Finite Group Databasebeta

← #11896071

Subgroups — #11896071

Order 1152.

782 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#634C211yesmaximal, characteristic, complemented
2#10593863C211yesmaximal, characteristic
3C3×#5031C211yesmaximal, characteristic, complemented
4C3×C4×#215C211yesmaximal, characteristic
5#10591612C212yesmaximal
6#10591582C212yesmaximal, complemented
7#10590541C212yesmaximal
8#10590497C212yesmaximal, complemented
9#10590296C211yesmaximal, characteristic, complemented
10#10590287C211yesmaximal, characteristic, complemented
11#10590268C211yesmaximal, characteristic
12#80464—31nomaximal
13#79626—31nomaximal
14C32×C4×Q8C2211yescharacteristic
15Q8×C3⋊Dic3C2212yes—
16#64097—22no—
17#64097C2212yes—
18#64078—21no—
19#64078C2211yescharacteristic
20C4×#634—21no—
21C4×#634C2211yescharacteristic
22C2×C3×#216C2212yes—
23C2×C3×#215C2211yescharacteristic
24C3×C4×D12C2211yescharacteristic
25C3×#869C2211yescharacteristic
26#63863C2212yes—
27#63858—22no—
28#63857—21no—
29#63843C2212yes—
30#63841—21no—
31C3×#835C2211yescharacteristic
32C3×#810C2212yes—
33C3×#809C2211yescharacteristic
34C3×#808C2211yescharacteristic
35C3×C4⋊(C3⋊C8)C2211yescharacteristic
36C3×C4×C3⋊C8C2211yescharacteristic
37#63573C2212yes—
38#63572C2212yes—
39#63566C2212yes—
40#63566C2212yes—
41#63563C2212yes—
42#63555C2212yes—
43#5732—31no—
44Q8×Dic6—31no—
45Q8×Dic6—31no—
46Q8×Dic6—61no—
47#5542—31no—
48C3×#462—31no—
49C3×C4×SD16—31no—
50#5185—32no—
51#5184—32no—
52#5166—32no—
53#5079—31no—
54#5031D311yescharacteristic, complemented
55C4×#215D311yescharacteristic
56#5008—31no—
57#4854—32no—
58#4842—31no—
59#4831—31no—
60#4794—32no—
61#4791—32no—
62#4789—32no—
63#4765—32no—
64#4681—31no—
65#4679—31no—
66C2×C32×Q8C2311yescharacteristic
67C2×#634—21no—
68C2×#634—22no—
69C2×#634D411yescharacteristic
70C2×C3×D12C2311yescharacteristic
71C2×C3×C4×D3—21no—
72C2×C3×Dic6C2312yes—
73C32×C4⋊C4—21no—
74C32×C4⋊C4C2311yescharacteristic
75C32×C42—21no—
76C32×C42C2311yescharacteristic
77C4⋊(C3⋊Dic3)—21no—
78C4⋊(C3⋊Dic3)—22no—
79C4⋊(C3⋊Dic3)D411yescharacteristic
80C4⋊(C3⋊Dic3)C2312yes—
81#3750—22no—
82#3750D412yes—
83C4×C3⋊Dic3—22no—
84C4×C3⋊Dic3C2312yes—
85C3×#216—24no—
86C3×#215—22no—
87C3×(C2×D3)⋊C4—21no—
88C3×C4⋊Dic3—21no—
89C3×C4⋊Dic3—21no—
90C3×C4⋊Dic3C2311yescharacteristic
91C2×C3×C3⋊C8C2311yescharacteristic
92C2×C3×C3⋊C8C2312yes—
93C2×C3×C3⋊C8C2311yescharacteristic
94Dic3⋊Dic3—22no—
95(C2×D3)⋊Dic3—22no—
96(C2×D3)⋊Dic3—22no—
97Dic32—22no—
98#3369—91noSylow
99C2×C3×Q16—32no—
100C2×C3×SD16—31no—
101C3×C4⋊Q8—31no—
102C3×C4×Q8—21no—
103C3×C4×Q8C2×D311yescharacteristic
104C3×C4×Q8C2×D311yescharacteristic
105C3×C4×D4—31no—
106#947—32no—
107Q8×Dic3—32no—
108Q8×Dic3—32no—
109Q8×Dic3—32no—
110Q8×Dic3—62no—
111C2×#216C2×D312yes—
112C2×#215C2×D311yescharacteristic
