Finite Group Databasebeta

← #11897662

Subgroups — #11897662

Order 1152.

786 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#634C211yesmaximal, characteristic, complemented
2C3×#5579C211yesmaximal, characteristic
3C3×#5024C211yesmaximal, characteristic, complemented
4#10590360C212yesmaximal
5C4×#3716C211yesmaximal, characteristic
6#10590311C211yesmaximal, characteristic, complemented
7#10590306C212yesmaximal, complemented
8#10590292C212yesmaximal, complemented
9#10590284C211yesmaximal, characteristic, complemented
10#10590272C212yesmaximal
11#10590268C211yesmaximal, characteristic
12#83631—31nomaximal
13#79976—31nomaximal
14C32×C4×Q8C2211yescharacteristic
15Q8×C3⋊Dic3—21no—
16#64097—21no—
17#64097—22no—
18#64078C2211yescharacteristic
19#64078C2212yes—
20C4×#634C2211yescharacteristic
21C4×#634C2212yes—
22C3×#946—21no—
23C3×#896—22no—
24C3×D3×C4⋊C4—21no—
25C3×C4×D12C2211yescharacteristic
26C3×C4×D12C2212yes—
27C3×#870C2212yes—
28C3×#869C2211yescharacteristic
29C2×#3716C2211yescharacteristic
30C3×C4⋊(C3⋊C8)C2211yescharacteristic
31C3×C4⋊(C3⋊C8)C2212yes—
32C3×#803C2212yes—
33C3×C4×C3⋊C8C2211yescharacteristic
34#63575C2212yes—
35#63575C2212yes—
36#63575C2212yes—
37#63573C2212yes—
38#63573C2211yescharacteristic
39#63567C2212yes—
40#63567C2211yescharacteristic
41#63563C2212yes—
42#63563C2211yescharacteristic
43C3×#554—31no—
44#5579D311yescharacteristic
45Q8×Dic6—31no—
46Q8×Dic6—31no—
47Q8×Dic6—61no—
48C3×#446—31no—
49#5091—31no—
50#5078—31no—
51#5064—32no—
52#5063—32no—
53#5024D311yescharacteristic, complemented
54#5018—32no—
55C4×#214—31no—
56#5008—31no—
57#5007—32no—
58#4850—31no—
59#4846—32no—
60#4842—31no—
61#4829—32no—
62#4710—32no—
63#4702—32no—
64C4×#204—31no—
65#4682—31no—
66C2×C32×Q8D411yescharacteristic
67C2×#634—22no—
68C2×#634—21no—
69C2×#634C2311yescharacteristic
70C2×C3×D12C2311yescharacteristic
71C2×C3×C4×D3—21no—
72C2×C3×C4×D3—22no—
73C2×C3×Dic6—21no—
74C32×C4⋊C4D411yescharacteristic
75C32×C4⋊C4D412yes—
76C32×C42C2311yescharacteristic
77C32×C42C2312yes—
78C4⋊(C3⋊Dic3)—22no—
79C4⋊(C3⋊Dic3)—21no—
80C4⋊(C3⋊Dic3)C2312yes—
81C4⋊(C3⋊Dic3)C2311yescharacteristic
82#3750—22no—
83#3750—21no—
84C4×C3⋊Dic3—22no—
85C4×C3⋊Dic3—21no—
86C3×(C2×D3)⋊C4—21no—
87C3×(C2×D3)⋊C4—22no—
88C3×C4⋊Dic3—21no—
89C3×C4⋊Dic3C2311yescharacteristic
90C3×C4⋊Dic3C2312yes—
91C3×Dic3⋊C4—22no—
92C3×Dic3⋊C4—22no—
93C2×C3×C3⋊C8C2312yes—
94C2×C3×C3⋊C8C2311yescharacteristic
95C2×C3×C3⋊C8C2311yescharacteristic
96#3716Q814yescomplemented
97#3385—91noSylow
98C3×C4⋊Q8—31no—
99C3×#125—32no—
100C3×C22⋊Q8—61no—
101C3×C22⋊Q8—62no—
102C3×C4×Q8—21no—
103C3×C4×Q8C2×D311yescharacteristic
104C3×C4×Q8C2×D311yescharacteristic
105C3×C4×D4—32no—
106C3×C4×D4—31no—
107C2×C3×C4⋊C4—61no—
108#946—21no—
109Q8×Dic3—62no—
110Q8×Dic3—61no—
111Q8×Dic3—61no—
112C2×#214—31no—
