Finite Group Databasebeta

← #11778304

Subgroups — #11778304

Order 1152.

597 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 597 conjugacy classes of subgroups — too many to draw. Only its 80 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#10596116C211yesmaximal, characteristic
2C3×#5050C211yesmaximal, characteristic, complemented
3#10591346C211yesmaximal, characteristic, complemented
4#10591174C211yesmaximal, characteristic
5C3×#4538C211yesmaximal, characteristic, complemented
6#10589151C211yesmaximal, characteristic
7#10589148C211yesmaximal, characteristic, complemented
8#70982—31nomaximal
9#68326—31nomaximal
10C22×C3×Dic6C2211yescharacteristic
11C32×C22⋊Q8C2211yescharacteristic
12#64155—21no—
13#64139C2211yescharacteristic
14C2×#3750C2211yescharacteristic
15#64107—21no—
16#64103—21no—
17#64099—21no—
18#64095—21no—
19#64090—21no—
20C2×C3×#216—21no—
21C2×Dic3⋊Dic3—21no—
22C2×#3717—21no—
23C3×C3⋊#95C2211yescharacteristic
24C3×#830C2211yescharacteristic
25C3×#830C2211yescharacteristic
26C3×#810—21no—
27#63579—21no—
28#63566—21no—
29#5645—31no—
30#5645—31no—
31#5645—61no—
32C3×#451—31no—
33#5230—31no—
34#5227—31no—
35#5050D311yescharacteristic, complemented
36#4738—31no—
37#4656—31no—
38C3×#328—31no—
39#4538D311yescharacteristic, complemented
40#4470—31no—
41#4469—31no—
42#4469—31no—
43#4451—31no—
44C2×C32×Q8D411yescharacteristic
45C22×C32×C4C2311yescharacteristic
46C2×#638—21no—
47C22×C3⋊Dic3D411yescharacteristic
48C22×C3⋊Dic3D411yescharacteristic
49C2×C4⋊(C3⋊D3)D411yescharacteristic
50C2×C4×C3⋊D3—21no—
51C22×C3×Dic3—21no—
52C22×C3×Dic3—21no—
53C2×C3×Dic6—21no—
54C2×C3×Dic6—21no—
55C2×C3×Dic6—42no—
56C32×C4⋊C4—21no—
57C32×C4⋊C4D411yescharacteristic
58C32×C22⋊C4—21no—
59#3752—21no—
60#3752—21no—
61#3752—21no—
62#3752—21no—
63C4⋊(C3⋊Dic3)—21no—
64C4⋊(C3⋊Dic3)—21no—
65C4⋊(C3⋊Dic3)D411yescharacteristic
66#3750—21no—
67#3750—21no—
68#3750—21no—
69C4×C3⋊Dic3—21no—
70C3×#216—42no—
71C2×C3×C3⋊C8—21no—
72C2×C3×C3⋊C8—21no—
73Dic3⋊Dic3—42no—
74#3717—42no—
75#1603—91noSylow
76C22×C3×Q8—31no—
77C22×Dic6C2×D311yescharacteristic
78C2×C3×Q16—61no—
79C3×C22⋊Q8—21no—
80C3×C22⋊Q8C2×D311yescharacteristic
81C3×C22⋊Q8C2×D311yescharacteristic
82#946—62no—
83#946—61no—
84#946—61no—
85C2×#216—21no—
86C2×#215—61no—
87#930—31no—
88#930—31no—
89#930—61no—
90C2×C4⋊Dic3—61no—
91C2×Dic3⋊C4—31no—
92C2×Dic3⋊C4—31no—
93C2×Dic3⋊C4—61no—
94C2×Dic3⋊C4—61no—
95C2×#204—61no—
96#898—62no—
97#898—61no—
98#898—61no—
99#894—61no—
100#894—61no—
101#894—62no—
102#890—61no—
103#890—61no—
104#890—62no—
105#886—61no—
106#886—61no—
107#886—62no—
108#67⋊C4—61no—
109#67⋊C4—61no—
110#67⋊C4—62no—
111C3×#103—61no—
112C3×C22⋊C8—31no—
