Finite Group Databasebeta

← #11909910

Subgroups — #11909910

Order 1152.

2124 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C3×C4×D3×Q8C211yesmaximal, characteristic, complemented
3C3×#5570C211yesmaximal, characteristic, complemented
4Q8×#626C212yesmaximal, complemented
5#10595339C212yesmaximal, complemented
6#10594553C216yesmaximal, complemented
7#10594355C216yesmaximal, complemented
8#10593870C213yesmaximal
9#10593865C213yesmaximal, complemented
10#10593860C213yesmaximal, complemented
11#10593859C213yesmaximal, complemented
12#87024—31nomaximal
13#85804—31nomaximal
14C2×Q8×C3⋊D3C2212yes—
15C2×C3×D3×Q8C2211yescharacteristic
16C32×C4×Q8C2211yescharacteristic
17#64106C2216yes—
18C4×C4⋊(C3⋊D3)C2213yes—
19#64078C2213yes—
20C3×Q8×Dic3C2211yescharacteristic
21C3×Q8×Dic3C2212yes—
22C3×D3×C4⋊C4C2213yes—
23C3×#890C2216yes—
24C3×#887C2213yes—
25C3×C42×D3C2213yes—
26C3×#869C2213yes—
27C3×C4×Dic6C2213yes—
28C3×C4×Dic6C2213yes—
29#63909C2211yescharacteristic
30C4×#626C2216yes—
31#63874C2216yes—
32#63871C2216yes—
33#63869C2216yes—
34#63869C2213yes—
35#63868C2216yes—
36#63866C2216yes—
37#63864C2213yes—
38#63857C2216yes—
39#63856C2213yes—
40#63850C2216yes—
41#63849C2216yes—
42#63848C22112yes—
43#63845C22112yes—
44#63844C2216yes—
45#63844C2216yes—
46Dic3×Dic6C2216yes—
47#63841C2216yes—
48C3×#557—31no—
49C2×C3×C4×Q8—31no—
50Q8×#67—32no—
51#5732—33no—
52#5728—33no—
53#5698—36no—
54#5694—36no—
55#5584—33no—
56#5583—36no—
57#5580—33no—
58Q8×D12—32no—
59Q8×D12—31no—
60Q8×D12—31no—
61Q8×D12—61no—
62#5572—32no—
63#5572—32no—
64C4×D3×Q8D311yescharacteristic, complemented
65#5570—33no—
66#5570D311yescharacteristic, complemented
67#5567—33no—
68#5513—36no—
69#5479—33no—
70#5470—33no—
71C2×C32×Q8C2311yescharacteristic
72Q8×C3⋊D3—28no—
73C2×C4⋊(C3⋊D3)C2311yescharacteristic
74C2×C4×C3⋊D3C2316yes—
75C2×#634C2316yes—
76C3×D3×Q8—24no—
77C2×C3×C4×D3C2313yes—
78C2×C3×C4×D3C2313yes—
79C2×C3×C4×D3C2311yescharacteristic
80C2×C3×Dic6C2313yes—
81C2×C3×Dic6C2313yes—
82C2×#626C2312yes—
83C32×C4⋊C4C2313yes—
84C32×C42C2313yes—
85#3752C2316yes—
86C4⋊(C3⋊Dic3)C2313yes—
87C4⋊(C3⋊Dic3)C2316yes—
88C3×C4⋊Dic3C2316yes—
89C3×C4⋊Dic3C2316yes—
90C3×C4⋊Dic3C2313yes—
91C3×C4⋊Dic3C2311yescharacteristic
92C3×Dic3⋊C4C2316yes—
93C3×Dic3⋊C4C2316yes—
94C3×C4×Dic3C2313yes—
95C3×C4×Dic3C2313yes—
96C3×C4×Dic3C2316yes—
97C3×C4×Dic3C2313yes—
98#3724C2316yes—
99#3724C2316yes—
100Dic3⋊Dic3C23112yes—
101Dic3⋊Dic3C2316yes—
102#3722C2316yes—
103#3722C2316yes—
104#3722C2312yes—
105(C2×D3)⋊Dic3C2316yes—
106Dic32C23112yes—
107#3523—91noSylow
108C22×C3×Q8—31no—
109C2×D3×Q8—32no—
110C2×D3×Q8—32no—
111C2×D3×Q8—62no—
112C2×D3×Q8C2×D311yescharacteristic
113C3×C4⋊Q8—33no—
114C3×#125—36no—
115C3×C4×Q8—21no—
116C3×C4×Q8—31no—
117C3×C4×Q8—32no—
118C3×C4×Q8—33no—
119C3×C4×Q8—33no—
120C3×C4×Q8—33no—
