Finite Group Databasebeta

← #11896084

Subgroups — #11896084

Order 1152.

730 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10593865C211yesmaximal, characteristic
2Dic62C211yesmaximal, characteristic, complemented
3#10592541C211yesmaximal, characteristic
4C3×#4849C211yesmaximal, characteristic, complemented
5C3×#4831C211yesmaximal, characteristic, complemented
6#10590893C212yesmaximal
7#10590812C212yesmaximal, complemented
8#10590803C212yesmaximal, complemented
9#10590801C212yesmaximal
10C4×#3717C211yesmaximal, characteristic
11#10590287C211yesmaximal, characteristic, complemented
12#80464—31nomaximal
13#79981—31nomaximal
14C4×C4⋊(C3⋊D3)C2211yescharacteristic
15#64078C2211yescharacteristic
16C2×C3×Dic12C2212yes—
17C3×#869C2211yescharacteristic
18C3×C4×Dic6C2211yescharacteristic
19C3×C4×Dic6C2211yescharacteristic
20#63918—21no—
21#63918C2212yes—
22#63871—21no—
23#63864C2212yes—
24#63849—21no—
25#63848—22no—
26Dic3×Dic6—21no—
27Dic3×Dic6C2212yes—
28Dic3×Dic6C2212yes—
29#63840—22no—
30C2×#3717C2211yescharacteristic
31C32×C4⋊C8C2211yescharacteristic
32#63648C2212yes—
33#63644C2212yes—
34C3×#816C2212yes—
35C3×#815C2212yes—
36C3×Dic3⋊C8C2212yes—
37C3×C4×C3⋊C8C2211yescharacteristic
38#63573C2211yescharacteristic
39#63566C2212yes—
40#63566C2212yes—
41#63566C2211yescharacteristic
42#63564C2211yescharacteristic
43#5732—31no—
44#5584—31no—
45Q8×Dic6—31no—
46Q8×Dic6—31no—
47C3×#462—31no—
48C3×#446—31no—
49#5185—32no—
50#5184—32no—
51#5079—31no—
52C4×#215—31no—
53#4851—32no—
54#4849D311yescharacteristic, complemented
55#4845—32no—
56#4842—31no—
57#4842—31no—
58#4842—61no—
59#4831D311yescharacteristic, complemented
60#4794—32no—
61#4791—32no—
62#4716—32no—
63#4714—32no—
64C4×#204—31no—
65#4681—31no—
66C2×C4⋊(C3⋊D3)C2311yescharacteristic
67C2×C4×C3⋊D3—21no—
68C2×#634—21no—
69C2×C3×Dic6D411yescharacteristic
70C2×C3×Dic6C2312yes—
71C2×C3×Dic6C2311yescharacteristic
72C2×#627—22no—
73C2×C32×C8C2312yes—
74C32×C42C2311yescharacteristic
75#3752—21no—
76C4⋊(C3⋊Dic3)—21no—
77C4⋊(C3⋊Dic3)C2312yes—
78C4⋊(C3⋊Dic3)C2311yescharacteristic
79C3×C4⋊Dic3—21no—
80C3×C4⋊Dic3—21no—
81C3×C4⋊Dic3D411yescharacteristic
82C3×C4⋊Dic3C2311yescharacteristic
83C3×Dic3⋊C4—21no—
84C3×Dic3⋊C4D412yes—
85C3×C4×Dic3—21no—
86C3×C4×Dic3C2312yes—
87C2×C3×C3⋊C8C2311yescharacteristic
88C2×C3×C3⋊C8C2311yescharacteristic
89C3×Dic12—24no—
90Dic3⋊Dic3—22no—
91Dic3⋊Dic3—22no—
92Dic3⋊Dic3—22no—
93#3722—22no—
94#3722—22no—
95Dic32—22no—
96Dic32—22no—
97#3717—22no—
98#3369—91noSylow
99C2×C3×Q16—32no—
100C3×C4⋊Q8—31no—
101C3×C4×Q8—31no—
102C3×C4×Q8—31no—
103#947—32no—
104#946—61no—
105Q8×Dic3—32no—
106Q8×Dic3—32no—
107Q8×Dic3—61no—
108C2×#215—31no—
109C2×Dic12C2×D312yes—
110C2×#204—31no—
111#898—62no—
