Finite Group Databasebeta

← #11909852

Subgroups — #11909852

Order 1152.

2976 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#519C211yesmaximal, characteristic, complemented
2Q8×#638C211yesmaximal, characteristic, complemented
3Q8×#638C211yesmaximal, characteristic, complemented
4#10596054C213yesmaximal, complemented
5#10596052C213yesmaximal, complemented
6#10596049C213yesmaximal, complemented
7Q8×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
8#10596043C211yesmaximal, characteristic, complemented
9C4×Q8×C3⋊D3C211yesmaximal, characteristic, complemented
10#10596040C211yesmaximal, characteristic, complemented
11#10596038C213yesmaximal, complemented
12#10596004C213yesmaximal, complemented
13#10596000C213yesmaximal, complemented
14#10595999C213yesmaximal
15#10595989C213yesmaximal, complemented
16#85884—34nomaximal
17C22×C32×Q8C2211yescharacteristic
18C2×Q8×C3⋊D3C2211yescharacteristic
19C2×Q8×C3⋊D3C2211yescharacteristic
20C32×C4×Q8C2213yes—
21C32×C4×Q8C2211yescharacteristic
22C32×C4×Q8C2211yescharacteristic
23C32×C4×Q8C2213yes—
24C32×#117C2213yes—
25C32×#117C2213yes—
26#64155C2213yes—
27#64155C2213yes—
28Q8×C3⋊Dic3C2211yescharacteristic
29Q8×C3⋊Dic3C2211yescharacteristic
30#64153C2213yes—
31#64153C2213yes—
32C4×#638C2213yes—
33C4×#638C2213yes—
34#64133C2213yes—
35#64133C2213yes—
36#64107C2213yes—
37#64106C2213yes—
38#64106C2213yes—
39#64105C2213yes—
40#64103C2213yes—
41#64101C2213yes—
42C4⋊C4×C3⋊D3C2213yes—
43#64099C2213yes—
44#64098C2216yes—
45#64098C2216yes—
46#64098C2213yes—
47#64097C2213yes—
48#64096C2213yes—
49#64096C2213yes—
50#64094C2213yes—
51#64093C2211yescharacteristic
52#64092C2213yes—
53#64088C2213yes—
54#64088C2216yes—
55#64085C2216yes—
56C4×C4⋊(C3⋊D3)C2213yes—
57#64081C2213yes—
58#64081C2216yes—
59C42×C3⋊D3C2213yes—
60#64079C2213yes—
61#64079C2216yes—
62#64078C2213yes—
63C4×#634C2213yes—
64C4×#634C2213yes—
65C4×#634C2213yes—
66C4×#634C2216yes—
67C3×#519D314yescomplemented
68Q8×#67—34no—
69Q8×#67—34no—
70#5583—312no—
71#5581—312no—
72#5578—312no—
73Q8×D12—34no—
74#5572—34no—
75C4×D3×Q8—34no—
76#5569—34no—
77#5567—312no—
78#5533—312no—
79#5529—312no—
80#5528—312no—
81#5518—312no—
82C2×C32×Q8—24no—
83C2×C32×Q8C2313yes—
84C2×C32×Q8C2311yescharacteristic
85C22×C32×C4C2313yes—
86C2×#638C2311yescharacteristic
87C2×#638C2311yescharacteristic
88Q8×C3⋊D3—24no—
89Q8×C3⋊D3—24no—
90C2×C4⋊(C3⋊D3)C2311yescharacteristic
91C2×C4×C3⋊D3C2311yescharacteristic
92C2×C4×C3⋊D3C2313yes—
93C2×C4×C3⋊D3C2313yes—
94C2×C4×C3⋊D3C2313yes—
95C2×#634C2313yes—
96C2×#634C2313yes—
97C2×#634C2313yes—
98C32×C4⋊C4C2313yes—
99C32×C4⋊C4C2313yes—
100C32×C4⋊C4C2313yes—
101C32×C4⋊C4C2313yes—
102C32×C22⋊C4C2311yescharacteristic
103C32×C22⋊C4C2313yes—
104C32×C42C2313yes—
105C32×C42C2313yes—
106C32×C42C2316yes—
107#3757C2313yes—
108#3757C2313yes—
109#3752C2311yescharacteristic
110#3752C2313yes—
111#3752C2313yes—
112#3752C2313yes—
113#3752C2313yes—
