Finite Group Databasebeta

← #11922931

Subgroups — #11922931

Order 1152.

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

normal subgroups only

This group has 796 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×#64266C211yesmaximal, characteristic, complemented
2C22×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3×#548⋊C3C211yesmaximal, characteristic, complemented
4#10593661C211yesmaximal, characteristic, complemented
5#10593653C211yesmaximal, characteristic, complemented
6#10593629C211yesmaximal, characteristic, complemented
7C4×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
8#85380—33nomaximal
9#83545—31nomaximal
10C2×C3×#41⋊C3C2211yescharacteristic, complemented
11C22×C3×SL(2,𝔽3)C2211yescharacteristic
12#64266—22no—
13C2×C3⋊GL(2,𝔽3)—22no—
14C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
15C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
16C2×C3⋊#226C2211yescharacteristic, complemented
17C22⋊C4×C3⋊D3—41nomaximal
18C3×C4×SL(2,𝔽3)C2211yescharacteristic, complemented
19C3⋊#859C2211yescharacteristic, complemented
20C2×#985—33no—
21C22×GL(2,𝔽3)—33no—
22C3×#548D311yescharacteristic, complemented
23C2×#950—31no—
24C22×#215—31no—
25#5550—31no—
26#548⋊C3D313yescomplemented
27#5427—33no—
28#5419—33no—
29#5395—33no—
30C4×GL(2,𝔽3)—33no—
31#5238—31no—
32#5228—31no—
33#5059—31no—
34#5057—31no—
35#5048—31no—
36#5047—31no—
37#5041—31no—
38#5036—31no—
39#5028—31no—
40C4×#215—31no—
41D4×C3⋊C8—31no—
42C23×C3⋊D3—41no—
43C2×C4×C3⋊D3—41no—
44C2×C4×C3⋊D3—41no—
45C3×#41⋊C3—21no—
46C2×C3×SL(2,𝔽3)—21no—
47C2×C3×SL(2,𝔽3)C2311yescharacteristic
48C3⋊GL(2,𝔽3)—21no—
49C3⋊GL(2,𝔽3)D412yescomplemented
50C3⋊#226—21no—
51C32×C22⋊C4—41no—
52#3757—41no—
53#3752—41no—
54#3752—41no—
55#3282—91noSylow
56C2×C3×#41C2×D311yescharacteristic, complemented
57C22×C3×Q8C2×D311yescharacteristic
58C2×#235—31no—
59C22×D12—31no—
60C2×#41⋊C3C2×D313yescomplemented
61C22×SL(2,𝔽3)C2×D313yes—
62#985—66no—
63C2×GL(2,𝔽3)—33no—
64C2×GL(2,𝔽3)—33no—
65C2×GL(2,𝔽3)—66no—
66C2×#226—33no—
67C3×#124—31no—
68C3×C22⋊Q8—31no—
69C3×#121—31no—
70C3×C4×Q8C2×D311yescharacteristic, complemented
71C3×C4×D4—31no—
72#950—62no—
73#950—62no—
74C2×#216—31no—
75C2×#216—31no—
76C2×#215—31no—
77C2×#215—31no—
78C2×#215—31no—
79C2×#215—31no—
80C2×#215—62no—
81C2×#215—62no—
82#937—61no—
83C2×#214—31no—
84C2×#213—31no—
85C2×#213—31no—
86#930—31no—
87#924—31no—
88C22×C3⋊C8—31no—
89C22×C3⋊C8—31no—
90#894—61no—
91#884—61no—
92D3×C22⋊C4—61no—
93D3×C22⋊C4—123no—
94#875—31no—
95C4×D12—31no—
96C4×SL(2,𝔽3)C2×D313yescomplemented
97#859—33no—
98#835—31no—
99#835—31no—
100#832—31no—
101#830—31no—