113C3×C22⋊C8—31no—
114C3×#95—31no—
115#835—61no—
116C3⋊#95—61no—
117C3⋊#95—61no—
118C3⋊#95C2×D311yescharacteristic
119#830C2×D311yescharacteristic
120#830C2×D311yescharacteristic
121#816—61no—
122#810—21no—
123C23×C32C2×D411yescharacteristic
124C22×C3⋊D3—21no—
125C32×Q8—41no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—41no—
129C2×C32×C4C2×D411yescharacteristic
130C2×C32×C4C2×D411yescharacteristic
131#638—41no—
132#638—41no—
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3—21no—
135C2×C3⋊Dic3—21no—
136C2×C3⋊Dic3—21no—
137C2×C3⋊Dic3—21no—
138C2×C3⋊Dic3—41no—
139C4⋊(C3⋊D3)—41no—
140C4×C3⋊D3—81no—
141C2×C3×Dic3—22no—
142C2×C3×Dic3—22no—
143C2×C3×Dic3—41no—
144C2×C3×Dic3—41no—
145C2×C3×Dic3—41no—
146C2×C3×Dic3—41no—
147C3×Dic6—22no—
148C3×Dic6—22no—
149C3×Dic6—41no—
150C3×Dic6—41no—
151C3×C3⋊C8—41no—
152C3×C3⋊C8—41no—
153#539—91no—
154#451—31no—
155#448—91no—
156#395—91no—
157#328—31no—
158#327—91no—
159#324—91no—
160C2×C3×Q8—21no—
161C2×C3×Q8—61no—
162C2×C3×Q8—61no—
163C2×C3×Q8—122no—
164C2×C3×Q8D1211yescharacteristic
165C2×C3×Q8#6711yescharacteristic
166C22×C3×C4—21no—
167C22×C3×C4—61no—
168C22×C3×C4—61no—
169C22×C3×C4C22×D311yescharacteristic
170C22×C3×C4C22×D311yescharacteristic
171C2×#67—62no—
172C2×#67—61no—
173C2×#67—61no—
174C22×Dic3—21no—
175C22×Dic3—21no—
176C22×Dic3—31no—
177C22×Dic3—31no—
178C22×Dic3—31no—
179C22×Dic3—31no—
180C22×Dic3—61no—
181C22×Dic3—61no—
182C2×D12—31no—
183C2×D12—31no—
184C2×D12—61no—
185C2×C4×D3—62no—
186C2×C4×D3—61no—
187C2×C4×D3—61no—
188C2×Dic6—21no—
189C2×Dic6—21no—
190C2×Dic6—42no—
191C3×Q16—122no—
192C2×C3×C8—61no—
193C2×C3×C8—61no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4D1211yescharacteristic
199C3×C4⋊C4#6711yescharacteristic
200C3×C22⋊C4—22no—
201C3×C22⋊C4—21no—
202C3×C22⋊C4—21no—
203#216—42no—
204#215—122no—
205(C2×D3)⋊C4—62no—
206(C2×D3)⋊C4—61no—
207(C2×D3)⋊C4—61no—
208(C2×D3)⋊C4—62no—
209(C2×D3)⋊C4—61no—
210(C2×D3)⋊C4—61no—
211(C2×D3)⋊C4—62no—
212(C2×D3)⋊C4—61no—
213(C2×D3)⋊C4—61no—
214(C2×D3)⋊C4—62no—
215(C2×D3)⋊C4—61no—
216(C2×D3)⋊C4—61no—
217C4⋊Dic3—31no—
218C4⋊Dic3—31no—
219C4⋊Dic3—61no—
220C4⋊Dic3—62no—
221C4⋊Dic3—61no—
222C4⋊Dic3—61no—
223C4⋊Dic3—62no—
224C4⋊Dic3—61no—
225C4⋊Dic3—61no—
226C4⋊Dic3—122no—
227Dic3⋊C4—62no—
228Dic3⋊C4—61no—
229Dic3⋊C4—61no—
230Dic3⋊C4—62no—
231Dic3⋊C4—62no—
232Dic3⋊C4—61no—
233Dic3⋊C4—61no—
234Dic3⋊C4—61no—
235Dic3⋊C4—61no—
236Dic3⋊C4—122no—
237C4×Dic3—62no—
238C4×Dic3—61no—
239C4×Dic3—61no—
240C2×C3⋊C8—21no—
241C2×C3⋊C8—21no—
242#204—122no—
243C22×C32—21no—
244C22×C32—21no—
245C22×C32#12011yescharacteristic
246C22×C32#13511yescharacteristic
247C22×C32#13711yescharacteristic
