Finite Group Databasebeta

← #11779316

Subgroups — #11779316

Order 1152.

682 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 92 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#10594004C211yesmaximal, characteristic
2#10592541C211yesmaximal, characteristic, complemented
3#10590369C211yesmaximal, characteristic, complemented
4#10590268C211yesmaximal, characteristic, complemented
5C3×#4485C211yesmaximal, characteristic, complemented
6C3×#4483C211yesmaximal, characteristic, complemented
7#10589082C211yesmaximal, characteristic, complemented
8#68391—31nomaximal
9#68195—31nomaximal
10C2×#3802—21no—
11C2×D3×D12—21no—
12C4×C4⋊(C3⋊D3)C2211yescharacteristic
13#64078C2211yescharacteristic, complemented
14C3×#874C2211yescharacteristic
15C3×C4×D12C2211yescharacteristic
16#63922—22no—
17#63916—21no—
18#63908—21no—
19C2×#3716—22no—
20C32×C4⋊C8C2211yescharacteristic, complemented
21#63648—21no—
22#63644—21no—
23C3×C4⋊(C3⋊C8)C2211yescharacteristic, complemented
24C3×C4×C3⋊C8C2211yescharacteristic, complemented
25#63573—21no—
26#63563—21no—
27#5717—31no—
28D4×D12—31no—
29#5076—31no—
30#5008—31no—
31#4842—31no—
32#4842—31no—
33#4842—31no—
34#4842—61no—
35#4693—31no—
36C3×#331—31no—
37C3×D4⋊C8—31no—
38#4485D311yescharacteristic, complemented
39#4483—31no—
40#4483D311yescharacteristic, complemented
41#4459—31no—
42C22×D32—22no—
43C2×C4⋊(C3⋊D3)D411yescharacteristic
44C2×C4×C3⋊D3—21no—
45C2×#634—21no—
46C2×C3×D12—21no—
47C2×C3×D12—21no—
48C2×C3×D12D411yescharacteristic
49C2×C3×D12D412yes—
50C2×C3×C4×D3—21no—
51C2×#626—22no—
52C2×#625—22no—
53#3802—42no—
54#3802—42no—
55D3×D12—42no—
56D3×D12—42no—
57C2×C32×C8—21no—
58C32×C42C2311yescharacteristic
59#3752—21no—
60C4⋊(C3⋊Dic3)—21no—
61C4⋊(C3⋊Dic3)—21no—
62C4⋊(C3⋊Dic3)D411yescharacteristic
63C3×(C2×D3)⋊C4—21no—
64C3×C4⋊Dic3D411yescharacteristic
65C2×C3×C3⋊C8—21no—
66C2×C3×C3⋊C8—21no—
67(C2×D3)⋊Dic3—22no—
68#3716—44no—
69#1629—91noSylow
70C2×D3×D4—61no—
71C2×D3×D4—61no—
72C2×D3×D4—61no—
73C22×D12—61no—
74C3×#127—31no—
75C3×C4×D4—31no—
76#938—61no—
77#938—61no—
78#937—62no—
79C2×#214—62no—
80#930—61no—
81C2×#204—62no—
82#895—61no—
83#882—62no—
84#874C2×D311yescharacteristic
85C4×D12—31no—
86C4×D12—31no—
87C4×D12—61no—
88C4×D12C2×D311yescharacteristic
89#869—31no—
90#869—31no—
91#869—61no—
92C3×C4⋊C8—21no—
93C3×C4⋊C8—31no—
94C3×C4⋊C8C2×D311yescharacteristic, complemented
95C3×C4⋊C8C2×D311yescharacteristic, complemented
96C3×C4×C8—31no—
97#832—61no—
98#821—61no—
99#821—61no—
100#821—62no—
101#821—61no—
102#817—61no—
103#817—61no—
104#817—62no—
105#817—61no—
106#808—61no—
107C4⋊(C3⋊C8)C2×D311yescharacteristic, complemented
