Finite Group Databasebeta

← #11887541

Subgroups — #11887541

Order 1152.

901 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×Q8×D12C211yesmaximal, characteristic, complemented
2#10593896C211yesmaximal, characteristic
3C4×#3755C211yesmaximal, characteristic, complemented
4C3×#5020C211yesmaximal, characteristic, complemented
5#10591584C212yesmaximal, complemented
6D3×#835C212yesmaximal, complemented
7#10590577C212yesmaximal, complemented
8#10590439C212yesmaximal, complemented
9#10590379C211yesmaximal, characteristic, complemented
10#10590329C211yesmaximal, characteristic, complemented
11#10590290C211yesmaximal, characteristic, complemented
12#83550—31nomaximal
13#81022—31nomaximal
14C2×C3×D3×Q8C2212yes—
15C2×#3796—21no—
16C32×C4×Q8C2211yescharacteristic
17C2×#3755C2211yescharacteristic
18C4×C4⋊(C3⋊D3)C2211yescharacteristic
19C3×#897—22no—
20C3×#897C2212yes—
21C3×C4×D12—21no—
22C3×C4×D12C2211yescharacteristic
23C3×#869—21no—
24C3×#869C2211yescharacteristic
25C3×#869C2211yescharacteristic
26#63909C2212yes—
27#63890—22no—
28D3×C4⋊Dic3C2212yes—
29#63871—21no—
30C2×#3717C2212yes—
31C2×D3×C3⋊C8C2212yes—
32#63662C2211yescharacteristic
33#63636C2211yescharacteristic
34#63635C2211yescharacteristic
35C4×C3⋊(C3⋊C8)C2211yescharacteristic
36C3×#835C2212yes—
37C3×#808C2212yes—
38C3×C4⋊(C3⋊C8)C2211yescharacteristic
39#63566C2212yes—
40#63565C2212yes—
41#63565C2212yes—
42(C2×D3)⋊(C3⋊C8)C2212yes—
43C3×D4×Q8—31no—
44#5726—31no—
45Q8×D12D311yescharacteristic, complemented
46#5555—31no—
47C3×#475—31no—
48#5227—32no—
49C2×#835—32no—
50#5093—31no—
51#5083—31no—
52#5079—31no—
53#5048—32no—
54#5043—32no—
55C4×#215—31no—
56C4×#215—31no—
57C4×#215—61no—
58#5020D311yescharacteristic, complemented
59D4×C3⋊C8—31no—
60#4864—32no—
61#4862—32no—
62#4841—31no—
63#4834—31no—
64#4806—32no—
65D3×#103—32no—
66C2×C32×Q8C2311yescharacteristic
67C2×C4⋊(C3⋊D3)C2311yescharacteristic
68C2×C4×C3⋊D3—21no—
69C3×D3×Q8—24no—
70C3×D3×Q8—42no—
71C2×C3×D12D411yescharacteristic
72C2×C3×C4×D3—22no—
73C2×C3×C4×D3C2312yes—
74C2×C3×Dic6—21no—
75C2×C3×Dic6—22no—
76C2×C3×Dic6C2312yes—
77C2×#626—22no—
78C2×D3×Dic3—22no—
79#3796—44no—
80C32×C4⋊C4—21no—
81C32×C4⋊C4C2311yescharacteristic
82C32×C42—21no—
83C32×C42C2311yescharacteristic
84#3755—22no—
85#3752—21no—
86C4⋊(C3⋊Dic3)C2311yescharacteristic
87C2×C3⋊(C3⋊C8)C2311yescharacteristic
88C2×C3⋊(C3⋊C8)C2311yescharacteristic
89C3×(C2×D3)⋊C4—22no—
90C3×(C2×D3)⋊C4D412yes—
91C3×C4⋊Dic3—21no—
