Finite Group Databasebeta

← #11897669

Subgroups — #11897669

Order 1152.

734 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5722C211yesmaximal, characteristic, complemented
2#10593958C211yesmaximal, characteristic
3#10592783C211yesmaximal, characteristic
4C3×#5078C211yesmaximal, characteristic, complemented
5#10590544C212yesmaximal, complemented
6#10590464C212yesmaximal
7#10590457C212yesmaximal
8#10590420C212yesmaximal, complemented
9C4×#3717C211yesmaximal, characteristic
10#10590310C211yesmaximal, characteristic, complemented
11#10590258C211yesmaximal, characteristic, complemented
12#83706—31nomaximal
13#81954—31nomaximal
14C32×C4⋊Q8C2211yescharacteristic
15C4×C4⋊(C3⋊D3)C2211yescharacteristic
16C3×#944—22no—
17C3×#888—21no—
18C3×#888C2212yes—
19C3×#887—21no—
20C3×#887C2212yes—
21C3×#869C2211yescharacteristic
22C3×C4×Dic6C2211yescharacteristic
23C3×C4×Dic6C2211yescharacteristic
24#63900—21no—
25#63894C2212yes—
26#63888—22no—
27#63871—21no—
28#63869C2212yes—
29C2×#3717C2211yescharacteristic
30#63636C2212yes—
31#63635C2212yes—
32C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
33C3×#807C2212yes—
34C3×#807C2212yes—
35C3×C4×C3⋊C8C2211yescharacteristic
36#63576C2212yes—
37#63573C2211yescharacteristic
38Dic3⋊(C3⋊C8)C2212yes—
39#63566C2211yescharacteristic
40#63565C2212yes—
41#63564C2211yescharacteristic
42#63554C2212yes—
43C3×Q82—31no—
44#5730—31no—
45#5722D311yescharacteristic, complemented
46#5579—31no—
47C3×#500—31no—
48#5091—31no—
49#5087—32no—
50#5084—31no—
51#5084—31no—
52#5084—61no—
53#5081—32no—
54#5078D311yescharacteristic, complemented
55#5026—32no—
56C4×#215—31no—
57#5024—31no—
58#5022—32no—
59#5021—31no—
60#4890—32no—
61#4889—32no—
62#4877—32no—
63#4876—32no—
64C4×#204—31no—
65#4679—31no—
66C2×C32×Q8—21no—
67C2×C4⋊(C3⋊D3)C2311yescharacteristic
68C2×C4×C3⋊D3—21no—
69C2×C3×Dic6—21no—
70C2×C3×Dic6—22no—
71C2×C3×Dic6D411yescharacteristic
72C2×C3×Dic6C2311yescharacteristic
73C2×#624—22no—
74C32×C4⋊C4—21no—
75C32×C4⋊C4C2312yes—
76C32×C42C2311yescharacteristic
77#3752—21no—
78C4⋊(C3⋊Dic3)C2311yescharacteristic
79C2×C3⋊(C3⋊C8)C2312yes—
80C3×C4⋊Dic3—21no—
81C3×C4⋊Dic3D411yescharacteristic
82C3×C4⋊Dic3C2311yescharacteristic
83C3×C4⋊Dic3C2312yes—
84C3×Dic3⋊C4—21no—
85C3×Dic3⋊C4—22no—
86C3×Dic3⋊C4—22no—
87C3×Dic3⋊C4D412yes—
88C3×C4×Dic3—21no—
89C3×C4×Dic3—22no—
90C3×C4×Dic3C2312yes—
91C2×C3×C3⋊C8C2311yescharacteristic
92C2×C3×C3⋊C8C2311yescharacteristic
93#3724—22no—
94Dic3⋊Dic3—22no—
95#3722—22no—
96#3717Q814yescomplemented
97#3385—91noSylow
98C3×C4⋊Q8—21no—
99C3×C4⋊Q8—32no—
100C3×C4⋊Q8—31no—
101C3×C4⋊Q8—61no—
102C3×C4⋊Q8—62no—
103C3×C4⋊Q8C2×D311yescharacteristic
104C3×C4⋊Q8C2×D311yescharacteristic
105C3×C4×Q8—31no—
106C3×C4×Q8—32no—