108C4×C3⋊C8C2×D311yescharacteristic, complemented
109C22×C3⋊D3—21no—
110C22×C3×D3—21no—
111C22×C3×D3—22no—
112C2×D32—44no—
113C2×D32—44no—
114C2×D32—82no—
115C2×C32×C4C2×D411yescharacteristic
116C2×C32×C4C2×D411yescharacteristic
117C2×C32×C4C2×D411yescharacteristic
118C2×C3⋊Dic3—21no—
119C2×C3⋊Dic3—41no—
120C4⋊(C3⋊D3)—21no—
121C4⋊(C3⋊D3)SD1612yescomplemented
122C4×C3⋊D3—41no—
123C4×C3⋊D3—41no—
124#634—42no—
125C2×C3×Dic3—21no—
126C3×D12—21no—
127C3×D12—42no—
128C3×D12—41no—
129C3×D12—41no—
130C3×D12—41no—
131C3×D12—41no—
132C3×D12SD1612yescomplemented
133C3×C4×D3—41no—
134C3×C4×D3—41no—
135#626—44no—
136#625—44no—
137C32×C8—41no—
138C3×C3⋊C8—22no—
139C3×C3⋊C8—41no—
140D42—91no—
141#492—91no—
142#476—91no—
143#476—91no—
144#331—31no—
145D4⋊C8—31no—
146D4⋊C8—91no—
147C23×D3—62no—
148C23×D3—62no—
149C2×C3×D4—31no—
150C2×C3×D4—32no—
151C2×C3×D4—61no—
152C2×C3×D4—61no—
153C22×C3×C4—61no—
154C2×#67—62no—
155C2×#67—62no—
156C2×#67—62no—
157D3×D4—122no—
158D3×D4—122no—
159D3×D4—122no—
160D3×D4—122no—
161D3×D4—122no—
162D3×D4—122no—
163C2×D12—21no—
164C2×D12—21no—
165C2×D12—31no—
166C2×D12—31no—
167C2×D12—61no—
168C2×D12—62no—
169C2×D12—122no—
170C2×D12—122no—
171C2×D12D1211yescharacteristic
172C2×D12D1212yes—
173C2×C4×D3—21no—
174C2×C4×D3—61no—
175C2×C4×D3—61no—
176C2×C4×D3—121no—
177C2×Dic6—61no—
178C2×Dic6—61no—
179C2×Dic6—121no—
180C2×C3×C8—22no—
181C2×C3×C8—21no—
182C2×C3×C8—21no—
183C2×C3×C8—61no—
184C2×C3×C8—61no—
185C3×C4⋊C4—31no—
186C3×C22⋊C4—61no—
187C3×C42—21no—
188C3×C42C22×D311yescharacteristic
189C3×C42C22×D311yescharacteristic
190#217—62no—
191#214—124no—
192(C2×D3)⋊C4—21no—
193(C2×D3)⋊C4—62no—
194(C2×D3)⋊C4—61no—
195(C2×D3)⋊C4—61no—
196(C2×D3)⋊C4—121no—
197C4⋊Dic3—31no—
198C4⋊Dic3—31no—
199C4⋊Dic3—61no—
200C4⋊Dic3—62no—
201C4⋊Dic3—61no—
202C4⋊Dic3—61no—
203C4⋊Dic3—61no—
204C4⋊Dic3—61no—
205C4⋊Dic3—121no—
206C4⋊Dic3D1211yescharacteristic
207C2×C3⋊C8—21no—
208C2×C3⋊C8—21no—
209#204—124no—
210C22×C32#12011yescharacteristic
211C2×C3⋊D3—22no—
212C2×C3⋊D3—41no—
213C2×C3⋊D3—41no—
214C2×C3×D3—22no—
215C2×C3×D3—41no—
216C2×C3×D3—41no—
217C2×C3×D3—42no—
218C2×C3×D3—42no—
219C2×C3×D3—42no—
220D32—88no—
221C32×C4—21no—
222C32×C4C2×SD1611yescharacteristic
223C32×C4C2×SD1611yescharacteristic
224C32×C4AGL(1,ℤ/8)11yescharacteristic
225C32×C4AGL(1,ℤ/8)11yescharacteristic
226C3⋊Dic3—41no—
227C3⋊Dic3—81no—
228C3×Dic3—41no—
229C22×D4—181no—
230C22×D4—181no—
231C2×SD16—182no—
232C4⋊Q8—91no—
233#127—31no—
234#121—181no—
235#121—181no—
236#120—182no—
237C4×D4—31no—
238C4×D4—91no—
239#106—181no—
240#106—181no—
241C4⋊C8—31no—
242C4⋊C8D3211yescharacteristic, complemented
243#102—181no—
244#102—181no—