92C3×C4⋊Dic3—21no—
93C3×C4⋊Dic3—21no—
94C3×C4⋊Dic3—22no—
95C3×C4⋊Dic3D411yescharacteristic
96C3×C4⋊Dic3C2312yes—
97C2×C3×C3⋊C8C2312yes—
98Dic3⋊Dic3—22no—
99#3722—22no—
100#3717—42no—
101D3×C3⋊C8—42no—
102#3298—91noSylow
103C22×C3×Q8—32no—
104C2×#239—61no—
105C2×D3×Q8C2×D312yes—
106C2×#237—61no—
107C3×C4⋊Q8—31no—
108C3×C4⋊Q8—31no—
109C3×C4⋊Q8—61no—
110C3×C22⋊Q8—32no—
111C3×C22⋊Q8—62no—
112C3×C4×Q8—21no—
113C3×C4×Q8C2×D311yescharacteristic
114C3×C4×Q8C2×D311yescharacteristic
115C3×C4×D4—31no—
116C3×C4×D4—61no—
117#946—61no—
118C2×#215—31no—
119C2×#215—31no—
120C2×#215—32no—
121C2×#215—61no—
122#930—32no—
123C2×C4⋊Dic3—32no—
124C22×C3⋊C8—32no—
125C2×#204—32no—
126C2×D3×C8—32no—
127#897—22no—
128#897—61no—
129#897C2×D312yes—
130#895—32no—
131#894—62no—
132D3×C4⋊C4—32no—
133#886—62no—
134C4×D12—21no—
135C4×D12—31no—
136C4×D12—31no—
137C4×D12—61no—
138C4×D12C2×D311yescharacteristic
139#869—21no—
140#869C2×D311yescharacteristic
141#869C2×D311yescharacteristic
142C3×#106—32no—
143C3×C4⋊C8—31no—
144C3×#103—32no—
145#835—31no—
146#835—31no—
147#835—32no—
148#835—32no—
149#835—32no—
150#835—61no—
151#835C2×D312yes—
152#830—32no—
153(C2×D3)⋊C8—32no—
154#816—32no—
155#810—32no—
156#810—32no—
157#809—31no—
158#809—31no—
159#809—61no—
160#808—31no—
161#808—31no—
162#808—61no—
163#808C2×D312yes—
164C4⋊(C3⋊C8)C2×D311yescharacteristic
165C4×C3⋊C8—31no—
166C4×C3⋊C8—31no—
167C4×C3⋊C8—61no—
168C22×C3⋊D3—21no—
169C22×C3×D3C2×D412yes—
170C32×Q8—21no—
171C32×Q8#4112yescomplemented
172C2×C32×C4—21no—
173C2×C32×C4—21no—
174C2×C32×C4C2×D411yescharacteristic
175C2×C32×C4C2×D411yescharacteristic
176C2×C32×C4C2411yescharacteristic
177C2×C3⋊Dic3—21no—
178C4⋊(C3⋊D3)—21no—
179C4⋊(C3⋊D3)—21no—
180C4×C3⋊D3—42no—
181C2×C3×Dic3—22no—
182C2×C3×Dic3—22no—
183C2×C3×Dic3C2×D412yes—
184C3×D12SD1614yescomplemented
185C3×C4×D3—22no—
186C3×C4×D3—22no—
187C3×C4×D3—44no—
188C3×Dic6—22no—
189C3×Dic6—22no—
190C3×Dic6—44no—
191C3×Dic6—42no—
192#626—44no—
193D3×Dic3—44no—
194C3⋊(C3⋊C8)—21no—
195C3⋊(C3⋊C8)—21no—
196C3×C3⋊C8—22no—
197D4×Q8—31no—
198#547—91no—
199#487—91no—
200#483—92no—
201#475—31no—
202#465—92no—
203#461—91no—
204#448—92no—
205C4×SD16—91no—
206C8×D4—91no—
207C2×#103—92no—
208C2×C3×Q8—21no—
209C2×C3×Q8—32no—
210C2×C3×Q8—62no—
211C2×C3×Q8—61no—
212C2×C3×Q8—64no—
213C2×C3×Q8—122no—
214C2×C3×Q8C22×D311yescharacteristic
215C2×C3×Q8C22×D311yescharacteristic
216C2×C3×D4—31no—
217C22×C3×C4—32no—
218C22×C3×C4—62no—