248C2×C3⋊D3—41no—
249C2×C3⋊D3—41no—
250C2×C3⋊D3—41no—
251C32×C4—21no—
252C32×C4—21no—
253C32×C4—41no—
254C32×C4—41no—
255C3⋊Dic3—41no—
256C3⋊Dic3—41no—
257C3⋊Dic3—41no—
258C3×Dic3—42no—
259C3×Dic3—42no—
260C22×Q8—31no—
261C2×Q16—61no—
262C2×SD16—181no—
263#126—181no—
264#125—181no—
265#123—181no—
266#123—181no—
267C22⋊Q8—181no—
268C22⋊Q8D3211yescharacteristic
269#121—91no—
270C4×D4—181no—
271C2×C4⋊C4—91no—
272C2×C4⋊C4—181no—
273#103—61no—
274#103—181no—
275C22⋊C8—31no—
276C22⋊C8—31no—
277#95—31no—
278#95—181no—
279C23×C3—21no—
280C23×C3D3×D411yescharacteristic
281C23×C3D3×D411yescharacteristic
282C22×D3—62no—
283C22×D3—61no—
284C22×D3—61no—
285C3×Q8—41no—
286C3×Q8—41no—
287C3×Q8—62no—
288C3×Q8—62no—
289C3×Q8—81no—
290C3×Q8—121no—
291C3×Q8—121no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—62no—
303C2×C3×C4—62no—
304C2×C3×C4—81no—
305C2×C3×C4—121no—
306C2×C3×C4—121no—
307C2×C3×C4—121no—
308C2×C3×C4—121no—
309C2×C3×C4C2×D1211yescharacteristic
310C2×C3×C4D3×D411yescharacteristic
311C2×C3×C4D3×D411yescharacteristic
312C2×C3×C4C2×#6711yescharacteristic
313#67—122no—
314#67—122no—
315#67—121no—
316#67—121no—
317#67—121no—
318#67—121no—
319C2×Dic3—22no—
320C2×Dic3—22no—
321C2×Dic3—41no—
322C2×Dic3—41no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—62no—
326C2×Dic3—62no—
327C2×Dic3—62no—
328C2×Dic3—61no—
329C2×Dic3—61no—
330C2×Dic3—61no—
331C2×Dic3—61no—
332C2×Dic3—61no—
333C2×Dic3—61no—
334C2×Dic3—61no—
335C2×Dic3—61no—
336C2×Dic3—61no—
337C2×Dic3—61no—
338C2×Dic3—121no—
339C2×Dic3—121no—
340C2×Dic3—121no—
341C2×Dic3—121no—
342C2×Dic3—241no—
343D12—121no—
344D12—121no—
345D12—241no—
346C4×D3—242no—
347C4×D3—241no—
348C4×D3—241no—
349Dic6—22no—
350Dic6—22no—
351Dic6—41no—
352Dic6—41no—
353C3×C8—121no—
354C3×C8—121no—
355C3⋊C8—41no—
356C3⋊C8—41no—
357C2×C32—21no—
358C2×C32—21no—
359C2×C32#45211yescharacteristic
360C2×C32#45611yescharacteristic
361C2×C32B(4,𝔽2)11yescharacteristic
362C3⋊D3—81no—
363C2×Q8—61no—
364C2×Q8—61no—
365C2×Q8—122no—
366C2×Q8#62611yescharacteristic
367C2×D4—91no—
368C2×D4—181no—
369C22×C4—61no—
370C22×C4—61no—
371C22×C4—91no—
372C22×C4—91no—
373C22×C4—181no—
374C22×C4C2×D3211yescharacteristic
375Q16—122no—
376SD16—362no—
377C2×C8—61no—
378C2×C8—61no—
379C4⋊C4—21no—
380C4⋊C4—91no—
381C4⋊C4—181no—
382C4⋊C4—181no—
383C4⋊C4—181no—
384C4⋊C4—181no—
385C4⋊C4—181no—
386C4⋊C4—362no—
387C4⋊C4#62611yescharacteristic
388C22⋊C4—21no—
389C22⋊C4—181no—
390C22⋊C4—181no—
391C22⋊C4—181no—
392C22⋊C4—181no—
393C42—181no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—41no—
402C22×C3—41no—
403C22×C3#88211yescharacteristic
404C22×C3#91611yescharacteristic
405C22×C3#91811yescharacteristic