107C3×C4×Q8—31no—
108C3×C4×Q8—61no—
109#946—61no—
110#946—62no—
111#944—22no—
112C2×#215—31no—
113C2×#204—31no—
114#897—61no—
115#896—62no—
116#893—32no—
117D3×C4⋊C4—61no—
118D3×C4⋊C4—61no—
119#890—32no—
120#888—21no—
121#888C2×D312yes—
122#887—21no—
123#887C2×D312yes—
124C4×D12—31no—
125C4×D12—31no—
126C4×D12—32no—
127C4×D12—61no—
128#870—32no—
129#869C2×D311yescharacteristic
130C4×Dic6C2×D311yescharacteristic
131C4×Dic6C2×D311yescharacteristic
132C3×#107—32no—
133C3×#107—32no—
134C3×C4×C8—31no—
135#835—32no—
136#835—31no—
137#821—31no—
138#817—31no—
139#816—31no—
140#815—32no—
141Dic3⋊C8—32no—
142#810—32no—
143#809—32no—
144#809—32no—
145#809—31no—
146#809—62no—
147#808—32no—
148#808—32no—
149#808—31no—
150#808—62no—
151#807—32no—
152#807—32no—
153#807C2×D312yes—
154#807C2×D312yes—
155C4⋊(C3⋊C8)—31no—
156C4⋊(C3⋊C8)—31no—
157C4⋊(C3⋊C8)—32no—
158C4⋊(C3⋊C8)—61no—
159#803—32no—
160C4×C3⋊C8C2×D311yescharacteristic
161C22×C3⋊D3—21no—
162C32×Q8—81no—
163C2×C32×C4—22no—
164C2×C32×C4C2×D411yescharacteristic
165C2×C32×C4C2×D411yescharacteristic
166C2×C32×C4C2411yescharacteristic
167C2×C3⋊Dic3—21no—
168C4⋊(C3⋊D3)—21no—
169C4⋊(C3⋊D3)C2×Q812yes—
170C4×C3⋊D3—42no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—22no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3C2×D412yes—
175C2×C3×Dic3C2×D412yes—
176C3×Dic6—22no—
177C3×Dic6—21no—
178C3×Dic6—82no—
179C3×Dic6—81no—
180C3×Dic6C2×Q812yes—
181#624—82no—
182C3⋊(C3⋊C8)—22no—
183C3×C3⋊C8—21no—
184C3×C3⋊C8C2×Q812yes—
185Q82—31no—
186#554—91no—
187#501—92no—
188#500—31no—
189#479—92no—
190#477—91no—
191#476—91no—
192#474—92no—
193#446—91no—
194#440—92no—
195C4×SD16—91no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—31no—
199C2×C3×Q8—31no—
200C2×C3×Q8—41no—
201C2×C3×Q8—62no—
202C2×C3×Q8—61no—
203C2×D12—31no—
204C2×D12—31no—
205C2×D12—61no—
206C2×C4×D3—61no—
207C2×C4×D3—61no—
208C2×C4×D3—62no—
209C2×C4×D3—62no—
210C2×C4×D3—121no—
211C2×Dic6—21no—
212C2×Dic6—22no—
213C2×Dic6#6711yescharacteristic
214C2×Dic6C22×D311yescharacteristic
215C2×C3×C8—31no—
216C2×C3×C8—31no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—31no—
221C3×C4⋊C4—32no—
222C3×C4⋊C4—31no—
223C3×C4⋊C4—32no—
224C3×C4⋊C4—41no—
225C3×C4⋊C4—62no—
226C3×C4⋊C4—62no—
227C3×C4⋊C4—61no—
228C3×C4⋊C4—61no—
229C3×C4⋊C4C22×D312yes—
230C3×C4⋊C4C22×D312yes—
231C3×C42—21no—
232C3×C42—32no—
233C3×C42—62no—
234C3×C42—61no—
235C3×C42C22×D311yescharacteristic
236C3×C42C22×D311yescharacteristic
237#215—34no—
238(C2×D3)⋊C4—62no—
239(C2×D3)⋊C4—61no—
240(C2×D3)⋊C4—61no—
241(C2×D3)⋊C4—62no—
242(C2×D3)⋊C4—121no—
243C4⋊Dic3—21no—
244C4⋊Dic3—31no—
245C4⋊Dic3—31no—
246C4⋊Dic3—61no—
247C4⋊Dic3—62no—
248C4⋊Dic3#6711yescharacteristic
249C4⋊Dic3C22×D311yescharacteristic