219C2×#67—62no—
220C22×Dic3—62no—
221#239—124no—
222D3×Q8—24no—
223D3×Q8—42no—
224#237—124no—
225C2×D12—31no—
226C2×D12—31no—
227C2×D12—61no—
228C2×D12—62no—
229C2×D12#6711yescharacteristic
230C2×C4×D3—22no—
231C2×C4×D3—62no—
232C2×C4×D3—61no—
233C2×C4×D3—61no—
234C2×C4×D3—121no—
235C2×C4×D3C22×D312yes—
236C2×Dic6—21no—
237C2×Dic6—22no—
238C2×Dic6C22×D312yes—
239C2×C3×C8—32no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—31no—
244C3×C4⋊C4—32no—
245C3×C4⋊C4—41no—
246C3×C4⋊C4—62no—
247C3×C4⋊C4—61no—
248C3×C4⋊C4—61no—
249C3×C4⋊C4—61no—
250C3×C4⋊C4C22×D311yescharacteristic
251C3×C4⋊C4C22×D311yescharacteristic
252C3×C22⋊C4—32no—
253C3×C22⋊C4—62no—
254C3×C42—21no—
255C3×C42—21no—
256C3×C42—21no—
257C3×C42—41no—
258C3×C42C22×D311yescharacteristic
259C3×C42C22×D311yescharacteristic
260#215—64no—
261#215—62no—
262#215—62no—
263#215—122no—
264(C2×D3)⋊C4—22no—
265(C2×D3)⋊C4—62no—
266(C2×D3)⋊C4—61no—
267(C2×D3)⋊C4—61no—
268(C2×D3)⋊C4—62no—
269(C2×D3)⋊C4—121no—
270(C2×D3)⋊C4#6712yes—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—22no—
275C4⋊Dic3—31no—
276C4⋊Dic3—31no—
277C4⋊Dic3—61no—
278C4⋊Dic3—62no—
279C4⋊Dic3#6711yescharacteristic
280C4⋊Dic3C22×D312yes—
281Dic3⋊C4—62no—
282C2×C3⋊C8—31no—
283C2×C3⋊C8—31no—
284C2×C3⋊C8—31no—
285C2×C3⋊C8—31no—
286C2×C3⋊C8—61no—
287C2×C3⋊C8—61no—
288C2×C3⋊C8—122no—
289C2×C3⋊C8C22×D312yes—
290#204—122no—
291D3×C8—122no—
292C22×C32C22×D411yescharacteristic
293C2×C3⋊D3—41no—
294C2×C3⋊D3—41no—
295C2×C3⋊D3—41no—
296C2×C3×D3—22no—
297C2×C3×D3—22no—
298C2×C3×D3C2×SD1614yes—
299C32×C4—22no—
300C32×C4—21no—
301C32×C4—41no—
302C32×C4C2×SD1612yes—
303C32×C4C2×#4111yescharacteristic
304C32×C4H(5,𝔽2)11yescharacteristic
305C3⋊Dic3—41no—
306C3×Dic3—22no—
307C3×Dic3—42no—
308C3×Dic3—42no—
309C2×#41—181no—
310C22×Q8—32no—
311C2×SD16—92no—
312C2×SD16—91no—
313C22×C8—92no—
314C4⋊Q8—31no—
315C4⋊Q8—31no—
316C4⋊Q8—61no—
317#123—182no—
318C22⋊Q8—32no—
319C22⋊Q8—62no—
320C22⋊Q8—181no—
321#121—92no—
322C4×Q8D3211yescharacteristic
323C4×D4—31no—
324C4×D4—61no—
325C4×D4—91no—
326C2×C4⋊C4—92no—
327#106—32no—
328#106—91no—
329C4⋊C8—31no—
330#103—32no—
331#103—92no—
332#103—92no—
333#103—92no—
334#103—91no—
335#102—91no—
336C22⋊C8—92no—
337C4×C8—91no—
338C23×C3—32no—
339C22×D3—61no—
340C22×D3—61no—
341C22×D3—121no—
342C22×D3C2×#6712yes—
343C3×Q8—21no—
344C3×Q8—21no—
345C3×Q8—22no—
346C3×Q8—41no—
347C3×Q8—62no—
348C3×Q8—62no—
349C3×Q8—124no—
350C3×Q8—122no—
351C3×Q8#23712yescomplemented