113C2×#204—31no—
114#896—22no—
115D3×C4⋊C4—21no—
116#888—62no—
117#888—64no—
118#888—61no—
119#888—62no—
120#888—61no—
121#888—62no—
122C4×D12C2×D311yescharacteristic
123C4×D12C2×D312yes—
124#870C2×D312yes—
125#869—31no—
126#869—32no—
127#869—32no—
128#869—31no—
129#869—62no—
130#869—61no—
131#869C2×D311yescharacteristic
132C4×Dic6—31no—
133C4×Dic6—32no—
134C4×Dic6—31no—
135C4×Dic6—32no—
136C4×Dic6—61no—
137C4×Dic6—62no—
138C3×C4⋊C8—32no—
139C3×C4⋊C8—31no—
140C3×#97—32no—
141C3×C4×C8—31no—
142#832—31no—
143#832—32no—
144#821—31no—
145#821—32no—
146#818—32no—
147#818—32no—
148#818—32no—
149#817—31no—
150#817—32no—
151#816—31no—
152#816—32no—
153#810—31no—
154#810—32no—
155#808—32no—
156#808—31no—
157#807—32no—
158#807—32no—
159#807—32no—
160C4⋊(C3⋊C8)C2×D312yes—
161C4⋊(C3⋊C8)C2×D311yescharacteristic
162#803C2×D312yes—
163C4×C3⋊C8C2×D311yescharacteristic
164C22×C3×D3—21no—
165C32×Q8—22no—
166C2×C32×C4C2×D412yes—
167C2×C32×C4C2×D412yes—
168C2×C32×C4C2×D411yescharacteristic
169C2×C32×C4C2×D411yescharacteristic
170C2×C32×C4C2411yescharacteristic
171C2×C3⋊Dic3—21no—
172C2×C3⋊Dic3—22no—
173C2×C3⋊Dic3—21no—
174#634—21no—
175#634—81no—
176#634—82no—
177#634C2×Q812yes—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—22no—
180C3×D12—21no—
181C3×D12C2×Q812yes—
182C3×C4×D3—42no—
183C3×C4×D3—82no—
184C3×Dic6—81no—
185C3×C3⋊C8—22no—
186C3×C3⋊C8—21no—
187C3×C3⋊C8C2×Q812yes—
188Q82—91no—
189#554—31no—
190#501—92no—
191#500—91no—
192#479—92no—
193#477—91no—
194#476—91no—
195#474—92no—
196#446—31no—
197#440—92no—
198C4×SD16—91no—
199C2×C3×Q8—21no—
200C2×C3×Q8—61no—
201C2×C3×Q8D1211yescharacteristic
202C2×C3×Q8#6711yescharacteristic
203C2×C3×D4—31no—
204C22×C3×C4—61no—
205C22×C3×C4—62no—
206C2×D12C22×D311yescharacteristic
207C2×C4×D3—21no—
208C2×C4×D3—22no—
209C2×Dic6—21no—
210C2×Dic6—31no—
211C2×Dic6—31no—
212C2×Dic6—61no—
213C2×Dic6—62no—
214C2×Dic6—64no—
215C2×Dic6—61no—
216C2×Dic6—62no—
217C2×Dic6—62no—
218C2×Dic6—61no—
219C2×C3×C8—31no—
220C2×C3×C8—32no—
221C2×C3×C8—31no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—32no—
225C3×C4⋊C4—31no—
226C3×C4⋊C4—62no—
227C3×C4⋊C4—62no—
228C3×C4⋊C4—61no—
229C3×C4⋊C4D1211yescharacteristic
230C3×C4⋊C4D1212yes—
231C3×C4⋊C4#6711yescharacteristic
232C3×C4⋊C4#6712yes—
233C3×C22⋊C4—61no—
234C3×C22⋊C4—62no—
235C3×C42—21no—
236C3×C42—22no—
237C3×C42C22×D311yescharacteristic
238C3×C42C22×D312yes—
239C3×C42C22×D311yescharacteristic
240C3×C42C22×D312yes—
241#214—34no—
242(C2×D3)⋊C4—21no—
243(C2×D3)⋊C4—22no—
244C4⋊Dic3—21no—
245C4⋊Dic3—32no—
246C4⋊Dic3—31no—
247C4⋊Dic3—31no—
248C4⋊Dic3—32no—
249C4⋊Dic3—61no—
250C4⋊Dic3—62no—
251C4⋊Dic3—64no—
252C4⋊Dic3—62no—
253C4⋊Dic3—62no—
254C4⋊Dic3—61no—
255C4⋊Dic3—62no—
256C4⋊Dic3—61no—
257C4⋊Dic3C22×D311yescharacteristic
258C4⋊Dic3C22×D312yes—
259Dic3⋊C4—22no—