121C3×C4×Q8C2×D311yescharacteristic
122C3×C4×Q8C2×D311yescharacteristic
123C2×C3×C4⋊C4—33no—
124C2×C3×C42—33no—
125#947—33no—
126#946—33no—
127#946—36no—
128Q8×Dic3—36no—
129Q8×Dic3C2×D312yes—
130Q8×Dic3C2×D311yescharacteristic
131#944—36no—
132#930—31no—
133#929—36no—
134C4×#67—36no—
135#926—36no—
136#924—33no—
137#924—36no—
138#898—312no—
139#898—312no—
140#898—36no—
141#897—36no—
142#897—36no—
143#897—36no—
144#897—36no—
145#897—66no—
146#895—31no—
147#894—36no—
148#892—36no—
149#892—36no—
150#892—36no—
151D3×C4⋊C4C2×D313yes—
152#890—312no—
153#890—36no—
154#890—33no—
155#890C2×D316yes—
156#889—312no—
157#887—36no—
158#887—36no—
159#887C2×D313yes—
160#876—36no—
161#875—33no—
162C4×D12—36no—
163C4×D12—33no—
164C4×D12—33no—
165C4×D12—63no—
166#872—36no—
167#872—36no—
168C42×D3C2×D313yes—
169#870—36no—
170#870—33no—
171#869—36no—
172#869—33no—
173#869—33no—
174#869—63no—
175#869C2×D313yes—
176C4×Dic6—36no—
177C4×Dic6—36no—
178C4×Dic6—36no—
179C4×Dic6C2×D313yes—
180C4×Dic6C2×D313yes—
181C22×C3⋊D3C2412yes—
182C22×C3×D3C2411yescharacteristic
183C32×Q8#4114yes—
184C2×C32×C4C2413yes—
185C2×C32×C4C2413yes—
186C2×C32×C4C2411yescharacteristic
187C2×C3⋊Dic3C2416yes—
188C4⋊(C3⋊D3)—22no—
189C4×C3⋊D3—212no—
190#634—212no—
191C2×C3×Dic3C2413yes—
192C2×C3×Dic3C2416yes—
193C2×C3×Dic3C2413yes—
194C2×C3×Dic3C2411yescharacteristic
195C2×C3×Dic3C2412yes—
196C3×C4×D3—26no—
197C3×C4×D3—26no—
198C3×C4×D3—22no—
199C3×Dic6—26no—
200C3×Dic6—26no—
201#626—24no—
202#557—31no—
203#557—93no—
204#556—93no—
205#552—96no—
206D4×Q8—92no—
207D4×Q8—91no—
208#541—96no—
209#533—93no—
210#529—93no—
211#519—92no—
212C2×C4×Q8—31no—
213C2×C3×Q8—21no—
214C2×C3×Q8—33no—
215C2×C3×Q8—33no—
216C2×C3×Q8—64no—
217C2×C3×Q8C22×D311yescharacteristic
218C2×C3×Q8C22×D311yescharacteristic
219C22×C3×C4—31no—
220C22×C3×C4—33no—
221C22×C3×C4—33no—
222C2×#67—32no—
223D3×Q8—24no—
224D3×Q8—68no—
225D3×Q8—68no—
226D3×Q8—128no—
227C2×D12—32no—
228C2×D12—31no—
229C2×D12—31no—
230C2×D12—61no—
231C2×C4×D3—36no—
232C2×C4×D3—36no—
233C2×C4×D3—66no—
234C2×C4×D3C22×D311yescharacteristic
235C2×C4×D3C22×D313yes—
236C2×C4×D3C22×D313yes—
237C2×Dic6—36no—
238C2×Dic6—36no—
239C2×Dic6—66no—
240C2×Dic6C22×D313yes—
241C2×Dic6C22×D313yes—
242C3×C4⋊C4—23no—
243C3×C4⋊C4—33no—
244C3×C4⋊C4—36no—
245C3×C4⋊C4—36no—
246C3×C4⋊C4—31no—
247C3×C4⋊C4—36no—
248C3×C4⋊C4—36no—
249C3×C4⋊C4C22×D313yes—
250C3×C4⋊C4C22×D313yes—
251C3×C42—23no—
252C3×C42—33no—
253C3×C42—33no—
254C3×C42—36no—
255C3×C42—33no—
256C3×C42C22×D313yes—
257C3×C42C22×D313yes—
258#217—36no—
259(C2×D3)⋊C4—36no—
260(C2×D3)⋊C4—36no—
261(C2×D3)⋊C4—36no—
262(C2×D3)⋊C4—32no—
263(C2×D3)⋊C4—32no—
264(C2×D3)⋊C4—36no—
265(C2×D3)⋊C4—36no—
266(C2×D3)⋊C4—36no—
267(C2×D3)⋊C4—36no—
268(C2×D3)⋊C4—66no—
269C4⋊Dic3—33no—
270C4⋊Dic3—33no—
271C4⋊Dic3—36no—
272C4⋊Dic3—36no—