352C3×Q8#23912yescomplemented
353C3×D4—34no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—32no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—62no—
365C2×C3×C4—62no—
366C2×C3×C4—62no—
367C2×C3×C4—62no—
368C2×C3×C4—124no—
369C2×C3×C4D3×D411yescharacteristic
370C2×C3×C4D3×D411yescharacteristic
371C2×C3×C4C2×#6711yescharacteristic
372C2×C3×C4C2×#6711yescharacteristic
373C2×C3×C4C23×D311yescharacteristic
374C2×C3×C4C23×D311yescharacteristic
375#67—124no—
376C2×Dic3—22no—
377C2×Dic3—22no—
378C2×Dic3—61no—
379C2×Dic3—61no—
380C2×Dic3—121no—
381C2×Dic3—124no—
382C2×Dic3C2×#6712yes—
383D12—62no—
384D12—61no—
385D12—61no—
386D12—61no—
387D12—61no—
388D12—121no—
389D12—124no—
390D12#21514yescomplemented
391C4×D3—22no—
392C4×D3—22no—
393C4×D3—44no—
394C4×D3—124no—
395C4×D3—122no—
396C4×D3—122no—
397C4×D3—242no—
398Dic6—22no—
399Dic6—22no—
400Dic6—44no—
401Dic6—42no—
402C3×C8—62no—
403C3⋊C8—22no—
404C3⋊C8—62no—
405C3⋊C8—61no—
406C3⋊C8—61no—
407C3⋊C8—61no—
408C3⋊C8—61no—
409C3⋊C8—121no—
410C2×C32#54611yescharacteristic
411C2×C32C22×SD1611yescharacteristic
412C2×C32#57811yescharacteristic
413C3⋊D3—81no—
414C3×D3—24no—
415#41—364no—
416C2×Q8—32no—
417C2×Q8—62no—
418C2×Q8—61no—
419C2×Q8—64no—
420C2×Q8—122no—
421C2×Q8C2×D3211yescharacteristic
422C2×D4—31no—
423C2×D4—91no—
424C2×D4—182no—
425C22×C4—32no—
426C22×C4—62no—
427C22×C4—182no—
428C22×C4—181no—
429SD16—182no—
430SD16—362no—
431C2×C8—32no—
432C2×C8—91no—
433C2×C8—91no—
434C2×C8—362no—
435C4⋊C4—21no—
436C4⋊C4—31no—
437C4⋊C4—32no—
438C4⋊C4—62no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—91no—
443C4⋊C4—182no—
444C4⋊C4C2×D3211yescharacteristic
445C22⋊C4—32no—
446C22⋊C4—62no—
447C22⋊C4—182no—
448C22⋊C4—181no—
449C42—21no—
450C42C2×D3211yescharacteristic
451C22×C3—21no—
452C22×C3—34no—
453C22×C3—62no—
454C22×C3—62no—
455C22×C3C2×D3×D411yescharacteristic
456C22×C3C22×#6711yescharacteristic
457C2×D3—22no—
458C2×D3—22no—
459C2×D3—122no—
460C2×D3—121no—
461C2×D3—121no—
462C2×D3—121no—
463C2×D3—121no—
464C2×D3—121no—
465C2×D3—121no—
466C2×D3—241no—
467C2×D3—241no—
468C2×D3C2×#21514yes—
469C3×C4—21no—
470C3×C4—22no—
471C3×C4—21no—
472C3×C4—22no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—22no—
476C3×C4—41no—
477C3×C4—42no—
478C3×C4—41no—
479C3×C4—41no—
480C3×C4—62no—
481C3×C4—81no—
482C3×C4—122no—
483C3×C4—122no—
484C3×C4D3×SD1612yes—
485C3×C4C2×#21512yes—
486C3×C4C2×#23711yescharacteristic
