Finite Group Databasebeta

← #11887672

Subgroups — #11887672

Order 1152.

843 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10593958C211yesmaximal, characteristic
2#10593903C211yesmaximal, characteristic
3C8×C4⋊(C3⋊D3)C211yesmaximal, characteristic
4C3×C4×Dic12C211yesmaximal, characteristic, complemented
5C3×#4681C211yesmaximal, characteristic, complemented
6#10590950C212yesmaximal, complemented
7#10590855C212yesmaximal, complemented
8#10590689C212yesmaximal, complemented
9#10590672C212yesmaximal, complemented
10#10590384C211yesmaximal, characteristic, complemented
11#10590310C211yesmaximal, characteristic, complemented
12#81613—31nomaximal
13#79735—31nomaximal
14C2×#3797C2212yes—
15C2×C8×C3⋊D3C2212yes—
16C4×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×Dic12C2211yescharacteristic
18C3×#869C2211yescharacteristic
19C3×#869C2211yescharacteristic
20C3×C4×Dic6C2211yescharacteristic
21C3×C4×Dic6C2211yescharacteristic
22#63918—21no—
23#63900C2212yes—
24#63895—22no—
25#63894—21no—
26#63894—21no—
27#63888—22no—
28#63871C2212yes—
29#63871C2212yes—
30#63869—21no—
31C2×#3718C2212yes—
32C32×C4×C8C2211yescharacteristic
33#63647C2212yes—
34C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
35C3×#818C2211yescharacteristic
36C3×#816C2212yes—
37C3×#816C2212yes—
38C3×#816C2211yescharacteristic
39C3×#816C2211yescharacteristic
40#63576C2212yes—
41#63565C2212yes—
42#63565C2212yes—
43#5730—31no—
44#5727—31no—
45#5579—31no—
46Q8×D12—31no—
47C3×#487—31no—
48C3×C4×Q16—31no—
49#5186—32no—
50#5091—31no—
51#5090—31no—
52#5031—31no—
53#5021—31no—
54#4884—32no—
55#4813—32no—
56#4813—32no—
57#4811—32no—
58#4809—32no—
59D3×#103—32no—
60D3×#103—32no—
61C4×Dic12D311yescharacteristic, complemented
62C8×D12—31no—
63C8×D12—31no—
64C8×D12—61no—
65#4681D311yescharacteristic, complemented
66C2×C4⋊(C3⋊D3)D411yescharacteristic
67C2×C4×C3⋊D3C2312yes—
68C2×C3×Dic6—21no—
69C2×C3×Dic6C2311yescharacteristic
70C2×C3×Dic6C2312yes—
71C2×C3×Dic6C2311yescharacteristic
72C2×#627—22no—
73C2×#624—22no—
74C2×#624—22no—
75#3797—24no—
76#3797—42no—
77C2×C32×C8C2311yescharacteristic
78C2×C32×C8C2311yescharacteristic
79C32×C42C2311yescharacteristic
80#3752D412yes—
81C4⋊(C3⋊Dic3)D411yescharacteristic
82C2×C3⋊(C3⋊C8)C2312yes—
83C8×C3⋊D3—42no—
84C3×C4⋊Dic3—21no—
85C3×C4⋊Dic3—21no—
86C3×C4⋊Dic3—21no—
87C3×C4⋊Dic3C2311yescharacteristic
88C3×C4⋊Dic3C2312yes—
89C3×C4⋊Dic3C2311yescharacteristic
90C3×Dic3⋊C4—21no—
91C3×Dic3⋊C4—21no—
92C3×C4×Dic3—21no—
93C3×C4×Dic3—21no—
94C3×Dic12—22no—
95#3724—22no—
96Dic3⋊Dic3—22no—
97Dic3⋊Dic3—22no—
98#3722—22no—
99#3722—22no—
100#3718—42no—
101#3299—91noSylow
102C2×D3×Q8—32no—
103C2×D3×Q8—32no—
104C2×C3×Q16—31no—
105C3×C4⋊Q8—31no—
106C3×C4⋊Q8—31no—
107C3×C4×Q8—31no—
108C3×C4×Q8—31no—
109#946—32no—
110#946—32no—
111#946—62no—
112C2×#216—32no—
113C2×#216—32no—
114C2×Dic12C2×D311yescharacteristic
115C2×D3×C8—32no—
116C2×D3×C8—32no—