250C4⋊Dic3C22×D312yes—
251Dic3⋊C4—21no—
252Dic3⋊C4—22no—
253Dic3⋊C4—22no—
254Dic3⋊C4—62no—
255Dic3⋊C4—62no—
256Dic3⋊C4—62no—
257Dic3⋊C4#6712yes—
258C4×Dic3—21no—
259C4×Dic3—22no—
260C4×Dic3C22×D312yes—
261C2×C3⋊C8—32no—
262C2×C3⋊C8—32no—
263C2×C3⋊C8—62no—
264C2×C3⋊C8C22×D311yescharacteristic
265C2×C3⋊C8C22×D311yescharacteristic
266#204—34no—
267C22×C32C22×D411yescharacteristic
268C2×C3⋊D3—22no—
269C2×C3⋊D3—41no—
270C2×C3⋊D3—41no—
271C32×C4—21no—
272C32×C4—42no—
273C32×C4#13712yes—
274C32×C4C22×Q811yescharacteristic
275C32×C42−1+411yescharacteristic
276C3⋊Dic3—41no—
277C3×Dic3—22no—
278C3×Dic3—22no—
279C3×Dic3—22no—
280C3×Dic3—42no—
281C3×Dic3—41no—
282C2×SD16—91no—
283C4⋊Q8—32no—
284C4⋊Q8—31no—
285C4⋊Q8—61no—
286C4⋊Q8—62no—
287C4⋊Q8D3211yescharacteristic
288#125—92no—
289C22⋊Q8—182no—
290C22⋊Q8—181no—
291C4×Q8—31no—
292C4×Q8—32no—
293C4×Q8—31no—
294C4×Q8—61no—
295C4×D4—92no—
296C4×D4—91no—
297C2×C4⋊C4—181no—
298#107—32no—
299#107—32no—
300#107—92no—
301#106—91no—
302#106—92no—
303C4⋊C8—92no—
304C4⋊C8—91no—
305#103—92no—
306#103—91no—
307#102—91no—
308#102—92no—
309#97—92no—
310C4×C8—31no—
311C22×D3—61no—
312C22×D3—61no—
313C22×D3—121no—
314C3×Q8—32no—
315C3×Q8—62no—
316C3×Q8—61no—
317C3×Q8—81no—
318C3×Q8—81no—
319C3×Q8—82no—
320C3×Q8—241no—
321C3×Q8—242no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—32no—
328C2×C3×C4—32no—
329C2×C3×C4—42no—
330C2×C3×C4—61no—
331C2×C3×C4—61no—
332C2×C3×C4—62no—
333C2×C3×C4D3×D411yescharacteristic
334C2×C3×C4D3×D411yescharacteristic
335C2×C3×C4C2×#6711yescharacteristic
336C2×C3×C4C2×#6711yescharacteristic
337C2×C3×C4C23×D311yescharacteristic
338C2×C3×C4C23×D311yescharacteristic
339C2×Dic3—21no—
340C2×Dic3—22no—
341C2×Dic3—21no—
342C2×Dic3—61no—
343C2×Dic3—61no—
344C2×Dic3—121no—
345C2×Dic3C2×#6712yes—
346C2×Dic3C2×#6712yes—
347D12—32no—
348D12—32no—
349D12—62no—
350D12—61no—
351D12—61no—
352D12—121no—
353C4×D3—122no—
354C4×D3—122no—
355C4×D3—242no—
356C4×D3—242no—
357C4×D3—242no—
358Dic6—22no—
359Dic6—21no—
360Dic6—82no—
361Dic6—81no—
362Dic6D3×Q812yes—
363C3×C8—32no—
364C3×C8—61no—
365C3⋊C8—21no—
366C3⋊C8—62no—
367C3⋊C8—62no—
368C3⋊C8—122no—
369C3⋊C8D3×Q812yes—
370C2×C32D4×Q811yescharacteristic
371C2×C32C2×#13711yescharacteristic
372C2×C32#57611yescharacteristic
373C3⋊D3—42no—
374C2×Q8—21no—
375C2×Q8—31no—
376C2×Q8—31no—
377C2×Q8—62no—
378C2×Q8—61no—
379C2×D4—91no—
380C22×C4—182no—
381C22×C4—181no—
382SD16—94no—
383C2×C8—31no—
384C2×C8—31no—
385C2×C8—92no—
386C4⋊C4—21no—
387C4⋊C4—31no—
388C4⋊C4—32no—
389C4⋊C4—31no—
390C4⋊C4—32no—
391C4⋊C4—62no—
392C4⋊C4—62no—
393C4⋊C4—61no—
394C4⋊C4—61no—
395C4⋊C4—91no—
396C4⋊C4—182no—
397C4⋊C4—182no—
398C4⋊C4C2×D3212yes—
399C22⋊C4—182no—
400C22⋊C4—181no—
401C42—32no—