102#830—31no—
103#810—31no—
104#809—31no—
105#809—31no—
106#808—31no—
107C4⋊(C3⋊C8)—31no—
108C4×C3⋊C8—31no—
109C23×C32—41no—
110C22×C3⋊D3—41no—
111C22×C3⋊D3—41no—
112C22×C3⋊D3—44no—
113C22×C3⋊D3—82no—
114C22×C3⋊D3—82no—
115C2×C32×C4—41no—
116C2×C32×C4—41no—
117C2×C3⋊Dic3—41no—
118C2×C3⋊Dic3—41no—
119C4×C3⋊D3—82no—
120C4×C3⋊D3—82no—
121C3×SL(2,𝔽3)C2×D411yescharacteristic
122C2×#135—91no—
123C22×SD16—91no—
124#548C3⋊D311yescharacteristic, complemented
125#546—91no—
126#495—91no—
127#465—91no—
128#465—91no—
129#464—91no—
130#463—91no—
131#452—91no—
132#451—91no—
133#449—91no—
134#447—91no—
135C4×SD16—91no—
136C8×D4—91no—
137C23×D3—61no—
138C23×D3—123no—
139C3×#41—21no—
140C3×#41—61no—
141C2×C3×Q8—21no—
142C2×C3×Q8—31no—
143C2×C3×Q8—61no—
144C2×C3×Q8C22×D311yescharacteristic
145C2×C3×D4—31no—
146C2×C3×D4—31no—
147C22×C3×C4—31no—
148C22×C3×C4—31no—
149C2×#67—61no—
150C2×#67—61no—
151#235—62no—
152#235—121no—
153C2×D12—31no—
154C2×D12—31no—
155C2×D12—62no—
156C2×D12—61no—
157C2×D12—121no—
158C2×C4×D3—61no—
159C2×C4×D3—61no—
160C2×C4×D3—123no—
161C2×C4×D3—123no—
162C2×Dic6—31no—
163#41⋊C3—23no—
164C2×SL(2,𝔽3)—23no—
165C2×SL(2,𝔽3)C22×D313yes—
166GL(2,𝔽3)—36no—
167GL(2,𝔽3)—63no—
168#226—63no—
169C3×C4⋊C4—31no—
170C3×C4⋊C4—31no—
171C3×C22⋊C4—31no—
172C3×C22⋊C4—43no—
173C3×C22⋊C4—61no—
174C3×C22⋊C4S411yescharacteristic
175C3×C42—31no—
176#217—61no—
177#217—123no—
178#216—61no—
179#216—61no—
180#216—121no—
181#215—32no—
182#215—32no—
183#215—62no—
184#215—62no—
185#215—61no—
186#215—61no—
187#214—62no—
188#213—62no—
189#213—121no—
190(C2×D3)⋊C4—61no—
191(C2×D3)⋊C4—61no—
192(C2×D3)⋊C4—61no—
193(C2×D3)⋊C4—61no—
194(C2×D3)⋊C4—123no—
195(C2×D3)⋊C4—123no—
196C4⋊Dic3—31no—
197Dic3⋊C4—61no—
198C2×C3⋊C8—31no—
199C2×C3⋊C8—31no—
200C2×C3⋊C8—31no—
201C2×C3⋊C8—31no—
202C2×C3⋊C8—62no—
203C2×C3⋊C8—62no—
204C22×C32—42no—
205C22×C32—41no—
206C22×C32—81no—
207C22×C32—81no—
208C2×C3⋊D3—42no—
209C2×C3⋊D3—42no—
210C2×C3⋊D3—42no—
211C2×C3⋊D3—42no—
212C2×C3⋊D3—81no—
213C2×C3⋊D3—81no—
214C2×C3⋊D3—88no—
215C32×C4—81no—
216C32×C4—81no—
217C3⋊Dic3—81no—
218C3⋊Dic3—81no—
219C2×#41—91no—
220C2×#41C2×C3⋊D311yescharacteristic, complemented
221C22×Q8C2×C3⋊D311yescharacteristic
222C22×D4—91no—
223#135—182no—
224#135—182no—
225C2×Q16—91no—
226C2×Q16—91no—
227C2×SD16—91no—
228C2×SD16—91no—
229C2×SD16—91no—
230C2×SD16—91no—
231C2×SD16—91no—
232C2×SD16—182no—
233C2×SD16—182no—
234C2×D8—91no—
235C2×D8—91no—
236C22×C8—91no—
237C22×C8—91no—
238#124—31no—
239#124—91no—
240#123—181no—
241C22⋊Q8—31no—
242C22⋊Q8—91no—
243#121—31no—
244#121—91no—
245#121—181no—
246#120—181no—
