32—41no—
111C22×C3⋊D3—41no—
112C22×C3⋊D3—41no—
113C22×C3⋊D3—82no—
114C22×C3⋊D3—82no—
115C22×C3⋊D3—82no—
116C32×D4—82no—
117C2×C32×C4—41no—
118#638—82no—
119#638—82no—
120C2×C3⋊Dic3—41no—
121C2×C3⋊Dic3—41no—
122C3×SL(2,𝔽3)C2×D411yescharacteristic
123C2×#137—91no—
124C22×SD16—91no—
125D4×Q8C3⋊D311yescharacteristic, complemented
126#546—91no—
127#497—91no—
128#468—91no—
129#467—91no—
130#464—91no—
131#461—91no—
132#452—91no—
133#451—91no—
134#450—91no—
135#448—91no—
136#440—91no—
137#435—91no—
138C23×D3—61no—
139C23×D3—123no—
140C2×C3×Q8—21no—
141C2×C3×Q8—21no—
142C2×C3×Q8—31no—
143C2×C3×Q8—31no—
144C2×C3×Q8—61no—
145C2×C3×Q8—61no—
146C2×C3×Q8C22×D311yescharacteristic
147C2×C3×D4—43no—
148C2×C3×D4S411yescharacteristic
149C22×C3×C4—31no—
150C22×C3×C4—31no—
151C2×#67—61no—
152C2×#67—61no—
153C2×#67—123no—
154C2×#67—123no—
155#235—62no—
156#235—121no—
157C2×D12—31no—
158C2×D12—31no—
159C2×D12—62no—
160C2×D12—61no—
161C2×D12—121no—
162C2×C4×D3—61no—
163C2×C4×D3—61no—
164C2×Dic6—31no—
165C2×SL(2,𝔽3)—23no—
166C2×SL(2,𝔽3)—23no—
167C2×SL(2,𝔽3)C22×D313yes—
168GL(2,𝔽3)—36no—
169GL(2,𝔽3)—63no—
170#226—63no—
171C3×C4⋊C4—31no—
172C3×C4⋊C4—31no—
173C3×C4⋊C4—61no—
174C3×C22⋊C4—31no—
175C3×C22⋊C4—31no—
176C3×C42—31no—
177#217—61no—
178#217—123no—
179#216—62no—
180#216—61no—
181#216—61no—
182#216—121no—
183#215—32no—
184#215—32no—
185#215—62no—
186#215—62no—
187#215—62no—
188#215—61no—
189#215—61no—
190#215—121no—
191(C2×D3)⋊C4—61no—
192(C2×D3)⋊C4—61no—
193(C2×D3)⋊C4—61no—
194(C2×D3)⋊C4—61no—
195(C2×D3)⋊C4—123no—
196(C2×D3)⋊C4—123no—
197C4⋊Dic3—31no—
198Dic3⋊C4—61no—
199#208—62no—
200C2×C3⋊C8—31no—
201C2×C3⋊C8—31no—
202C2×C3⋊C8—31no—
203C2×C3⋊C8—31no—
204C2×C3⋊C8—62no—
205C22×C32—42no—
206C22×C32—41no—
207C22×C32—81no—
208C22×C32—81no—
209C22×C32—81no—
210C22×C32—81no—
211C22×C32—81no—
212C2×C3⋊D3—42no—
213C2×C3⋊D3—42no—
214C2×C3⋊D3—81no—
215C2×C3⋊D3—81no—
216C2×C3⋊D3—81no—
217C2×C3⋊D3—81no—
218C2×C3⋊D3—88no—
219C32×C4—81no—
220C3⋊Dic3—81no—
221C3⋊Dic3—81no—
222C2×#41—91no—
223C22×Q8C2×C3⋊D311yescharacteristic
224C22×Q8C2×C3⋊D311yescharacteristic, complemented
225C22×D4—91no—
226#137—182no—
227#137—182no—
228C2×Q16—91no—
229C2×Q16—91no—
230C2×Q16—91no—
231C2×SD16—91no—
232C2×SD16—91no—
233C2×SD16—91no—
234C2×SD16—91no—
235C2×SD16—91no—
236C2×SD16—91no—
237C2×SD16—182no—
238C2×SD16—182no—
239C2×M16—91no—
240C22×C8—91no—
241C4⋊Q8—31no—
242#124—91no—
243#123—181no—
244C22⋊Q8—31no—
245C22⋊Q8—31no—
246C22⋊Q8—91no—
247#121—91no—
248#121—181no—
249#120—181no—
250C4×Q8C2×C3⋊D311yescharacteristic, complemented
251C4×D4—31no—
252C4×D4—91no—