113#936—32no—
114#926—61no—
115#924—61no—
116#898—62no—
117#897—61no—
118#892—61no—
119#888—32no—
120#888—32no—
121#888—62no—
122#888—62no—
123#888—62no—
124#888—64no—
125#887—32no—
126#879—62no—
127C4×D12C2×D311yescharacteristic
128#869—31no—
129#869—31no—
130#869—61no—
131#869—61no—
132#869—61no—
133#869—62no—
134#869C2×D311yescharacteristic
135C4×Dic6—31no—
136C4×Dic6—31no—
137C4×Dic6—61no—
138C4×Dic6—61no—
139C4×Dic6—61no—
140C4×Dic6—62no—
141C3×#106—31no—
142C3×C4⋊C8—31no—
143C3×#103—31no—
144C3×#103—32no—
145C3×#102—31no—
146C3×C4×C8—31no—
147#835—32no—
148#835—32no—
149#835C2×D311yescharacteristic
150#832—32no—
151#821—32no—
152#817—32no—
153#816—32no—
154#816—32no—
155#815—32no—
156#815—32no—
157Dic3⋊C8—32no—
158Dic3⋊C8—32no—
159#810C2×D312yes—
160#809C2×D311yescharacteristic
161#808—32no—
162#808C2×D311yescharacteristic
163C4⋊(C3⋊C8)C2×D311yescharacteristic
164C4×C3⋊C8C2×D311yescharacteristic
165C22×C3×D3—21no—
166C32×Q8—21no—
167C32×Q8#4112yes—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4C2×D411yescharacteristic
171C2×C32×C4C2×D411yescharacteristic
172C2×C32×C4C2411yescharacteristic
173C2×C3⋊Dic3—22no—
174C2×C3⋊Dic3C2×D412yes—
175C2×C3⋊Dic3C2×D412yes—
176#634—22no—
177#634—82no—
178#634—81no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—22no—
181C3×D12—21no—
182C3×D12—21no—
183C3×C4×D3—42no—
184C3×Dic6—22no—
185C3×Dic6—22no—
186C3×C3⋊C8—21no—
187C3×C3⋊C8—22no—
188C3×C3⋊C8—21no—
189Q82—91no—
190#556—91no—
191#489—92no—
192#487—91no—
193#476—91no—
194#475—92no—
195#464—92no—
196#462—31no—
197#446—91no—
198#441—92no—
199C4×SD16—31no—
200C2×C3×Q8—21no—
201C2×C3×Q8—32no—
202C2×C3×Q8C22×D311yescharacteristic
203C2×C3×Q8C22×D311yescharacteristic
204C2×C3×D4—31no—
205C22×C3×C4—61no—
206C2×D12C22×D311yescharacteristic
207C2×C4×D3—21no—
208C2×Dic6—31no—
209C2×Dic6—31no—
210C2×Dic6—61no—
211C2×Dic6—62no—
212C2×Dic6—62no—
213C2×Dic6—64no—
214C2×Dic6—62no—
215C2×Dic6—61no—
216C2×Dic6—61no—
217C2×Dic6C22×D312yes—
218C3×Q16—64no—
219C3×SD16—62no—
220C2×C3×C8—31no—
221C2×C3×C8—32no—
222C2×C3×C8—31no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—31no—
228C3×C4⋊C4—61no—
229C3×C4⋊C4—61no—
230C3×C4⋊C4C22×D311yescharacteristic
231C3×C4⋊C4C22×D311yescharacteristic
232C3×C22⋊C4—61no—
233C3×C42—22no—
234C3×C42—21no—
235C3×C42—21no—
236C3×C42—21no—
237C3×C42C22×D311yescharacteristic
238C3×C42C22×D311yescharacteristic
239#217—62no—
240#217—62no—
241#216—24no—
242#215—22no—
243(C2×D3)⋊C4—21no—
244(C2×D3)⋊C4—62no—
245(C2×D3)⋊C4—62no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—31no—
249C4⋊Dic3—31no—
250C4⋊Dic3—32no—
251C4⋊Dic3—32no—
252C4⋊Dic3—61no—
253C4⋊Dic3—62no—
254C4⋊Dic3—62no—
255C4⋊Dic3—62no—
256C4⋊Dic3—62no—
257C4⋊Dic3—62no—
258C4⋊Dic3—64no—
259C4⋊Dic3—61no—