112#898—62no—
113#897—61no—
114#892—61no—
115#892—61no—
116#888—32no—
117#888—61no—
118#888—62no—
119#888—62no—
120#875—32no—
121C4×D12—31no—
122C4×D12—31no—
123C4×D12—61no—
124#869—31no—
125#869—31no—
126#869—32no—
127#869—61no—
128#869—61no—
129#869C2×D311yescharacteristic
130C4×Dic6—32no—
131C4×Dic6—32no—
132C4×Dic6—61no—
133C4×Dic6C2×D311yescharacteristic
134C4×Dic6C2×D311yescharacteristic
135C3×C4⋊C8—21no—
136C3×C4⋊C8—32no—
137C3×C4⋊C8C2×D311yescharacteristic
138C3×C4⋊C8C2×D311yescharacteristic
139C3×#103—32no—
140C3×#97—32no—
141C3×C4×C8—31no—
142#835—31no—
143#835—32no—
144#835—32no—
145#821—32no—
146#821—32no—
147#821—31no—
148#821—62no—
149#817—32no—
150#817—32no—
151#817—31no—
152#817—62no—
153#816—32no—
154#816—32no—
155#816—31no—
156#816C2×D312yes—
157#815C2×D312yes—
158Dic3⋊C8C2×D312yes—
159#809—31no—
160#808—31no—
161C4×C3⋊C8C2×D311yescharacteristic
162C22×C3⋊D3—21no—
163C2×C32×C4C2×D411yescharacteristic
164C2×C32×C4C2×D411yescharacteristic
165C2×C32×C4C2411yescharacteristic
166C2×C3⋊Dic3—21no—
167C2×C3⋊Dic3—22no—
168C4⋊(C3⋊D3)—21no—
169C4⋊(C3⋊D3)—21no—
170C4×C3⋊D3—42no—
171#634—81no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—22no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3C2×D412yes—
176C2×C3×Dic3C2×D412yes—
177C3×Dic6—22no—
178C3×Dic6—21no—
179C3×Dic6—22no—
180C3×Dic6—22no—
181C3×Dic6#4112yes—
182#627—82no—
183C32×C8—22no—
184C3×C3⋊C8—21no—
185C3×C3⋊C8—21no—
186Q82—91no—
187#556—91no—
188#489—92no—
189#487—91no—
190#476—91no—
191#475—92no—
192#464—92no—
193#462—31no—
194#446—31no—
195#441—92no—
196C4×SD16—91no—
197C2×C3×Q8—31no—
198C2×C3×Q8—31no—
199C2×C3×Q8—32no—
200C2×D12—31no—
201C2×D12—31no—
202C2×D12—61no—
203C2×C4×D3—61no—
204C2×C4×D3—61no—
205C2×C4×D3—121no—
206C2×Dic6—62no—
207C2×Dic6—61no—
208C2×Dic6—61no—
209C2×Dic6—62no—
210C2×Dic6—121no—
211C2×Dic6D1211yescharacteristic
212C2×Dic6C22×D312yes—
213C2×Dic6C22×D311yescharacteristic
214C3×Q16—64no—
215C2×C3×C8—22no—
216C2×C3×C8—31no—
217C2×C3×C8—31no—
218C2×C3×C8C22×D312yes—
219C2×C3×C8C22×D312yes—
220C3×C4⋊C4—31no—
221C3×C4⋊C4—32no—
222C3×C4⋊C4—31no—
223C3×C4⋊C4—61no—
224C3×C4⋊C4—61no—
225C3×C4⋊C4—61no—
226C3×C42—21no—
227C3×C42—32no—
228C3×C42—61no—
229C3×C42C22×D311yescharacteristic
230C3×C42C22×D311yescharacteristic
231#215—62no—
232(C2×D3)⋊C4—62no—
233(C2×D3)⋊C4—62no—
234(C2×D3)⋊C4—61no—
235(C2×D3)⋊C4—61no—
236(C2×D3)⋊C4—62no—
237(C2×D3)⋊C4—62no—
238(C2×D3)⋊C4—121no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—31no—
242C4⋊Dic3—32no—
243C4⋊Dic3—31no—
244C4⋊Dic3—32no—
245C4⋊Dic3—61no—
246C4⋊Dic3—62no—
247C4⋊Dic3—62no—
248C4⋊Dic3—61no—
249C4⋊Dic3—61no—
250C4⋊Dic3—62no—
251C4⋊Dic3—62no—
252C4⋊Dic3—121no—
253C4⋊Dic3D1211yescharacteristic
254C4⋊Dic3C22×D311yescharacteristic