117C2×D3×C8—62no—
118#897—32no—
119#897—32no—
120#897—62no—
121#896—62no—
122#896—62no—
123#893—61no—
124#893—61no—
125D3×C4⋊C4—32no—
126D3×C4⋊C4—32no—
127#890—61no—
128#888—61no—
129C4×D12—31no—
130C4×D12—31no—
131C4×D12—61no—
132C4×D12—61no—
133C4×D12—61no—
134#870—61no—
135#869—61no—
136#869C2×D311yescharacteristic
137#869C2×D311yescharacteristic
138C4×Dic6C2×D311yescharacteristic
139C4×Dic6C2×D311yescharacteristic
140C3×#107—31no—
141C3×#103—31no—
142C3×#103—32no—
143C3×#103—31no—
144C3×#103—32no—
145C3×C4×C8—21no—
146C3×C4×C8C2×D311yescharacteristic
147C3×C4×C8C2×D311yescharacteristic
148#835—32no—
149#835—32no—
150(C2×D3)⋊C8—32no—
151(C2×D3)⋊C8—32no—
152(C2×D3)⋊C8—62no—
153#818C2×D311yescharacteristic
154#816C2×D312yes—
155#816C2×D312yes—
156#816C2×D311yescharacteristic
157#816C2×D311yescharacteristic
158#810—32no—
159#810—32no—
160#807—32no—
161#807—32no—
162C4⋊(C3⋊C8)—31no—
163C4⋊(C3⋊C8)—31no—
164C4⋊(C3⋊C8)—61no—
165C22×C3⋊D3C2×D412yes—
166C2×C32×C4C2×D411yescharacteristic
167C2×C32×C4C2×D411yescharacteristic
168C2×C32×C4C2411yescharacteristic
169C2×C3⋊Dic3C2×D412yes—
170C4⋊(C3⋊D3)Q1614yescomplemented
171C4×C3⋊D3—22no—
172C4×C3⋊D3—22no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—22no—
177C2×C3×Dic3—22no—
178C2×C3×Dic3—21no—
179C3×Dic6—21no—
180C3×Dic6—21no—
181C3×Dic6—22no—
182C3×Dic6—21no—
183C3×Dic6—22no—
184C3×Dic6—81no—
185C3×Dic6#4112yes—
186#627—44no—
187#624—44no—
188#624—82no—
189C32×C8—21no—
190C32×C8—21no—
191C3⋊(C3⋊C8)—22no—
192#554—91no—
193D4×Q8—91no—
194#487—31no—
195#484—92no—
196#477—91no—
197#467—92no—
198#462—91no—
199#451—92no—
200C4×Q16—31no—
201C8×D4—91no—
202C2×#103—92no—
203C2×C3×Q8—31no—
204C2×C3×Q8—32no—
205C2×C3×Q8—31no—
206C2×C3×Q8—61no—
207D3×Q8—64no—
208D3×Q8—64no—
209D3×Q8—122no—
210D3×Q8—122no—
211C2×D12—31no—
212C2×D12—31no—
213C2×D12—61no—
214C2×C4×D3—32no—
215C2×C4×D3—32no—
216C2×C4×D3—62no—
217C2×C4×D3—62no—
218C2×C4×D3—62no—
219C2×C4×D3—62no—
220C2×C4×D3—62no—
221C2×Dic6—21no—
222C2×Dic6—62no—
223C2×Dic6—62no—
224C2×Dic6C22×D312yes—
225C2×Dic6C22×D311yescharacteristic
226C2×Dic6C22×D311yescharacteristic
227C3×Q16—62no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8C22×D311yescharacteristic
231C2×C3×C8C22×D311yescharacteristic
232C2×C3×C8C22×D311yescharacteristic
233C2×C3×C8C22×D311yescharacteristic
234C3×C4⋊C4—31no—
235C3×C4⋊C4—32no—
236C3×C4⋊C4—31no—
237C3×C4⋊C4—61no—
238C3×C4⋊C4—61no—
239C3×C4⋊C4—61no—
240C3×C4⋊C4—61no—
241C3×C4⋊C4—61no—
242C3×C42—21no—
243C3×C42—61no—
244C3×C42—61no—
245C3×C42C22×D311yescharacteristic
246C3×C42C22×D311yescharacteristic
247#216—122no—
248#216—122no—
249(C2×D3)⋊C4—32no—
250(C2×D3)⋊C4—32no—
251(C2×D3)⋊C4—62no—
252(C2×D3)⋊C4—62no—
253(C2×D3)⋊C4—62no—
254(C2×D3)⋊C4—62no—
255(C2×D3)⋊C4—62no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—31no—
260C4⋊Dic3—31no—
261C4⋊Dic3—61no—
262C4⋊Dic3—62no—
263C4⋊Dic3—62no—