487C3×C4#100411yescharacteristic
488C3×C4C2×#23911yescharacteristic
489C3×C4#100911yescharacteristic
490Dic3—22no—
491Dic3—42no—
492Dic3—42no—
493Dic3—121no—
494Dic3—121no—
495Dic3—241no—
496C32#329811yescharacteristic, complemented, Sylow
497C23—32no—
498C23—181no—
499Q8—21no—
500Q8—62no—
501Q8—62no—
502Q8—124no—
503Q8—122no—
504Q8#379612yescomplemented
505D4—34no—
506D4—181no—
507D4—181no—
508D4—364no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—32no—
512C2×C4—62no—
513C2×C4—62no—
514C2×C4—62no—
515C2×C4—62no—
516C2×C4—124no—
517C2×C4—181no—
518C2×C4—364no—
519C2×C4—362no—
520C2×C4D3×#6711yescharacteristic
521C2×C4D3×#6711yescharacteristic
522C2×C4C22×D3211yescharacteristic
523C8—62no—
524C8—181no—
525C8—181no—
526C2×C3—21no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—64no—
530C2×C3#563011yescharacteristic
531C2×C3C2×D3×SD1611yescharacteristic
532C2×C3#577911yescharacteristic
533C2×C3#580611yescharacteristic
534C2×C3C22×#21511yescharacteristic
535C2×C3#583811yescharacteristic
536D3—24no—
537D3—242no—
538D3—241no—
539D3—241no—
540C22—34no—
541C22—62no—
542C22—62no—
543C22—361no—
544C22—361no—
545C22—361no—
546C22C2×D3×#6711yescharacteristic
547C4—22no—
548C4—21no—
549C4—41no—
550C4—62no—
551C4—122no—
552C4—122no—
553C4—361no—
554C4D3×#21512yes—
555C4C2×#379611yescharacteristic
556C4#6430711yescharacteristic
557C3—21no—
558C3#8102211yescharacteristic, complemented
559C3#8355011yescharacteristic, complemented
560C2—64no—
561C2—721no—
562C2#1059488911yescharacteristic
563C2C2×D3×#21511yescharacteristic
564C2#1059546511yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4×Q8C22
socleC22×C32C22×D4
prime radical#10593896C2
hypercenterC4×Q8D32
hypocenterC32#3298
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×Q8×D12C2semidirect
3. C4×#3755C2semidirect
4. C3×#5020C2semidirect
5. #10591584C2semidirect
6. D3×#835C2semidirect
7. #10590577C2semidirect
8. #10590439C2semidirect
9. #10590379C2semidirect
10. #10590329C2semidirect
11. #10590290C2semidirect
45. Q8×D12D3semidirect
58. #5020D3semidirect
171. C32×Q8#41semidirect
184. C3×D12SD16semidirect
351. C3×Q8#237semidirect
352. C3×Q8#239semidirect
390. D12#215semidirect
496. C32#3298semidirect
504. Q8#3796semidirect
558. C3#81022semidirect
559. C3#83550semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×D122
#105938962
C4×#37552
C3×#50202
#105915842
D3×#8352
#105905772
#105904392
#105903792
#105903292
#105902902
#835503
#810223