260C4⋊Dic3—61no—
261C4⋊Dic3C22×D311yescharacteristic
262Dic3⋊C4—32no—
263Dic3⋊C4—32no—
264Dic3⋊C4—62no—
265Dic3⋊C4—62no—
266Dic3⋊C4—62no—
267Dic3⋊C4—62no—
268Dic3⋊C4—64no—
269C4×Dic3—32no—
270C4×Dic3—32no—
271C4×Dic3—62no—
272C4×Dic3—62no—
273C4×Dic3—62no—
274C4×Dic3—62no—
275C4×Dic3—62no—
276C4×Dic3—64no—
277C2×C3⋊C8C22×D311yescharacteristic
278C2×C3⋊C8C22×D311yescharacteristic
279C2×C3⋊C8C22×D312yes—
280C22×C32C22×D411yescharacteristic
281C2×C3×D3—41no—
282C2×C3×D3—41no—
283C2×C3×D3—41no—
284C32×C4—22no—
285C32×C4—21no—
286C32×C4—41no—
287C32×C4#13712yes—
288C32×C4C2×#4111yescharacteristic
289C32×C42−1+411yescharacteristic
290C3⋊Dic3—22no—
291C3⋊Dic3—22no—
292C3⋊Dic3—42no—
293C3×Dic3—42no—
294C3×Dic3—41no—
295C2×Q16—32no—
296C2×SD16—31no—
297C4⋊Q8—31no—
298C4⋊Q8—92no—
299C4⋊Q8—91no—
300C4⋊Q8—181no—
301C4⋊Q8—182no—
302#126—182no—
303#124—92no—
304C22⋊Q8—181no—
305C4×Q8—91no—
306C4×Q8—92no—
307C4×Q8—92no—
308C4×Q8—181no—
309C4×Q8D3211yescharacteristic
310C4×D4—31no—
311#117—181no—
312#106—31no—
313#106—92no—
314C4⋊C8—31no—
315C4⋊C8—92no—
316#103—31no—
317#103—32no—
318#103—92no—
319#103—92no—
320#102—31no—
321#102—92no—
322#97—92no—
323C4×C8—31no—
324C23×C3—61no—
325C22×D3—21no—
326C3×Q8—22no—
327C3×Q8—21no—
328C3×Q8—21no—
329C3×Q8—22no—
330C3×Q8—62no—
331C3×Q8—62no—
332C3×Q8#23512yes—
333C3×Q8#23712yes—
334C3×D4—61no—
335C3×D4—61no—
336C2×C3×C4—22no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—22no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—61no—
346C2×C3×C4—62no—
347C2×C3×C4—122no—
348C2×C3×C4D3×D411yescharacteristic
349C2×C3×C4D3×D411yescharacteristic
350C2×C3×C4D3×D411yescharacteristic
351C2×C3×C4D3×D411yescharacteristic
352C2×C3×C4C23×D311yescharacteristic
353C2×C3×C4C23×D311yescharacteristic
354C2×Dic3—21no—
355C2×Dic3—22no—
356C2×Dic3—32no—
357C2×Dic3—32no—
358C2×Dic3—32no—
359C2×Dic3—32no—
360C2×Dic3—62no—
361C2×Dic3—62no—
362C2×Dic3—64no—
363C2×Dic3—62no—
364C2×Dic3—62no—
365D12—21no—
366D12—21no—
367C4×D3—42no—
368Dic6—22no—
369Dic6—22no—
370Dic6—64no—
371Dic6—62no—
372Dic6—62no—
373Dic6—242no—
374Dic6—242no—
375Dic6—241no—
376Dic6—241no—
377Dic6—242no—
378Dic6—244no—
379C3×C8—62no—
380C3×C8—61no—
381C3×C8—61no—
382C3⋊C8—21no—
383C3⋊C8—22no—
384C3⋊C8—21no—
385C2×C32#54811yescharacteristic
386C2×C32C2×#13711yescharacteristic
387C2×C32#57711yescharacteristic
388C3×D3—81no—
389C2×Q8—32no—
390C2×Q8—91no—
391C2×Q8—182no—
392C2×Q8—181no—
393C2×Q8C2×D3211yescharacteristic
394C2×D4—31no—
395C22×C4—61no—
396Q16—64no—
397SD16—62no—
398C2×C8—31no—
399C2×C8—31no—
400C2×C8—32no—
401C4⋊C4—21no—
402C4⋊C4—31no—
403C4⋊C4—61no—
404C4⋊C4—61no—
405C4⋊C4—91no—
406C4⋊C4—92no—
407C4⋊C4—92no—
408C4⋊C4—182no—
409C4⋊C4—182no—
410C4⋊C4—182no—
411C4⋊C4—181no—
412C4⋊C4C2×D3211yescharacteristic
413C22⋊C4—61no—