260Dic3⋊C4—22no—
261Dic3⋊C4—62no—
262Dic3⋊C4—64no—
263Dic3⋊C4—61no—
264Dic3⋊C4—62no—
265Dic3⋊C4—62no—
266Dic3⋊C4—61no—
267C4×Dic3—64no—
268C4×Dic3—62no—
269C4×Dic3—62no—
270C4×Dic3—61no—
271C4×Dic3—62no—
272C4×Dic3—61no—
273C2×C3⋊C8C22×D311yescharacteristic
274C2×C3⋊C8C22×D312yes—
275C2×C3⋊C8C22×D311yescharacteristic
276#204—34no—
277C22×C32C22×D411yescharacteristic
278C2×C3×D3—22no—
279C2×C3×D3—41no—
280C2×C3×D3—41no—
281C32×C4—22no—
282C32×C4—21no—
283C32×C4—22no—
284C32×C4#13712yes—
285C32×C4C22×Q811yescharacteristic
286C32×C42−1+411yescharacteristic
287C3⋊Dic3—22no—
288C3⋊Dic3—42no—
289C3⋊Dic3—41no—
290C3×Dic3—41no—
291C3×Dic3—42no—
292C2×SD16—91no—
293C4⋊Q8—31no—
294C4⋊Q8—92no—
295C4⋊Q8—91no—
296C4⋊Q8—181no—
297C4⋊Q8—182no—
298#125—32no—
299C22⋊Q8—61no—
300C22⋊Q8—62no—
301C4×Q8—91no—
302C4×Q8—92no—
303C4×Q8—181no—
304C4×Q8D3211yescharacteristic
305C4×D4—31no—
306C4×D4—32no—
307C2×C4⋊C4—61no—
308#107—92no—
309#107—92no—
310#107—92no—
311#106—91no—
312#106—92no—
313C4⋊C8—32no—
314C4⋊C8—31no—
315#103—91no—
316#103—92no—
317#102—91no—
318#102—92no—
319#97—32no—
320C4×C8—31no—
321C23×C3—61no—
322C22×D3—21no—
323C3×Q8—24no—
324C3×Q8—22no—
325C3×Q8—22no—
326C3×Q8—241no—
327C3×D4—32no—
328C3×D4—61no—
329C2×C3×C4—21no—
330C2×C3×C4—22no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—22no—
334C2×C3×C4—61no—
335C2×C3×C4—62no—
336C2×C3×C4—122no—
337C2×C3×C4—242no—
338C2×C3×C4C2×D1212yes—
339C2×C3×C4C2×D1211yescharacteristic
340C2×C3×C4C2×D1212yes—
341C2×C3×C4C2×D1211yescharacteristic
342C2×C3×C4C2×#6712yes—
343C2×C3×C4C2×#6712yes—
344C2×C3×C4C2×#6711yescharacteristic
345C2×C3×C4C2×#6711yescharacteristic
346C2×C3×C4C23×D311yescharacteristic
347C2×C3×C4C23×D311yescharacteristic
348C2×Dic3—21no—
349C2×Dic3—22no—
350C2×Dic3—62no—
351C2×Dic3—61no—
352C2×Dic3—61no—
353C2×Dic3—64no—
354C2×Dic3—62no—
355C2×Dic3—62no—
356C2×Dic3—61no—
357C2×Dic3—62no—
358C2×Dic3—61no—
359D12—21no—
360D12D3×Q812yes—
361C4×D3—42no—
362C4×D3—82no—
363Dic6—32no—
364Dic6—32no—
365Dic6—62no—
366Dic6—62no—
367Dic6—61no—
368Dic6—61no—
369Dic6—81no—
370Dic6—242no—
371Dic6—241no—
372Dic6—241no—
373Dic6—242no—
374Dic6—244no—
375Dic6—242no—
376C3×C8—32no—
377C3×C8—62no—
378C3×C8—61no—
379C3⋊C8—21no—
380C3⋊C8—22no—
381C3⋊C8D3×Q812yes—
382C2×C32D4×Q811yescharacteristic
383C2×C32C2×#13711yescharacteristic
384C2×C32#57611yescharacteristic
385C3×D3—42no—
386C2×Q8—61no—
387C2×Q8—91no—
388C2×Q8—181no—
389C2×Q8—182no—
390C2×Q8#62611yescharacteristic
391C2×D4—31no—
392C22×C4—61no—
393C22×C4—62no—
394SD16—94no—
395C2×C8—31no—
396C2×C8—31no—
397C2×C8—32no—
398C4⋊C4—31no—
399C4⋊C4—32no—
400C4⋊C4—62no—
401C4⋊C4—61no—
402C4⋊C4—62no—
403C4⋊C4—91no—
404C4⋊C4—92no—
405C4⋊C4—181no—
406C4⋊C4—182no—
407C4⋊C4—182no—
408C4⋊C4—181no—
409C4⋊C4#62611yescharacteristic