253C2×C22⋊C4—181no—
254#107—91no—
255C4⋊C8—91no—
256#103—91no—
257#103—91no—
258#103—91no—
259#103—91no—
260#102—91no—
261#102—91no—
262C22⋊C8—91no—
263C22⋊C8—91no—
264#97—91no—
265C23×C3—43no—
266C23×C3—43no—
267C23×C3C2×S411yescharacteristic
268C23×C3C2×S411yescharacteristic
269C22×D3—61no—
270C22×D3—61no—
271C22×D3—123no—
272C22×D3—123no—
273C22×D3—122no—
274C22×D3—122no—
275C22×D3—246no—
276C22×D3—246no—
277C22×D3—241no—
278C22×D3—246no—
279C3×Q8—31no—
280C3×Q8—61no—
281C3×Q8—61no—
282C3×Q8—61no—
283C3×Q8—61no—
284C3×Q8D3×D411yescharacteristic
285C3×D4—22no—
286C3×D4—86no—
287C2×C3×C4—31no—
288C2×C3×C4—31no—
289C2×C3×C4—31no—
290C2×C3×C4—31no—
291C2×C3×C4—43no—
292C2×C3×C4—61no—
293C2×C3×C4—61no—
294C2×C3×C4—61no—
295C2×C3×C4—61no—
296C2×C3×C4C2×S411yescharacteristic
297#67—122no—
298#67—246no—
299#67—241no—
300#67—246no—
301C2×Dic3—61no—
302C2×Dic3—61no—
303C2×Dic3—123no—
304C2×Dic3—123no—
305D12—32no—
306D12—61no—
307D12—61no—
308D12—61no—
309D12—122no—
310C4×D3—122no—
311C4×D3—241no—
312Dic6—61no—
313Dic6—61no—
314SL(2,𝔽3)D3×D413yes—
315C3⋊C8—32no—
316C3⋊C8—61no—
317C3⋊C8—61no—
318C3⋊C8—61no—
319C2×C32—41no—
320C2×C32—41no—
321C2×C32—41no—
322C2×C32—82no—
323C2×C32—161no—
324C3⋊D3—82no—
325C3⋊D3—82no—
326C24—181no—
327#41—182no—
328#41—361no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—31no—
332C2×Q8—31no—
333C2×Q8—61no—
334C2×Q8—61no—
335C2×Q8—91no—
336C2×Q8C22×C3⋊D311yescharacteristic
337C2×D4—91no—
338C2×D4—91no—
339C2×D4—181no—
340C2×D4—182no—
341C2×D4—181no—
342C2×D4—181no—
343C2×D4—361no—
344C2×D4C3⋊S411yescharacteristic
345C22×C4—31no—
346C22×C4—31no—
347C22×C4—181no—
348C22×C4—181no—
349Q16—182no—
350Q16—181no—
351Q16—181no—
352Q16—361no—
353SD16—92no—
354SD16—92no—
355SD16—182no—
356SD16—182no—
357SD16—182no—
358SD16—181no—
359SD16—181no—
360SD16—361no—
361M16—182no—
362C2×C8—91no—
363C2×C8—91no—
364C2×C8—91no—
365C2×C8—91no—
366C2×C8—182no—
367C4⋊C4—31no—
368C4⋊C4—31no—
369C4⋊C4—61no—
370C4⋊C4—91no—
371C4⋊C4—181no—
372C22⋊C4—31no—
373C22⋊C4—31no—
374C22⋊C4—181no—
375C22⋊C4—181no—
376C22⋊C4—181no—
377C22⋊C4—181no—
378C22⋊C4—181no—
379C42—31no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—46no—
386C22×C3—43no—
387C22×C3—83no—
388C22×C3—83no—
389C22×C3—83no—
390C22×C3—83no—
391C22×C3—83no—
392C22×C3#98312yescomplemented
393C22×C3C22×S411yescharacteristic
394C2×D3—62no—
395C2×D3—126no—
396C2×D3—121no—
397C2×D3—121no—
398C2×D3—126no—
399C2×D3—121no—
400C2×D3—121no—
401C2×D3—121no—
402C2×D3—243no—
403C2×D3—243no—
404C2×D3—243no—
405C2×D3—243no—
406C2×D3—244no—
407C2×D3—2424no—
408C3×C4—21no—