414C22⋊C4—182no—
415C22⋊C4—182no—
416C42—21no—
417C42—92no—
418C42—182no—
419C42—182no—
420C42C2×D3211yescharacteristic
421C22×C3—21no—
422C22×C3—121no—
423C22×C3—121no—
424C22×C3—121no—
425C22×C3C2×D3×D411yescharacteristic
426C22×C3C2×D3×D411yescharacteristic
427C2×D3—41no—
428C2×D3—41no—
429C2×D3—41no—
430C3×C4—24no—
431C3×C4—21no—
432C3×C4—21no—
433C3×C4—22no—
434C3×C4—22no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—22no—
438C3×C4—41no—
439C3×C4—42no—
440C3×C4—41no—
441C3×C4—41no—
442C3×C4—121no—
443C3×C4—122no—
444C3×C4#91512yes—
445C3×C4#91812yes—
446C3×C4C2×#23511yescharacteristic
447C3×C4C2×#23711yescharacteristic
448C3×C4#101011yescharacteristic
449C3×C4#101011yescharacteristic
450Dic3—41no—
451Dic3—42no—
452Dic3—64no—
453Dic3—62no—
454Dic3—62no—
455Dic3—62no—
456Dic3—62no—
457Dic3—122no—
458Dic3—124no—
459Dic3—122no—
460Dic3—122no—
461C32#336911yescharacteristic, complemented, Sylow
462C23—61no—
463Q8—21no—
464Q8—62no—
465Q8—62no—
466Q8—182no—
467Q8—721no—
468Q8—722no—
469Q8#379512yes—
470D4—61no—
471D4—61no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—61no—
475C2×C4—62no—
476C2×C4—92no—
477C2×C4—92no—
478C2×C4—122no—
479C2×C4—182no—
480C2×C4#381111yescharacteristic
481C2×C4#381111yescharacteristic
482C2×C4C22×D3211yescharacteristic
483C8—61no—
484C8—62no—
485C8—61no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—241no—
490C2×C3#555211yescharacteristic
491C2×C3#559811yescharacteristic
492C2×C3C2×#91511yescharacteristic
493C2×C3C2×#91811yescharacteristic
494C2×C3#577011yescharacteristic
495C2×C3#577811yescharacteristic
496D3—81no—
497C22—121no—
498C22—121no—
499C22—121no—
500C22C2×#381111yescharacteristic
501C4—22no—
502C4—21no—
503C4—41no—
504C4—121no—
505C4—122no—
506C4—182no—
507C4—182no—
508C4—362no—
509C4#6395112yes—
510C4C2×#379511yescharacteristic
511C4#6429811yescharacteristic
512C3—21no—
513C3#7962611yescharacteristic, complemented
514C3#8046411yescharacteristic, complemented
515C2—241no—
516C2#1059491811yescharacteristic
517C2C2×#6395111yescharacteristic
518C2#1059547311yescharacteristic
subgroupG/H
centerC22C2×#3811
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4×Q8C22
socleC22×C32C22×D4
prime radicalC2×C3×D12C23
hypercenterC4×Q8D32
hypocenterC32#3369
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. Q8×#634C2semidirect
3. C3×#5031C2semidirect
6. #10591582C2semidirect
8. #10590497C2semidirect
9. #10590296C2semidirect
10. #10590287C2semidirect
54. #5031D3semidirect
461. C32#3369semidirect
513. C3#79626semidirect
514. C3#80464semidirect

Maximal subgroups

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

subgroupindex
Q8×#6342
#105938632
C3×#50312
C3×C4×#2152
#105916122
#105915822
#105905412
#105904972
#105902962
#105902872
#105902682
#804643
#796263