410C4⋊C4#62612yes—
411C22⋊C4—61no—
412C22⋊C4—62no—
413C42—182no—
414C42—181no—
415C42C2×D3211yescharacteristic
416C42C2×D3212yes—
417C22×C3—21no—
418C22×C3—62no—
419C22×C3—121no—
420C22×C3—121no—
421C22×C3C22×D1211yescharacteristic
422C22×C3C22×#6711yescharacteristic
423C2×D3—22no—
424C2×D3—41no—
425C2×D3—41no—
426C3×C4—24no—
427C3×C4—22no—
428C3×C4—22no—
429C3×C4—21no—
430C3×C4—22no—
431C3×C4—21no—
432C3×C4—22no—
433C3×C4—21no—
434C3×C4—21no—
435C3×C4—22no—
436C3×C4—42no—
437C3×C4—41no—
438C3×C4—121no—
439C3×C4—122no—
440C3×C4#90912yes—
441C3×C4#95112yes—
442C3×C4C2×D3×Q811yescharacteristic
443C3×C4C2×D3×Q811yescharacteristic
444C3×C4#100711yescharacteristic
445C3×C4#101011yescharacteristic
446Dic3—41no—
447Dic3—42no—
448Dic3—64no—
449Dic3—62no—
450Dic3—62no—
451Dic3—122no—
452Dic3—124no—
453Dic3—121no—
454Dic3—122no—
455Dic3—122no—
456Dic3—121no—
457C32#338511yescharacteristic, complemented, Sylow
458C23—61no—
459Q8—22no—
460Q8—92no—
461Q8—181no—
462Q8—241no—
463Q8—722no—
464Q8—721no—
465D4—32no—
466D4—61no—
467C2×C4—61no—
468C2×C4—62no—
469C2×C4—122no—
470C2×C4—181no—
471C2×C4—182no—
472C2×C4—181no—
473C2×C4—242no—
474C2×C4C2×#62612yes—
475C2×C4C2×#62611yescharacteristic
476C2×C4C2×#62612yes—
477C2×C4C2×#62611yescharacteristic
478C2×C4C22×D3211yescharacteristic
479C8—32no—
480C8—61no—
481C8—62no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—122no—
486C2×C3Q8×D1211yescharacteristic
487C2×C3C2×#90911yescharacteristic
488C2×C3#575211yescharacteristic
489C2×C3Q8×#6711yescharacteristic
490C2×C3C2×#95111yescharacteristic
491C2×C3#583511yescharacteristic
492D3—42no—
493C22—62no—
494C22—121no—
495C22—121no—
496C22C22×#62611yescharacteristic
497C4—22no—
498C4—21no—
499C4—22no—
500C4—122no—
501C4—121no—
502C4—182no—
503C4—361no—
504C4—362no—
505C4#6383112yes—
506C4#6431511yescharacteristic
507C4D32×Q811yescharacteristic
508C3—21no—
509C3#7997611yescharacteristic, complemented
510C3#8363111yescharacteristic, complemented
511C2—122no—
512C2C2×#6383111yescharacteristic
513C2Q8×#62611yescharacteristic
514C2#1059544911yescharacteristic
subgroupG/H
centerC22C22×#626
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4×Q8C22
socleC22×C32C22×D4
prime radicalC2×C3×D12C23
hypercenterC4×Q8D32
hypocenterC32#3385
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. Q8×#634C2semidirect
3. C3×#5024C2semidirect
6. #10590311C2semidirect
7. #10590306C2semidirect
8. #10590292C2semidirect
9. #10590284C2semidirect
53. #5024D3semidirect
96. #3716Q8semidirect
457. C32#3385semidirect
509. C3#79976semidirect
510. C3#83631semidirect

Maximal subgroups

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

subgroupindex
Q8×#6342
C3×#55792
C3×#50242
#105903602
C4×#37162
#105903112
#105903062
#105902922
#105902842
#105902722
#105902682
#836313
#799763