255Dic3⋊C4—21no—
256Dic3⋊C4—62no—
257Dic3⋊C4—62no—
258Dic3⋊C4—62no—
259Dic3⋊C4D1212yes—
260C4×Dic3—21no—
261C4×Dic3—62no—
262C4×Dic3—62no—
263C4×Dic3—62no—
264C4×Dic3—62no—
265C4×Dic3C22×D312yes—
266C2×C3⋊C8C22×D311yescharacteristic
267C2×C3⋊C8C22×D311yescharacteristic
268Dic12—24no—
269#204—62no—
270C22×C32C22×D411yescharacteristic
271C2×C3⋊D3—41no—
272C2×C3⋊D3—41no—
273C2×C3⋊D3—41no—
274C32×C4—21no—
275C32×C4#13712yes—
276C32×C4C2×#4111yescharacteristic
277C32×C42−1+411yescharacteristic
278C3⋊Dic3—41no—
279C3⋊Dic3—42no—
280C3×Dic3—22no—
281C3×Dic3—22no—
282C3×Dic3—22no—
283C3×Dic3—42no—
284C3×Dic3—41no—
285C2×Q16—32no—
286C2×SD16—91no—
287C4⋊Q8—31no—
288C4⋊Q8—92no—
289C4⋊Q8—91no—
290C4⋊Q8—181no—
291C4⋊Q8—182no—
292#126—182no—
293#124—92no—
294C22⋊Q8—181no—
295C4×Q8—31no—
296C4×Q8—31no—
297C4×Q8—92no—
298C4×Q8—92no—
299C4×Q8—181no—
300C4×D4—91no—
301#117—181no—
302#106—91no—
303#106—92no—
304C4⋊C8—32no—
305C4⋊C8D3211yescharacteristic
306#103—32no—
307#103—91no—
308#103—92no—
309#103—92no—
310#102—91no—
311#102—92no—
312#97—32no—
313C4×C8—31no—
314C22×D3—61no—
315C22×D3—61no—
316C22×D3—121no—
317C3×Q8—32no—
318C3×Q8—62no—
319C3×Q8—61no—
320C3×Q8—62no—
321C3×Q8—62no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—32no—
326C2×C3×C4—32no—
327C2×C3×C4—61no—
328C2×C3×C4—62no—
329C2×C3×C4—61no—
330C2×C3×C4C2×D1211yescharacteristic
331C2×C3×C4C2×D1211yescharacteristic
332C2×C3×C4D3×D411yescharacteristic
333C2×C3×C4D3×D411yescharacteristic
334C2×C3×C4C23×D311yescharacteristic
335C2×C3×C4C23×D311yescharacteristic
336C2×Dic3—21no—
337C2×Dic3—21no—
338C2×Dic3—22no—
339C2×Dic3—61no—
340C2×Dic3—62no—
341C2×Dic3—61no—
342C2×Dic3—62no—
343C2×Dic3—122no—
344C2×Dic3—121no—
345C2×Dic3C2×D1212yes—
346C2×Dic3C2×D1212yes—
347D12—62no—
348D12—61no—
349D12—61no—
350D12—61no—
351D12—61no—
352D12—121no—
353C4×D3—122no—
354C4×D3—122no—
355C4×D3—242no—
356Dic6—22no—
357Dic6—21no—
358Dic6—22no—
359Dic6—22no—
360Dic6—242no—
361Dic6—242no—
362Dic6—241no—
363Dic6—241no—
364Dic6—242no—
365Dic6#23712yes—
366C3×C8—22no—
367C3×C8—22no—
368C3×C8—42no—
369C3×C8—61no—
370C3×C8—61no—
371C3⋊C8—21no—
372C3⋊C8—21no—
373C2×C32#54811yescharacteristic
374C2×C32C2×#13711yescharacteristic
375C2×C32#57711yescharacteristic
376C3⋊D3—81no—
377C2×Q8—31no—
378C2×Q8—31no—
379C2×Q8—32no—
380C2×Q8—182no—
381C2×Q8—181no—
382C2×D4—91no—
383C22×C4—181no—
384Q16—64no—
385SD16—182no—
386C2×C8—31no—
387C2×C8—31no—
388C2×C8C2×D3212yes—
389C4⋊C4—31no—
390C4⋊C4—32no—
391C4⋊C4—31no—
392C4⋊C4—61no—
393C4⋊C4—61no—
394C4⋊C4—61no—
395C4⋊C4—92no—
396C4⋊C4—91no—
397C4⋊C4—182no—
398C4⋊C4—182no—
399C4⋊C4—182no—
400C4⋊C4—181no—
401C22⋊C4—182no—
402C22⋊C4—182no—
403C22⋊C4—181no—
404C42—32no—
405C42—61no—