114#3752C2313yes—
115C4⋊(C3⋊Dic3)C2316yes—
116C4⋊(C3⋊Dic3)C2313yes—
117C4⋊(C3⋊Dic3)C2313yes—
118C4⋊(C3⋊Dic3)C2313yes—
119C4⋊(C3⋊Dic3)C2313yes—
120#3750C2311yescharacteristic
121#3750C2313yes—
122#3750C2313yes—
123#3750C2313yes—
124#3750C2316yes—
125#3750C2316yes—
126#3750C2313yes—
127#3750C2313yes—
128#3750C2316yes—
129#3750C2316yes—
130C4×C3⋊Dic3C2313yes—
131C4×C3⋊Dic3C2313yes—
132C4×C3⋊Dic3C2316yes—
133C4×C3⋊Dic3C2313yes—
134C4×C3⋊Dic3C2313yes—
135#3523—91noSylow
136C22×C3×Q8C2×D314yes—
137C2×D3×Q8—34no—
138C2×D3×Q8—34no—
139C3×C4×Q8C2×D3112yes—
140C3×C4×Q8C2×D314yes—
141C3×C4×Q8C2×D314yes—
142C3×C4×Q8C2×D3112yes—
143C3×#117C2×D3112yes—
144C3×#117C2×D3112yes—
145#946—312no—
146#946—312no—
147Q8×Dic3—34no—
148Q8×Dic3—34no—
149#944—312no—
150#944—312no—
151C4×#67—312no—
152C4×#67—312no—
153#924—312no—
154#924—312no—
155#898—312no—
156#897—312no—
157#897—312no—
158#896—312no—
159#894—312no—
160#892—312no—
161D3×C4⋊C4—312no—
162#890—312no—
163#889—324no—
164#889—324no—
165#889—312no—
166#888—312no—
167#887—312no—
168#887—312no—
169#885—312no—
170#884—34no—
171#883—312no—
172#879—312no—
173#879—324no—
174#876—324no—
175C4×D12—312no—
176#872—312no—
177#872—324no—
178C42×D3—312no—
179#870—312no—
180#870—324no—
181#869—312no—
182C4×Dic6—312no—
183C4×Dic6—312no—
184C4×Dic6—312no—
185C4×Dic6—324no—
186C23×C32C2411yescharacteristic
187C22×C3⋊D3C2411yescharacteristic
188C22×C3⋊D3C2411yescharacteristic
189C32×Q8—26no—
190C32×Q8#4114yes—
191C2×C32×C4—26no—
192C2×C32×C4C2411yescharacteristic
193C2×C32×C4C2413yes—
194C2×C32×C4C2413yes—
195C2×C32×C4C2411yescharacteristic
196C2×C32×C4C2413yes—
197C2×C32×C4C2413yes—
198#638—22no—
199#638—22no—
200C2×C3⋊Dic3C2411yescharacteristic
201C2×C3⋊Dic3C2413yes—
202C2×C3⋊Dic3C2413yes—
203C2×C3⋊Dic3C2413yes—
204C2×C3⋊Dic3C2413yes—
205C2×C3⋊Dic3C2411yescharacteristic
206C4⋊(C3⋊D3)—22no—
207C4×C3⋊D3—26no—
208C4×C3⋊D3—26no—
209C4×C3⋊D3—26no—
210C4×C3⋊D3—22no—
211#634—26no—
212#634—26no—
213#634—26no—
214#557—93no—
215#557—91no—
216#556—93no—
217#552—93no—
218#552—93no—
219D4×Q8—91no—
220D4×Q8—91no—
221D4×Q8—91no—
222#541—93no—
223#541—93no—
224#533—93no—
225#529—93no—
226#519—91no—
227#519C3⋊D311yescharacteristic, complemented
228C2×C4×Q8—91no—
229C2×C3×Q8—216no—
230C2×C3×Q8C22×D3112yes—
231C2×C3×Q8C22×D314yes—
232C22×C3×C4C22×D3112yes—
233C2×#67—34no—
234C2×#67—34no—
235D3×Q8—616no—
236D3×Q8—616no—
237C2×D12—34no—
238C2×C4×D3—34no—
239C2×C4×D3—312no—
240C2×C4×D3—312no—
241C2×C4×D3—312no—
242C2×Dic6—312no—
243C2×Dic6—312no—
244C2×Dic6—312no—
245C3×C4⋊C4C22×D3112yes—
246C3×C4⋊C4C22×D3112yes—
247C3×C4⋊C4C22×D3112yes—
248C3×C4⋊C4C22×D3112yes—
249C3×C22⋊C4C22×D314yes—
250C3×C22⋊C4C22×D3112yes—
251C3×C42C22×D3112yes—
252C3×C42C22×D3112yes—
253C3×C42C22×D3124yes—
254#217—312no—
255#217—312no—
256(C2×D3)⋊C4—34no—
257(C2×D3)⋊C4—312no—
258(C2×D3)⋊C4—312no—
259(C2×D3)⋊C4—312no—
260(C2×D3)⋊C4—312no—