245C4×C8—31no—
246C23×C3—61no—
247C23×C3—62no—
248C22×D3—21no—
249C22×D3—22no—
250C22×D3—61no—
251C22×D3—61no—
252C22×D3—121no—
253C22×D3—124no—
254C22×D3—124no—
255C22×D3—124no—
256C22×D3—124no—
257C22×D3—242no—
258C22×D3—242no—
259C3×D4—32no—
260C3×D4—61no—
261C3×D4—122no—
262C3×D4—121no—
263C3×D4—121no—
264C3×D4—121no—
265C3×D4—121no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—61no—
270C2×C3×C4—121no—
271C2×C3×C4—121no—
272C2×C3×C4C2×D1211yescharacteristic
273C2×C3×C4C2×D1211yescharacteristic
274C2×C3×C4D3×D411yescharacteristic
275C2×C3×C4D3×D411yescharacteristic
276C2×C3×C4D3×D411yescharacteristic
277C2×C3×C4D3×D411yescharacteristic
278#67—124no—
279#67—124no—
280#67—124no—
281C2×Dic3—21no—
282C2×Dic3—61no—
283C2×Dic3—61no—
284C2×Dic3—121no—
285C2×Dic3—122no—
286C2×Dic3—121no—
287C2×Dic3—121no—
288D12—21no—
289D12—32no—
290D12—32no—
291D12—42no—
292D12—41no—
293D12—41no—
294D12—41no—
295D12—41no—
296D12—62no—
297D12—61no—
298D12—61no—
299D12—121no—
300D12—124no—
301D12#20412yescomplemented
302C4×D3—41no—
303C4×D3—41no—
304C4×D3—121no—
305C4×D3—121no—
306C4×D3—121no—
307C4×D3—121no—
308C4×D3—241no—
309C4×D3—241no—
310Dic6—122no—
311Dic6—122no—
312Dic6—242no—
313C3×C8—42no—
314C3×C8—41no—
315C3×C8—41no—
316C3×C8—62no—
317C3×C8—121no—
318C3⋊C8—22no—
319C3⋊C8—41no—
320C2×C32#45011yescharacteristic
321C2×C32#45011yescharacteristic
322C2×C32#45311yescharacteristic
323C3⋊D3—42no—
324C3×D3—42no—
325C3×D3—82no—
326C24—182no—
327C2×Q8—181no—
328C2×D4—31no—
329C2×D4—32no—
330C2×D4—61no—
331C2×D4—61no—
332C2×D4—91no—
333C2×D4—182no—
334C2×D4—182no—
335C2×D4—362no—
336C2×D4—362no—
337C2×D4—362no—
338C2×D4—362no—
339C22×C4—61no—
340C22×C4—181no—
341SD16—364no—
342C2×C8—21no—
343C2×C8—61no—
344C2×C8—61no—
345C4⋊C4—31no—
346C4⋊C4—91no—
347C4⋊C4—181no—
348C4⋊C4—181no—
349C22⋊C4—61no—
350C22⋊C4—182no—
351C22⋊C4—181no—
352C42C2×D3211yescharacteristic
353C22×C3—21no—
354C22×C3—62no—
355C22×C3—121no—
356C22×C3—121no—
357C22×C3—122no—
358C22×C3—122no—
359C22×C3—122no—
360C22×C3#88211yescharacteristic
361C22×C3#88211yescharacteristic
362C2×D3—22no—
363C2×D3—41no—
364C2×D3—41no—
365C2×D3—42no—
366C2×D3—42no—
367C2×D3—42no—
368C2×D3—62no—
369C2×D3—62no—
370C2×D3—122no—
371C2×D3—121no—
372C2×D3—121no—
373C2×D3—121no—
374C2×D3—121no—
375C2×D3—241no—
376C2×D3—241no—
377C2×D3—248no—
378C2×D3—248no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—41no—
386C3×C4—121no—
387C3×C4C2×#20411yescharacteristic
388C3×C4#90811yescharacteristic
389C3×C4#91111yescharacteristic
390C3×C4#91111yescharacteristic
391C3×C4#91111yescharacteristic
392C3×C4D3×SD1611yescharacteristic