406C42—182no—
407C42—182no—
408C42C2×D3211yescharacteristic
409C22×C3—21no—
410C22×C3C22×D1211yescharacteristic
411C22×C3C2×D3×D411yescharacteristic
412C2×D3—122no—
413C2×D3—121no—
414C2×D3—121no—
415C2×D3—121no—
416C2×D3—121no—
417C2×D3—121no—
418C2×D3—121no—
419C2×D3—241no—
420C2×D3—241no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—22no—
425C3×C4—21no—
426C3×C4—41no—
427C3×C4—62no—
428C3×C4—62no—
429C3×C4—62no—
430C3×C4—121no—
431C3×C4—122no—
432C3×C4#90912yes—
433C3×C4#91812yes—
434C3×C4C2×#23711yescharacteristic
435C3×C4#100711yescharacteristic
436C3×C4D3×#4111yescharacteristic
437C3×C4#101011yescharacteristic
438Dic3—22no—
439Dic3—22no—
440Dic3—22no—
441Dic3—41no—
442Dic3—42no—
443Dic3—121no—
444Dic3—122no—
445Dic3—121no—
446Dic3—122no—
447Dic3—242no—
448Dic3—241no—
449C32#336911yescharacteristic, complemented, Sylow
450C23—181no—
451Q8—32no—
452Q8—62no—
453Q8—61no—
454Q8—62no—
455Q8—62no—
456Q8—721no—
457Q8—722no—
458D4—181no—
459D4—181no—
460C2×C4—32no—
461C2×C4—32no—
462C2×C4—61no—
463C2×C4—61no—
464C2×C4—62no—
465C2×C4—181no—
466C2×C4—182no—
467C2×C4—362no—
468C2×C4D3×D1211yescharacteristic
469C2×C4D3×D1211yescharacteristic
470C2×C4C22×D3211yescharacteristic
471C8—22no—
472C8—61no—
473C8—61no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3#557611yescharacteristic
478C2×C3#559811yescharacteristic
479C2×C3C2×#90911yescharacteristic
480C2×C3#575111yescharacteristic
481C2×C3C2×#91811yescharacteristic
482C2×C3#577811yescharacteristic
483D3—242no—
484D3—241no—
485D3—241no—
486C22—361no—
487C22—361no—
488C22—361no—
489C22C2×D3×D1211yescharacteristic
490C4—21no—
491C4—62no—
492C4—62no—
493C4—62no—
494C4—122no—
495C4—121no—
496C4—361no—
497C4—362no—
498C4#6380112yes—
499C4D3×#23711yescharacteristic
500C4#6431511yescharacteristic
501C3—21no—
502C3#7998111yescharacteristic, complemented
503C3#8046411yescharacteristic, complemented
504C2—721no—
505C2#1059438211yescharacteristic
506C2C2×#6380111yescharacteristic
507C2#1059471511yescharacteristic
subgroupG/H
centerC22C2×D3×D12
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4⋊C8C22
socleC22×C32C22×D4
prime radicalC2×C4⋊(C3⋊D3)C23
hypercenterC4⋊C8D32
hypocenterC32#3369
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
2. Dic62C2semidirect
4. C3×#4849C2semidirect
5. C3×#4831C2semidirect
7. #10590812C2semidirect
8. #10590803C2semidirect
11. #10590287C2semidirect
54. #4849D3semidirect
59. #4831D3semidirect
449. C32#3369semidirect
502. C3#79981semidirect
503. C3#80464semidirect

Maximal subgroups

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

subgroupindex
#105938652
Dic622
#105925412
C3×#48492
C3×#48312
#105908932
#105908122
#105908032
#105908012
C4×#37172
#105902872
#804643
#799813