261(C2×D3)⋊C4—312no—
262C4⋊Dic3—324no—
263C4⋊Dic3—312no—
264C4⋊Dic3—312no—
265C4⋊Dic3—312no—
266C4⋊Dic3—312no—
267Dic3⋊C4—34no—
268Dic3⋊C4—312no—
269Dic3⋊C4—312no—
270Dic3⋊C4—312no—
271Dic3⋊C4—324no—
272Dic3⋊C4—324no—
273Dic3⋊C4—312no—
274Dic3⋊C4—312no—
275Dic3⋊C4—324no—
276Dic3⋊C4—324no—
277C4×Dic3—312no—
278C4×Dic3—312no—
279C4×Dic3—324no—
280C4×Dic3—312no—
281C4×Dic3—312no—
282C22×C32—21no—
283C22×C32—21no—
284C22×C32—21no—
285C22×C32C2511yescharacteristic
286C2×C3⋊D3—21no—
287C2×C3⋊D3—21no—
288C2×C3⋊D3—21no—
289C2×C3⋊D3—21no—
290C2×C3⋊D3—21no—
291C2×C3⋊D32−1+412yes—
292C32×C4—23no—
293C32×C4—23no—
294C32×C4—23no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4C2×#4116yes—
298C3⋊Dic3—23no—
299C3⋊Dic3—23no—
300C3⋊Dic3—23no—
301C3⋊Dic3—23no—
302C3⋊Dic3—21no—
303C3⋊Dic32−1+412yes—
304C22×Q8—91no—
305C22×Q8—91no—
306C22×Q8C2×C3⋊D311yescharacteristic
307C4⋊Q8—93no—
308C4⋊Q8—93no—
309C4⋊Q8—93no—
310C4⋊Q8—93no—
311#126—93no—
312#126—93no—
313#126—96no—
314#126—96no—
315#125—93no—
316#125—96no—
317#125—96no—
318#125—93no—
319#125—96no—
320#125—93no—
321#124—93no—
322#123—93no—
323#123—93no—
324C22⋊Q8—93no—
325C22⋊Q8—93no—
326C22⋊Q8—93no—
327C22⋊Q8—93no—
328C22⋊Q8—93no—
329C22⋊Q8—93no—
330C22⋊Q8—93no—
331#121—91no—
332C4×Q8—91no—
333C4×Q8—91no—
334C4×Q8—93no—
335C4×Q8—93no—
336C4×Q8—93no—
337C4×Q8—93no—
338C4×Q8—93no—
339C4×Q8—96no—
340C4×Q8C2×C3⋊D311yescharacteristic
341C4×Q8C2×C3⋊D313yes—
342C4×Q8C2×C3⋊D311yescharacteristic
343C4×Q8C2×C3⋊D313yes—
344C4×D4—93no—
345C4×D4—93no—
346C4×D4—93no—
347#117—93no—
348#117—93no—
349#117—96no—
350#117C2×C3⋊D313yes—
351#117C2×C3⋊D313yes—
352C2×C4⋊C4—93no—
353C2×C42—93no—
354C23×C3C23×D314yes—
355C22×D3—34no—
356C22×D3—34no—
357C3×Q8—224no—
358C3×Q8#235116yes—
359C2×C3×C4—224no—
360C2×C3×C4C23×D314yes—
361C2×C3×C4C23×D3112yes—
362C2×C3×C4C23×D3112yes—
363C2×C3×C4C23×D314yes—
364C2×C3×C4C23×D3112yes—
365C2×C3×C4C23×D3112yes—
366#67—68no—
367#67—68no—
368C2×Dic3—34no—
369C2×Dic3—34no—
370C2×Dic3—312no—
371C2×Dic3—312no—
372C2×Dic3—312no—
373C2×Dic3—312no—
374D12—68no—
375C4×D3—68no—
376C4×D3—624no—
377C4×D3—624no—
378C4×D3—624no—
379Dic6—624no—
380Dic6—624no—
381Dic6—624no—
382C2×C32—41no—
383C2×C32C22×#4111yescharacteristic
384C2×C32C2×2−1+411yescharacteristic
385C2×C32C2×2−1+411yescharacteristic
386C3⋊D3—22no—
387C3⋊D3—41no—
388C2×Q8—24no—
389C2×Q8—93no—
390C2×Q8—93no—
391C2×Q8—93no—
392C2×Q8—184no—
393C2×Q8—184no—
394C2×Q8C22×C3⋊D313yes—
395C2×Q8C22×C3⋊D311yescharacteristic
396C2×D4—91no—
397C2×D4—91no—
398C2×D4—91no—
399C22×C4—91no—
400C22×C4—93no—
401C22×C4—93no—
402C22×C4—93no—
403C22×C4C22×C3⋊D313yes—
404C4⋊C4—93no—
405C4⋊C4—96no—
406C4⋊C4—93no—
407C4⋊C4—93no—
408C4⋊C4—93no—
409C4⋊C4—96no—
410C4⋊C4—96no—
411C4⋊C4—93no—
412C4⋊C4—93no—
413C4⋊C4—93no—
414C4⋊C4—91no—
415C4⋊C4—96no—
416C4⋊C4—96no—
417C4⋊C4—93no—
418C4⋊C4—93no—
419C4⋊C4C22×C3⋊D313yes—
420C4⋊C4C22×C3⋊D313yes—