247C4×Q8C2×C3⋊D311yescharacteristic, complemented
248C4×D4—31no—
249C4×D4—91no—
250C2×C22⋊C4—181no—
251#106—91no—
252C4⋊C8—91no—
253#103—91no—
254#103—91no—
255#103—91no—
256#102—91no—
257#102—91no—
258#102—91no—
259C22⋊C8—91no—
260C22⋊C8—91no—
261C4×C8—91no—
262C23×C3—31no—
263C23×C3—43no—
264C23×C3C2×S411yescharacteristic
265C22×D3—61no—
266C22×D3—61no—
267C22×D3—123no—
268C22×D3—123no—
269C22×D3—122no—
270C22×D3—1212no—
271C22×D3—122no—
272C22×D3—246no—
273C22×D3—241no—
274C22×D3—246no—
275C3×Q8—31no—
276C3×Q8—61no—
277C3×Q8—61no—
278C3×Q8D3×D411yescharacteristic
279C3×D4—61no—
280C3×D4—61no—
281C3×D4—62no—
282C2×C3×C4—31no—
283C2×C3×C4—31no—
284C2×C3×C4—31no—
285C2×C3×C4—43no—
286C2×C3×C4—43no—
287C2×C3×C4—61no—
288C2×C3×C4—61no—
289C2×C3×C4—61no—
290C2×C3×C4—61no—
291C2×C3×C4C2×S411yescharacteristic
292C2×C3×C4C2×S411yescharacteristic
293#67—122no—
294#67—241no—
295C2×Dic3—61no—
296C2×Dic3—61no—
297C2×Dic3—123no—
298C2×Dic3—123no—
299D12—32no—
300D12—61no—
301D12—61no—
302D12—61no—
303D12—122no—
304C4×D3—122no—
305C4×D3—246no—
306C4×D3—241no—
307C4×D3—246no—
308Dic6—61no—
309Dic6—61no—
310SL(2,𝔽3)D3×D413yes—
311C3⋊C8—32no—
312C3⋊C8—61no—
313C3⋊C8—61no—
314C3⋊C8—61no—
315C2×C32—41no—
316C2×C32—41no—
317C2×C32—41no—
318C2×C32—82no—
319C3⋊D3—44no—
320C3⋊D3—82no—
321C24—181no—
322#41—21no—
323#41—61no—
324#41—182no—
325#41—361no—
326C2×Q8—21no—
327C2×Q8—31no—
328C2×Q8—61no—
329C2×Q8—91no—
330C2×Q8C22×C3⋊D311yescharacteristic
331C2×D4—31no—
332C2×D4—31no—
333C2×D4—91no—
334C2×D4—91no—
335C2×D4—181no—
336C2×D4—182no—
337C2×D4—181no—
338C2×D4—181no—
339C2×D4—361no—
340C22×C4—31no—
341C22×C4—31no—
342C22×C4—181no—
343C22×C4—181no—
344Q16—181no—
345Q16—181no—
346Q16—361no—
347SD16—92no—
348SD16—92no—
349SD16—182no—
350SD16—182no—
351SD16—182no—
352SD16—181no—
353SD16—181no—
354D8—182no—
355D8—361no—
356C2×C8—91no—
357C2×C8—91no—
358C2×C8—91no—
359C2×C8—91no—
360C2×C8—182no—
361C2×C8—182no—
362C4⋊C4—31no—
363C4⋊C4—31no—
364C4⋊C4—91no—
365C4⋊C4—181no—
366C22⋊C4—31no—
367C22⋊C4—61no—
368C22⋊C4—181no—
369C22⋊C4—181no—
370C22⋊C4—181no—
371C22⋊C4—181no—
372C22⋊C4—181no—
373C22⋊C4C3⋊S411yescharacteristic
374C42—31no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—46no—
378C22×C3—43no—
379C22×C3—61no—
380C22×C3—61no—
381C22×C3—61no—
382C22×C3—83no—
383C22×C3—83no—
384C22×C3#98512yescomplemented
385C22×C3C22×S411yescharacteristic
386C2×D3—62no—
387C2×D3—126no—
388C2×D3—126no—
389C2×D3—126no—
390C2×D3—121no—
391C2×D3—121no—
392C2×D3—126no—
393C2×D3—121no—
394C2×D3—121no—
395C2×D3—121no—
396C2×D3—243no—
397C2×D3—243no—
398C2×D3—244no—
399C2×D3—2424no—