409C3×C4—31no—
410C3×C4—31no—
411C3×C4—61no—
412C3×C4—61no—
413C3×C4—61no—
414C3×C4—83no—
415Dic3—121no—
416Dic3—121no—
417Dic3—243no—
418Dic3—243no—
419C32—41noSylow
420C23—181no—
421C23—181no—
422C23—362no—
423C23—362no—
424C23—721no—
425C23C2×C3⋊S411yescharacteristic
426C23C2×C3⋊S411yescharacteristic
427Q8—31no—
428Q8—61no—
429Q8—61no—
430Q8—61no—
431Q8—61no—
432Q8—181no—
433Q8—181no—
434Q8D4×C3⋊D311yescharacteristic
435D4—22no—
436D4—92no—
437D4—181no—
438D4—181no—
439D4—181no—
440D4—362no—
441D4—362no—
442D4—721no—
443C2×C4—31no—
444C2×C4—31no—
445C2×C4—31no—
446C2×C4—31no—
447C2×C4—61no—
448C2×C4—61no—
449C2×C4—61no—
450C2×C4—61no—
451C2×C4—181no—
452C2×C4—181no—
453C2×C4—362no—
454C2×C4—721no—
455C2×C4C2×C3⋊S411yescharacteristic
456C8—92no—
457C8—181no—
458C8—181no—
459C8—181no—
460C2×C3—22no—
461C2×C3—41no—
462C2×C3—43no—
463C2×C3—43no—
464C2×C3—43no—
465C2×C3—86no—
466C2×C3—163no—
467C2×C3D4×S411yescharacteristic
468C2×C3C2×#98311yescharacteristic
469C2×C3#592611yescharacteristic
470D3—122no—
471D3—246no—
472D3—246no—
473D3—241no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—182no—
480C22—361no—
481C22—361no—
482C22—361no—
483C22—361no—
484C22—361no—
485C22—724no—
486C22#6426412yescomplemented
487C22C22×C3⋊S411yescharacteristic
488C4—21no—
489C4—31no—
490C4—31no—
491C4—61no—
492C4—61no—
493C4—61no—
494C4—361no—
495C4—361no—
496C3—43no—
497C3#8539811yescharacteristic, complemented
498C2—22no—
499C2—41no—
500C2—362no—
501C2—721no—
502C2D4×C3⋊S411yescharacteristic
503C2C2×#6426411yescharacteristic
504C2#1059658911yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×D4×Q8D3
socleC22×C3C22×S4
hypercenterC2×D4C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#64264C2semidirect
2. C22×C3⋊GL(2,𝔽3)C2semidirect
3. C3×D4×SL(2,𝔽3)C2semidirect
4. #10593655C2semidirect
5. #10593653C2semidirect
6. #10593629C2semidirect
7. #10593628C2semidirect
11. C22×C3×SL(2,𝔽3)C22semidirect
16. C2×C3⋊#226C22semidirect
18. C3×C4×SL(2,𝔽3)C22semidirect
19. C3⋊#859C22semidirect
22. C3×D4×Q8D3semidirect
26. D4×SL(2,𝔽3)D3semidirect
50. C3⋊GL(2,𝔽3)D4semidirect
57. C22×C3×Q8C2×D3semidirect
61. C22×SL(2,𝔽3)C2×D3semidirect
70. C3×C4×Q8C2×D3semidirect
96. C4×SL(2,𝔽3)C2×D3semidirect
125. D4×Q8C3⋊D3semidirect
224. C22×Q8C2×C3⋊D3semidirect
250. C4×Q8C2×C3⋊D3semidirect
392. C22×C3#983semidirect
486. C22#64264semidirect
497. C3#85398semidirect

Maximal subgroups

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

subgroupindex
C2×#642642
C22×C3⋊GL(2,𝔽3)2
C3×D4×SL(2,𝔽3)2
#105936552
#105936532
#105936292
#105936282
#853983
#835593
#641464