421C4⋊C4C22×C3⋊D313yes—
422C4⋊C4C22×C3⋊D313yes—
423C22⋊C4—93no—
424C22⋊C4—93no—
425C22⋊C4—91no—
426C22⋊C4—93no—
427C22⋊C4—93no—
428C22⋊C4—93no—
429C22⋊C4—93no—
430C22⋊C4—93no—
431C22⋊C4C22×C3⋊D311yescharacteristic
432C22⋊C4C22×C3⋊D313yes—
433C42—93no—
434C42—93no—
435C42—93no—
436C42—96no—
437C42—93no—
438C42C22×C3⋊D313yes—
439C42C22×C3⋊D316yes—
440C42C22×C3⋊D313yes—
441C22×C3—24no—
442C22×C3—24no—
443C22×C3—24no—
444C22×C3C24×D314yes—
445C2×D3—38no—
446C2×D3—64no—
447C2×D3—64no—
448C2×D3—64no—
449C2×D3—64no—
450C2×D3—64no—
451C3×C4—212no—
452C3×C4—212no—
453C3×C4—24no—
454C3×C4—212no—
455C3×C4—24no—
456C3×C4C2×#235124yes—
457Dic3—38no—
458Dic3—612no—
459Dic3—612no—
460Dic3—612no—
461Dic3—612no—
462Dic3—64no—
463C32#352311yescharacteristic, complemented, Sylow
464C23—91no—
465C23—91no—
466C23C23×C3⋊D311yescharacteristic
467Q8—26no—
468Q8—186no—
469Q8—186no—
470Q8—186no—
471Q8#382814yes—
472D4—182no—
473D4—182no—
474D4—182no—
475C2×C4—26no—
476C2×C4—91no—
477C2×C4—93no—
478C2×C4—91no—
479C2×C4—93no—
480C2×C4—93no—
481C2×C4—93no—
482C2×C4—182no—
483C2×C4—186no—
484C2×C4—186no—
485C2×C4—186no—
486C2×C4C23×C3⋊D313yes—
487C2×C4C23×C3⋊D311yescharacteristic
488C2×C4C23×C3⋊D313yes—
489C2×C4C23×C3⋊D311yescharacteristic
490C2×C4C23×C3⋊D313yes—
491C2×C4C23×C3⋊D313yes—
492C2×C3—44no—
493C2×C3C22×#23514yes—
494C2×C3D3×2−1+414yes—
495C2×C3D3×2−1+414yes—
496D3—68no—
497D3—124no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—92no—
502C22—181no—
503C22—181no—
504C22—181no—
505C22—181no—
506C22—181no—
507C22C24×C3⋊D311yescharacteristic
508C4—23no—
509C4—21no—
510C4—23no—
511C4—21no—
512C4—23no—
513C4—92no—
514C4—183no—
515C4—183no—
516C4—183no—
517C4—181no—
518C4—183no—
519C4C2×#382816yes—
520C3#8588414yescomplemented
521C2—41no—
522C2—182no—
523C2—361no—
524C2C22×#382811yescharacteristic
525C2C3⋊D3×2−1+411yescharacteristic
526C2C3⋊D3×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×C3⋊D3
commutator, socle, cosocleC22×C32C25
FittingC32×#519C2
prime radical#64093C22
hypercenter#519C3⋊D3
hypocenterC32#3523
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #519

Complements

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

NH ≅ G/Ntype
1. C32×#519C2semidirect
2. Q8×#638C2semidirect
3. Q8×#638C2semidirect
4. #10596054C2semidirect
5. #10596052C2semidirect
6. #10596049C2semidirect
7. Q8×C4⋊(C3⋊D3)C2semidirect
8. #10596043C2semidirect
9. C4×Q8×C3⋊D3C2semidirect
10. #10596040C2semidirect
11. #10596038C2semidirect
12. #10596004C2semidirect
13. #10596000C2semidirect
15. #10595989C2semidirect
67. C3×#519D3semidirect
227. #519C3⋊D3semidirect
463. C32#3523semidirect
520. C3#85884semidirect

Maximal subgroups

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

subgroupindex
C32×#5192
Q8×#6382
#105960542
#105960522
#105960492
Q8×C4⋊(C3⋊D3)2
#105960432
C4×Q8×C3⋊D32
#105960402
#105960382
#105960042
#105960002
#105959992
#105959892
#858843