406C22×C3#91811yescharacteristic
407C22×C3#91911yescharacteristic
408C22×C3#93711yescharacteristic
409C2×D3—122no—
410C2×D3—122no—
411C2×D3—122no—
412C2×D3—121no—
413C2×D3—121no—
414C2×D3—121no—
415C2×D3—121no—
416C2×D3—121no—
417C2×D3—121no—
418C3×C4—21no—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—41no—
423C3×C4—41no—
424C3×C4—42no—
425C3×C4—42no—
426C3×C4—41no—
427C3×C4—41no—
428C3×C4—41no—
429C3×C4—41no—
430C3×C4—122no—
431C3×C4—122no—
432Dic3—42no—
433Dic3—42no—
434Dic3—122no—
435Dic3—122no—
436Dic3—122no—
437Dic3—121no—
438Dic3—121no—
439Dic3—121no—
440Dic3—121no—
441Dic3—121no—
442Dic3—121no—
443C32#160311yescharacteristic, complemented, Sylow
444C23—181no—
445C23#381111yescharacteristic
446Q8—41no—
447Q8—62no—
448Q8—62no—
449Q8—121no—
450Q8—121no—
451D4—361no—
452D4—361no—
453D4—361no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—41no—
457C2×C4—62no—
458C2×C4—62no—
459C2×C4—121no—
460C2×C4—121no—
461C2×C4—121no—
462C2×C4—121no—
463C2×C4—181no—
464C2×C4—181no—
465C2×C4—181no—
466C2×C4—181no—
467C2×C4—181no—
468C2×C4—361no—
469C2×C4—721no—
470C2×C4C2×#62611yescharacteristic
471C2×C4#381111yescharacteristic
472C8—121no—
473C8—121no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—41no—
482C2×C3—41no—
483C2×C3#474111yescharacteristic
484C2×C3#480811yescharacteristic
485C2×C3#482611yescharacteristic
486C2×C3#503211yescharacteristic
487C2×C3#518711yescharacteristic
488C2×C3#520511yescharacteristic
489D3—242no—
490D3—241no—
491D3—241no—
492C22—21no—
493C22—21no—
494C22—361no—
495C22—361no—
496C22—361no—
497C22#6394511yescharacteristic
498C22#6395011yescharacteristic
499C22#6398111yescharacteristic
500C4—21no—
501C4—21no—
502C4—41no—
503C4—41no—
504C4—122no—
505C4—122no—
506C4—361no—
507C4—361no—
508C4—361no—
509C3—21no—
510C3#6832611yescharacteristic, complemented
511C3#7098211yescharacteristic, complemented
512C2—21no—
513C2—21no—
514C2—721no—
515C2#1059153211yescharacteristic
516C2#1059160211yescharacteristic
517C2#1059166811yescharacteristic
subgroupG/H
centerC22#63981
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC32×C22⋊Q8C22
socleC22×C32#120
prime radical#64139C22
hypercenterC22⋊Q8D32
hypocenterC32#1603
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C3×#5050C2semidirect
3. #10591346C2semidirect
5. C3×#4538C2semidirect
7. #10589148C2semidirect
35. #5050D3semidirect
39. #4538D3semidirect
443. C32#1603semidirect
510. C3#68326semidirect
511. C3#70982semidirect

Maximal subgroups

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

subgroupindex
#105961162
C3×#50502
#105913462
#105911742
C3×#45382
#105891512
#105891482
#709823
#683263