264C4⋊Dic3C22×D311yescharacteristic
265C4⋊Dic3C22×D312yes—
266C4⋊Dic3C22×D311yescharacteristic
267Dic3⋊C4—21no—
268Dic3⋊C4—21no—
269Dic3⋊C4—62no—
270Dic3⋊C4—62no—
271Dic3⋊C4—62no—
272Dic3⋊C4—62no—
273C4×Dic3—21no—
274C4×Dic3—21no—
275C2×C3⋊C8—32no—
276C2×C3⋊C8—32no—
277C2×C3⋊C8—62no—
278Dic12—22no—
279D3×C8—122no—
280D3×C8—122no—
281D3×C8—124no—
282C22×C32C22×D411yescharacteristic
283C2×C3⋊D3—22no—
284C2×C3⋊D3—22no—
285C2×C3⋊D3C2×Q1614yes—
286C32×C4—21no—
287C32×C4C2×Q1612yes—
288C32×C4C2×#4111yescharacteristic
289C32×C4H(5,𝔽2)11yescharacteristic
290C3⋊Dic3—22no—
291C3×Dic3—22no—
292C3×Dic3—42no—
293C3×Dic3—41no—
294C3×Dic3—41no—
295C3×Dic3—42no—
296C3×Dic3—41no—
297C22×Q8—92no—
298C2×Q16—31no—
299C2×Q16—92no—
300C22×C8—92no—
301C4⋊Q8—31no—
302C4⋊Q8—31no—
303C4⋊Q8—181no—
304#125—181no—
305C22⋊Q8—92no—
306C22⋊Q8—92no—
307C22⋊Q8—182no—
308C22⋊Q8—182no—
309C4×Q8—31no—
310C4×Q8—31no—
311C4×D4—91no—
312C4×D4—181no—
313C4×D4—181no—
314C2×C4⋊C4—92no—
315#107—31no—
316#107—92no—
317C4⋊C8—91no—
318#103—31no—
319#103—32no—
320#103—31no—
321#103—32no—
322#103—92no—
323#103—92no—
324C22⋊C8—92no—
325C4×C8D3211yescharacteristic
326C22×D3—32no—
327C22×D3—32no—
328C22×D3—62no—
329C3×Q8—32no—
330C3×Q8—61no—
331C3×Q8—62no—
332C3×Q8—61no—
333C3×Q8—61no—
334C3×Q8—62no—
335C3×Q8—241no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—61no—
340C2×C3×C4—61no—
341C2×C3×C4—61no—
342C2×C3×C4—62no—
343C2×C3×C4—62no—
344C2×C3×C4—61no—
345C2×C3×C4D3×D411yescharacteristic
346C2×C3×C4D3×D411yescharacteristic
347C2×C3×C4D3×D411yescharacteristic
348C2×C3×C4D3×D411yescharacteristic
349C2×C3×C4C23×D311yescharacteristic
350C2×C3×C4C23×D311yescharacteristic
351C2×Dic3—21no—
352C2×Dic3—21no—
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—22no—
356C2×Dic3—22no—
357C2×Dic3—32no—
358C2×Dic3—32no—
359C2×Dic3—62no—
360D12—34no—
361D12—34no—
362D12—64no—
363C4×D3—62no—
364C4×D3—62no—
365C4×D3—62no—
366C4×D3—62no—
367C4×D3—122no—
368C4×D3—122no—
369C4×D3—124no—
370C4×D3—124no—
371C4×D3—242no—
372C4×D3—242no—
373Dic6—21no—
374Dic6—21no—
375Dic6—22no—
376Dic6—21no—
377Dic6—22no—
378Dic6—81no—
379Dic6—124no—
380Dic6—124no—
381Dic6#23512yes—
382C3×C8—22no—
383C3×C8—21no—
384C3×C8—21no—
385C3×C8—21no—
386C3×C8—21no—
387C3×C8—41no—
388C3⋊C8—62no—
389C3⋊C8—62no—
390C3⋊C8—122no—
391C2×C32#54611yescharacteristic
392C2×C32C22×Q1611yescharacteristic
393C2×C32#57711yescharacteristic
394C3⋊D3—24no—
395C2×Q8—31no—
396C2×Q8—32no—
397C2×Q8—31no—
398C2×Q8—61no—
399C2×Q8—182no—
400C2×Q8—184no—
401C2×Q8—362no—
402C2×D4—91no—
403C22×C4—92no—
404C22×C4—182no—
405C22×C4—182no—
406Q16—62no—
407Q16—362no—
408C2×C8—92no—
409C2×C8—362no—
410C2×C8C2×D3211yescharacteristic
411C2×C8C2×D3211yescharacteristic
412C4⋊C4—32no—
413C4⋊C4—31no—
414C4⋊C4—31no—
415C4⋊C4—61no—
416C4⋊C4—61no—
417C4⋊C4—61no—
418C4⋊C4—61no—
419C4⋊C4—61no—
420C4⋊C4—91no—
421C4⋊C4—182no—
422C4⋊C4—182no—
423C4⋊C4—182no—