273C4⋊Dic3—36no—
274C4⋊Dic3—312no—
275C4⋊Dic3—63no—
276C4⋊Dic3—66no—
277C4⋊Dic3C22×D313yes—
278C4⋊Dic3C22×D316yes—
279C4⋊Dic3C22×D311yescharacteristic
280C4⋊Dic3C22×D316yes—
281Dic3⋊C4—36no—
282Dic3⋊C4—36no—
283Dic3⋊C4—312no—
284Dic3⋊C4—36no—
285Dic3⋊C4—36no—
286Dic3⋊C4—36no—
287Dic3⋊C4C22×D316yes—
288Dic3⋊C4C22×D316yes—
289C4×Dic3—312no—
290C4×Dic3—312no—
291C4×Dic3C22×D313yes—
292C4×Dic3C22×D316yes—
293C4×Dic3C22×D313yes—
294C4×Dic3C22×D313yes—
295C22×C32C2511yescharacteristic
296C2×C3⋊D3—22no—
297C2×C3⋊D3—22no—
298C2×C3⋊D3—22no—
299C2×C3×D3—21no—
300C2×C3×D3—21no—
301C2×C3×D32−1+412yes—
302C32×C4—23no—
303C32×C4C2×#4116yes—
304C32×C42−1+412yes—
305C3⋊Dic3—26no—
306C3×Dic3—23no—
307C3×Dic3—22no—
308C3×Dic3—26no—
309C3×Dic3—21no—
310C3×Dic3—23no—
311C22×Q8—31no—
312C22×Q8—92no—
313C4⋊Q8—33no—
314C4⋊Q8—96no—
315C4⋊Q8—93no—
316#126—96no—
317#126—912no—
318#125—36no—
319#125—93no—
320#125—96no—
321#125—912no—
322#124—93no—
323#123—96no—
324C22⋊Q8—96no—
325C22⋊Q8—96no—
326C22⋊Q8—93no—
327C22⋊Q8—96no—
328#121—91no—
329C4×Q8—31no—
330C4×Q8—32no—
331C4×Q8—33no—
332C4×Q8—33no—
333C4×Q8—33no—
334C4×Q8—96no—
335C4×Q8—96no—
336C4×Q8—96no—
337C4×Q8D3211yescharacteristic
338C4×D4—96no—
339C4×D4—93no—
340#117—96no—
341#117—96no—
342#117—96no—
343C2×C4⋊C4—33no—
344C2×C42—33no—
345C23×C3—31no—
346C22×D3—32no—
347C22×D3—32no—
348C22×D3—62no—
349C22×D3C23×D311yescharacteristic
350C3×Q8—24no—
351C3×Q8—66no—
352C3×Q8—66no—
353C3×Q8#23514yes—
354C3×Q8#23914yes—
355C2×C3×C4—21no—
356C2×C3×C4—23no—
357C2×C3×C4—23no—
358C2×C3×C4—33no—
359C2×C3×C4—36no—
360C2×C3×C4—31no—
361C2×C3×C4—33no—
362C2×C3×C4—32no—
363C2×C3×C4—66no—
364C2×C3×C4—66no—
365C2×C3×C4—62no—
366C2×C3×C4C23×D313yes—
367C2×C3×C4C23×D313yes—
368C2×C3×C4C23×D313yes—
369C2×C3×C4C23×D313yes—
370C2×C3×C4C23×D311yescharacteristic
371C2×C3×C4C23×D311yescharacteristic
372#67—64no—
373C2×Dic3—36no—
374C2×Dic3—36no—
375C2×Dic3—66no—
376C2×Dic3C23×D313yes—
377C2×Dic3C23×D316yes—
378C2×Dic3C23×D311yescharacteristic
379C2×Dic3C23×D312yes—
380C2×Dic3C23×D313yes—
381D12—64no—
382D12—64no—
383D12—62no—
384D12—62no—
385C4×D3—26no—
386C4×D3—26no—
387C4×D3—22no—
388C4×D3—612no—
389C4×D3—612no—
390C4×D3—1212no—
391Dic6—26no—
392Dic6—26no—
393Dic6—612no—
394Dic6—612no—
395Dic6—1212no—
396C2×C32C22×#4111yescharacteristic
397C2×C32C2×2−1+411yescharacteristic
398C2×C32C2×2−1+411yescharacteristic
399C3⋊D3—42no—
400C3×D3—22no—
401C2×Q8—33no—
402C2×Q8—33no—
403C2×Q8—64no—
404C2×Q8—96no—
405C2×Q8—188no—
406C2×Q8C2×D3211yescharacteristic
407C2×D4—92no—
408C2×D4—91no—
409C22×C4—31no—
410C22×C4—33no—
411C22×C4—33no—
412C22×C4—96no—
413C4⋊C4—33no—
414C4⋊C4—36no—
415C4⋊C4—36no—
416C4⋊C4—36no—
417C4⋊C4—31no—
418C4⋊C4—36no—
419C4⋊C4—96no—
420C4⋊C4—96no—
421C4⋊C4—96no—
422C4⋊C4—912no—
423C4⋊C4—96no—
424C4⋊C4—93no—
425C4⋊C4C2×D3213yes—