402C42—62no—
403C42—61no—
404C42C2×D3211yescharacteristic
405C22×C3—21no—
406C22×C3C2×D3×D411yescharacteristic
407C22×C3C22×#6711yescharacteristic
408C2×D3—62no—
409C2×D3—62no—
410C2×D3—122no—
411C2×D3—121no—
412C2×D3—121no—
413C2×D3—121no—
414C2×D3—121no—
415C2×D3—241no—
416C2×D3—241no—
417C3×C4—21no—
418C3×C4—21no—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—22no—
422C3×C4—41no—
423C3×C4—42no—
424C3×C4—42no—
425C3×C4—62no—
426C3×C4—62no—
427C3×C4—62no—
428C3×C4—82no—
429C3×C4—121no—
430C3×C4—122no—
431C3×C4#91812yes—
432C3×C4#94212yes—
433C3×C4C2×D3×Q811yescharacteristic
434C3×C4C2×D3×Q811yescharacteristic
435C3×C4#100711yescharacteristic
436C3×C4#100711yescharacteristic
437Dic3—22no—
438Dic3—22no—
439Dic3—22no—
440Dic3—42no—
441Dic3—41no—
442Dic3—121no—
443Dic3—121no—
444Dic3—241no—
445C32#338511yescharacteristic, complemented, Sylow
446C23—181no—
447Q8—32no—
448Q8—61no—
449Q8—62no—
450Q8—81no—
451Q8—241no—
452Q8—242no—
453D4—92no—
454D4—181no—
455C2×C4—22no—
456C2×C4—32no—
457C2×C4—32no—
458C2×C4—61no—
459C2×C4—61no—
460C2×C4—62no—
461C2×C4—181no—
462C2×C4—362no—
463C2×C4—722no—
464C2×C4D3×#6711yescharacteristic
465C2×C4D3×#6711yescharacteristic
466C2×C4C22×D3211yescharacteristic
467C8—32no—
468C8—61no—
469C8—182no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3#572711yescharacteristic
474C2×C3#575711yescharacteristic
475C2×C3C2×#91811yescharacteristic
476C2×C3C2×#94211yescharacteristic
477C2×C3Q8×#6711yescharacteristic
478C2×C3#583511yescharacteristic
479D3—122no—
480D3—122no—
481D3—242no—
482C22—182no—
483C22—361no—
484C22—361no—
485C22C2×D3×#6711yescharacteristic
486C4—21no—
487C4—42no—
488C4—62no—
489C4—62no—
490C4—62no—
491C4—122no—
492C4—121no—
493C4—361no—
494C4#6394812yes—
495C4#6431611yescharacteristic
496C4D32×Q811yescharacteristic
497C3—21no—
498C3#8195411yescharacteristic, complemented
499C3#8370611yescharacteristic, complemented
500C2—362no—
501C2#1059509611yescharacteristic
502C2C2×#6394811yescharacteristic
503C2#1059540311yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4⋊Q8C22
socleC22×C32C22×D4
prime radicalC2×C4⋊(C3⋊D3)C23
hypercenterC4⋊Q8D32
hypocenterC32#3385
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×#5722C2semidirect
4. C3×#5078C2semidirect
5. #10590544C2semidirect
8. #10590420C2semidirect
10. #10590310C2semidirect
11. #10590258C2semidirect
45. #5722D3semidirect
54. #5078D3semidirect
96. #3717Q8semidirect
445. C32#3385semidirect
498. C3#81954semidirect
499. C3#83706semidirect

Maximal subgroups

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

subgroupindex
C3×#57222
#105939582
#105927832
C3×#50782
#105905442
#105904642
#105904572
#105904202
C4×#37172
#105903102
#105902582
#837063
#819543