424C22⋊C4—92no—
425C22⋊C4—182no—
426C22⋊C4—182no—
427C42—61no—
428C42—61no—
429C42C2×D3211yescharacteristic
430C22×C3—21no—
431C22×C3C2×D3×D411yescharacteristic
432C22×C3C2×D3×D411yescharacteristic
433C2×D3—34no—
434C2×D3—34no—
435C2×D3—64no—
436C2×D3—62no—
437C2×D3—62no—
438C2×D3—62no—
439C2×D3—62no—
440C2×D3—122no—
441C2×D3—122no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—22no—
447C3×C4—41no—
448C3×C4—62no—
449C3×C4—121no—
450C3×C4—121no—
451C3×C4—121no—
452C3×C4—122no—
453C3×C4—122no—
454C3×C4D3×Q1612yes—
455C3×C4D3×Q1612yes—
456C3×C4C2×#23511yescharacteristic
457C3×C4D3×#4111yescharacteristic
458C3×C4#100911yescharacteristic
459C3×C4#100911yescharacteristic
460Dic3—22no—
461Dic3—42no—
462Dic3—42no—
463Dic3—41no—
464Dic3—41no—
465Dic3—41no—
466Dic3—62no—
467Dic3—62no—
468Dic3—122no—
469C32#329911yescharacteristic, complemented, Sylow
470C23—92no—
471Q8—32no—
472Q8—61no—
473Q8—61no—
474Q8—62no—
475Q8—61no—
476Q8—62no—
477Q8—241no—
478Q8—364no—
479D4—94no—
480C2×C4—61no—
481C2×C4—61no—
482C2×C4—61no—
483C2×C4—61no—
484C2×C4—62no—
485C2×C4—62no—
486C2×C4—92no—
487C2×C4—182no—
488C2×C4—182no—
489C2×C4—364no—
490C2×C4—722no—
491C2×C4#380211yescharacteristic
492C2×C4#380211yescharacteristic
493C2×C4C22×D3211yescharacteristic
494C8—21no—
495C8—21no—
496C8—182no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3#555011yescharacteristic
501C2×C3#567611yescharacteristic
502C2×C3#576211yescharacteristic
503C2×C3C2×D3×Q1611yescharacteristic
504C2×C3C2×D3×Q1611yescharacteristic
505C2×C3#577011yescharacteristic
506D3—64no—
507D3—64no—
508D3—124no—
509C22—94no—
510C22—182no—
511C22—182no—
512C22C2×#380211yescharacteristic
513C4—21no—
514C4—62no—
515C4—121no—
516C4—122no—
517C4—121no—
518C4—122no—
519C4—121no—
520C4—182no—
521C4#6380212yes—
522C4D3×#23511yescharacteristic
523C4#6430811yescharacteristic
524C3—21no—
525C3#7973511yescharacteristic, complemented
526C3#8161311yescharacteristic, complemented
527C2—184no—
528C2#1059400711yescharacteristic
529C2#1059468611yescharacteristic
530C2C2×#6380211yescharacteristic
subgroupG/H
centerC22C2×#3802
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4×C8C22
socleC22×C32C22×D4
prime radicalC2×C4⋊(C3⋊D3)D4
hypercenterC4×C8D32
hypocenterC32#3299
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
4. C3×C4×Dic12C2semidirect
5. C3×#4681C2semidirect
6. #10590950C2semidirect
7. #10590855C2semidirect
8. #10590689C2semidirect
9. #10590672C2semidirect
10. #10590384C2semidirect
11. #10590310C2semidirect
61. C4×Dic12D3semidirect
65. #4681D3semidirect
170. C4⋊(C3⋊D3)Q16semidirect
469. C32#3299semidirect
525. C3#79735semidirect
526. C3#81613semidirect

Maximal subgroups

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

subgroupindex
#105939582
#105939032
C8×C4⋊(C3⋊D3)2
C3×C4×Dic122
C3×#46812
#105909502
#105908552
#105906892
#105906722
#105903842
#105903102
#816133
#797353