393C3×C4D3×SD1611yescharacteristic
394C3×C4D3×SD1611yescharacteristic
395Dic3—41no—
396Dic3—121no—
397Dic3—121no—
398Dic3—241no—
399Dic3—242no—
400Dic3—241no—
401Dic3—241no—
402C32#162911yescharacteristic, complemented, Sylow
403C23—61no—
404C23—62no—
405C23—181no—
406C23—364no—
407C23—364no—
408C23—722no—
409Q8—362no—
410D4—32no—
411D4—61no—
412D4—92no—
413D4—122no—
414D4—121no—
415D4—121no—
416D4—121no—
417D4—121no—
418D4—181no—
419D4—364no—
420D4—364no—
421C2×C4—61no—
422C2×C4—121no—
423C2×C4—121no—
424C2×C4—181no—
425C2×C4—361no—
426C2×C4—361no—
427C2×C4—361no—
428C2×C4D3×D1211yescharacteristic
429C2×C4D3×D1211yescharacteristic
430C2×C4#381111yescharacteristic
431C8—41no—
432C8—62no—
433C8—121no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—122no—
438C2×C3—242no—
439C2×C3#473111yescharacteristic
440C2×C3#474711yescharacteristic
441C2×C3#477611yescharacteristic
442C2×C3#477611yescharacteristic
443C2×C3#477611yescharacteristic
444C2×C3#482211yescharacteristic
445D3—42no—
446D3—82no—
447D3—122no—
448D3—122no—
449D3—242no—
450C22—62no—
451C22—121no—
452C22—121no—
453C22—122no—
454C22—122no—
455C22—122no—
456C22—182no—
457C22—361no—
458C22—361no—
459C22—728no—
460C22#6392311yescharacteristic
461C4—21no—
462C4—121no—
463C4—361no—
464C4—721no—
465C4D3×#20411yescharacteristic
466C4#6379511yescharacteristic
467C4#6392711yescharacteristic
468C4#6394011yescharacteristic
469C3—21no—
470C3#6819511yescharacteristic, complemented
471C3#6839111yescharacteristic, complemented
472C2—122no—
473C2—242no—
474C2—362no—
475C2#1059059711yescharacteristic
476C2#1059094611yescharacteristic
477C2#1059101511yescharacteristic
subgroupG/H
centerC22#63923
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊C8C22
socleC22×C32#120
prime radical#10594004C2
hypercenterC4⋊C8D32
hypocenterC32#1629
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
2. #10592541C2semidirect
3. #10590369C2semidirect
4. #10590268C2semidirect
5. C3×#4485C2semidirect
6. C3×#4483C2semidirect
7. #10589082C2semidirect
13. #64078C22semidirect
20. C32×C4⋊C8C22semidirect
23. C3×C4⋊(C3⋊C8)C22semidirect
24. C3×C4×C3⋊C8C22semidirect
38. #4485D3semidirect
40. #4483D3semidirect
94. C3×C4⋊C8C2×D3semidirect
95. C3×C4⋊C8C2×D3semidirect
107. C4⋊(C3⋊C8)C2×D3semidirect
108. C4×C3⋊C8C2×D3semidirect
121. C4⋊(C3⋊D3)SD16semidirect
132. C3×D12SD16semidirect
242. C4⋊C8D32semidirect
301. D12#204semidirect
402. C32#1629semidirect
470. C3#68195semidirect
471. C3#68391semidirect

Maximal subgroups

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

subgroupindex
#105940042
#105925412
#105903692
#105902682
C3×#44852
C3×#44832
#105890822
#683913
#681953