426C22⋊C4—92no—
427C22⋊C4—96no—
428C22⋊C4—96no—
429C22⋊C4—96no—
430C22⋊C4—96no—
431C42—33no—
432C42—33no—
433C42—33no—
434C42—36no—
435C42—912no—
436C42C2×D3213yes—
437C22×C3—21no—
438C22×C3—32no—
439C22×C3—61no—
440C22×C3—61no—
441C22×C3C24×D311yescharacteristic
442C22×C3C24×D311yescharacteristic
443C2×D3—21no—
444C2×D3—21no—
445C2×D3—64no—
446C2×D3—62no—
447C2×D3—62no—
448C2×D3—62no—
449C2×D3—62no—
450C2×D3—62no—
451C2×D3—62no—
452C2×D3—122no—
453C2×D3—122no—
454C2×D3#101012yes—
455C3×C4—23no—
456C3×C4—23no—
457C3×C4—26no—
458C3×C4—22no—
459C3×C4—43no—
460C3×C4—63no—
461C3×C4—66no—
462C3×C4—62no—
463C3×C4—61no—
464C3×C4—63no—
465C3×C4C2×#23516yes—
466C3×C4C2×#23916yes—
467C3×C4#100712yes—
468C3×C4#101012yes—
469Dic3—23no—
470Dic3—21no—
471Dic3—26no—
472Dic3—23no—
473Dic3—22no—
474Dic3—66no—
475Dic3—66no—
476Dic3—126no—
477C32#352311yescharacteristic, complemented, Sylow
478C23—31no—
479C23—92no—
480Q8—66no—
481Q8—66no—
482Q8—1812no—
483Q8#379914yes—
484D4—184no—
485D4—182no—
486C2×C4—31no—
487C2×C4—32no—
488C2×C4—33no—
489C2×C4—33no—
490C2×C4—36no—
491C2×C4—66no—
492C2×C4—66no—
493C2×C4—62no—
494C2×C4—96no—
495C2×C4—1812no—
496C2×C4C22×D3213yes—
497C2×C4C22×D3213yes—
498C2×C4C22×D3211yescharacteristic
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—62no—
503C2×C3C22×#23511yescharacteristic
504C2×C3C22×#23911yescharacteristic
505C2×C3C2×#100711yescharacteristic
506C2×C3C2×#101011yescharacteristic
507C2×C3C2×#101011yescharacteristic
508C2×C3D3×2−1+411yescharacteristic
509D3—22no—
510D3—124no—
511D3—122no—
512D3—122no—
513C22—32no—
514C22—61no—
515C22—61no—
516C22—182no—
517C22—182no—
518C22—182no—
519C22C23×D3211yescharacteristic
520C4—23no—
521C4—63no—
522C4—61no—
523C4—66no—
524C4—63no—
525C4—62no—
526C4—186no—
527C4C2×#379916yes—
528C4#6431512yes—
529C3—21no—
530C3#8580411yescharacteristic, complemented
531C3#8702411yescharacteristic, complemented
532C2—62no—
533C2—362no—
534C2C22×#379911yescharacteristic
535C2C2×#6431511yescharacteristic
536C2D3×#101011yescharacteristic
subgroupG/H
center, FrattiniC22C23×D32
commutator, socle, cosocleC22×C32C25
FittingC32×C4×Q8C22
prime radical#63909C22
hypercenterC4×Q8D32
hypocenterC32#3523
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. Q8×C4⋊(C3⋊D3)C2semidirect
2. C3×C4×D3×Q8C2semidirect
3. C3×#5570C2semidirect
4. Q8×#626C2semidirect
5. #10595339C2semidirect
6. #10594553C2semidirect
7. #10594355C2semidirect
9. #10593865C2semidirect
10. #10593860C2semidirect
11. #10593859C2semidirect
64. C4×D3×Q8D3semidirect
66. #5570D3semidirect
477. C32#3523semidirect
530. C3#85804semidirect
531. C3#87024semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊(C3⋊D3)2
C3×C4×D3×Q82
C3×#55702
Q8×#6262
#105953392
#105945532
#105943552
#105938702
#105938652
#105938602
#105938592
#870243
#858043