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—31no—
403C3×C4—31no—
404C3×C4—61no—
405C3×C4—61no—
406C3×C4—83no—
407C3×C4—83no—
408Dic3—121no—
409Dic3—121no—
410Dic3—243no—
411Dic3—243no—
412C32—41noSylow
413C23—31no—
414C23—181no—
415C23—181no—
416C23—362no—
417C23—362no—
418C23—721no—
419C23C2×C3⋊S411yescharacteristic
420Q8—31no—
421Q8—61no—
422Q8—61no—
423Q8—181no—
424Q8—181no—
425Q8D4×C3⋊D311yescharacteristic
426D4—61no—
427D4—61no—
428D4—62no—
429D4—92no—
430D4—181no—
431D4—181no—
432D4—181no—
433D4—362no—
434D4—362no—
435D4—721no—
436C2×C4—31no—
437C2×C4—31no—
438C2×C4—31no—
439C2×C4—61no—
440C2×C4—61no—
441C2×C4—61no—
442C2×C4—61no—
443C2×C4—181no—
444C2×C4—181no—
445C2×C4—362no—
446C2×C4—721no—
447C2×C4C2×C3⋊S411yescharacteristic
448C2×C4C2×C3⋊S411yescharacteristic
449C8—92no—
450C8—181no—
451C8—181no—
452C8—181no—
453C2×C3—22no—
454C2×C3—43no—
455C2×C3—43no—
456C2×C3—43no—
457C2×C3—86no—
458C2×C3—121no—
459C2×C3D4×S411yescharacteristic
460C2×C3C2×#98511yescharacteristic
461C2×C3#592611yescharacteristic
462D3—1212no—
463D3—122no—
464D3—246no—
465D3—241no—
466C22—21no—
467C22—21no—
468C22—61no—
469C22—61no—
470C22—61no—
471C22—182no—
472C22—361no—
473C22—361no—
474C22—361no—
475C22—361no—
476C22—361no—
477C22—724no—
478C22#6426612yescomplemented
479C22C22×C3⋊S411yescharacteristic
480C4—21no—
481C4—21no—
482C4—31no—
483C4—31no—
484C4—61no—
485C4—61no—
486C4—361no—
487C4—361no—
488C3—43no—
489C3#8538011yescharacteristic, complemented
490C2—22no—
491C2—121no—
492C2—362no—
493C2—721no—
494C2D4×C3⋊S411yescharacteristic
495C2C2×#6426611yescharacteristic
496C2#1059658911yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×#548D3
socleC22×C3C22×S4
hypercenterC22⋊C4C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#64266C2semidirect
2. C22×C3⋊GL(2,𝔽3)C2semidirect
3. C3×#548⋊C3C2semidirect
4. #10593661C2semidirect
5. #10593653C2semidirect
6. #10593629C2semidirect
7. C4×C3⋊GL(2,𝔽3)C2semidirect
10. C2×C3×#41⋊C3C22semidirect
16. C2×C3⋊#226C22semidirect
18. C3×C4×SL(2,𝔽3)C22semidirect
19. C3⋊#859C22semidirect
22. C3×#548D3semidirect
26. #548⋊C3D3semidirect
49. C3⋊GL(2,𝔽3)D4semidirect
56. C2×C3×#41C2×D3semidirect
60. C2×#41⋊C3C2×D3semidirect
70. C3×C4×Q8C2×D3semidirect
96. C4×SL(2,𝔽3)C2×D3semidirect
124. #548C3⋊D3semidirect
220. C2×#41C2×C3⋊D3semidirect
247. C4×Q8C2×C3⋊D3semidirect
384. C22×C3#985semidirect
478. C22#64266semidirect
489. C3#85380semidirect

Maximal subgroups

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

subgroupindex
C2×#642662
C22×C3⋊GL(2,𝔽3)2
C3×#548⋊C32
#105936612
#105936532
#105936292
C4×C3⋊GL(2,𝔽3)2
#853803
#835453
C22⋊C4×C3⋊D34