Finite Group Databasebeta

← #11864209

Subgroups — #11864209

Order 1152.

1953 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 1953 conjugacy classes of subgroups — too many to draw. Only its 192 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#3832C211yesmaximal, characteristic
2C2×#63945C212yesmaximal, complemented
3#10594925C211yesmaximal, characteristic, complemented
4C3×#5228C212yesmaximal, complemented
5#10591602C214yesmaximal, complemented
6#10591554C212yesmaximal, complemented
7#10591183C212yesmaximal, complemented
8#10591159C211yesmaximal, characteristic, complemented
9#80184—32nomaximal
10C22×C32×Q8C2211yescharacteristic
11C2×#3832—28no—
12C2×#3832—21no—
13C2×#3832—21no—
14C2×#3832C2212yes—
15C2×#3832C2212yes—
16C22×C4⋊(C3⋊D3)—21no—
17C22×C4⋊(C3⋊D3)C2211yescharacteristic
18C22×C4×C3⋊D3—21no—
19C22×C4×C3⋊D3C2211yescharacteristic
20C2×#3797C2211yescharacteristic, complemented
21C2×C3×#216C2214yescomplemented
22C3×#924C2212yes—
23#63982—21no—
24#63974—21no—
25#63945—28no—
26#63900C2211yescharacteristic, complemented
27#63883—21no—
28#63871C2212yescomplemented
29#63865—21no—
30C2×#3717C2214yescomplemented
31C2×#3712C2212yescomplemented
32C3×#835C2214yescomplemented
33C3×#830C2212yes—
34#63564C2214yescomplemented
35#63557C2212yescomplemented
36C22×#239—32no—
37C22×#239—61no—
38C2×#918—34no—
39D3×C22⋊Q8—32no—
40C3×#451—32no—
41#5228D312yescomplemented
42#5187—34no—
43#5046—32no—
44#4808—34no—
45#4803—34no—
46#4731—32no—
47#4726—32no—
48C23×C3⋊D3—21no—
49C23×C3⋊D3D411yescharacteristic
50C2×C32×Q8—24no—
51C2×C32×Q8—21no—
52C2×C32×Q8C2312yes—
53C22×C32×C4—21no—
54C22×C32×C4C2311yescharacteristic
55C22×C3⋊Dic3D411yescharacteristic
56#3832—28no—
57#3832—28no—
58#3832—24no—
59#3832—24no—
60#3832—24no—
61#3832D418yes—
62C2×C4⋊(C3⋊D3)—28no—
63C2×C4⋊(C3⋊D3)—22no—
64C2×C4⋊(C3⋊D3)—22no—
65C2×C4⋊(C3⋊D3)—21no—
66C2×C4⋊(C3⋊D3)—24no—
67C2×C4⋊(C3⋊D3)C2312yes—
68C2×C4×C3⋊D3—28no—
69C2×C4×C3⋊D3—22no—
70C2×C4×C3⋊D3—22no—
71C2×C4×C3⋊D3—24no—
72C2×C4×C3⋊D3D411yescharacteristic
73C2×C4×C3⋊D3D411yescharacteristic
74C2×C4×C3⋊D3C2311yescharacteristic
75C2×C4×C3⋊D3C2311yescharacteristic
76C2×C3×Dic6C2312yes—
77C2×#627—21no—
78C2×#624—21no—
79C2×#624—21no—
80#3797—24no—
81C3×#217—22no—
82C3×#216—28no—
83C3×C4⋊Dic3C2312yes—
84C3×Dic3⋊C4—22no—
85C2×C3×C3⋊C8C2314yes—
86#3724—21no—
87Dic3⋊Dic3—22no—
88#3722—22no—
89#3717—28no—
90#3712—24no—
91#3007—91noSylow
92C22×C3×Q8—21no—
93C22×C3×Q8C2×D312yes—
94C2×#239—34no—
95C2×#239—34no—
96C2×#239—62no—
97C2×#239—62no—
98C2×#239—62no—
99C2×#239—62no—
100C2×#239—616no—
101C2×#239—62no—
102C2×#239—62no—
103C2×#239—616no—
104C2×D3×Q8—32no—
105C22×D12—32no—
106C22×D12—61no—
107C22×D12—62no—
108C22×D12—62no—
109C22×C4×D3—32no—
110C22×C4×D3—61no—
111C22×C4×D3—62no—
112C22×C4×D3—62no—
113C2×C3×Q16—34no—
114C3×C22⋊Q8—32no—
115#946—32no—
116C2×#216C2×D314yescomplemented
117C2×#215—34no—
118#924C2×D312yes—
119#918—616no—
120C2×#204—34no—
121C2×ΓL(1,𝔽25)—34no—
122#897—32no—
123#896—62no—
124D3×C4⋊C4—32no—
125D3×C4⋊C4—62no—
126D3×C22⋊C4—62no—
127#878—62no—
128C3×#103—34no—
129C3×C22⋊C8—32no—
130#835C2×D314yescomplemented
131#830C2×D312yes—
132#821—34no—
133(C2×D3)⋊C8—34no—
134#809—34no—
135C23×C32C2×D411yescharacteristic
136C22×C3⋊D3—24no—
137C22×C3⋊D3—22no—
138C22×C3⋊D3—22no—
139C22×C3⋊D3—22no—
140C22×C3⋊D3—41no—
141C22×C3⋊D3—42no—
142C22×C3⋊D3—42no—
143C22×C3⋊D3C2×D411yescharacteristic
144C22×C3⋊D3C2×D411yescharacteristic
145C32×Q8—24no—
146C32×Q8—22no—
147C32×Q8C2×D414yes—
148C2×C32×C4—24no—
149C2×C32×C4—22no—
150C2×C32×C4—22no—
151C2×C32×C4C2×D411yescharacteristic
152C2×C32×C4C2411yescharacteristic
153C2×C3⋊Dic3—22no—
154C2×C3⋊Dic3C2×D411yescharacteristic
155C2×C3⋊Dic3C2×D411yescharacteristic
156C4⋊(C3⋊D3)—28no—
157C4⋊(C3⋊D3)—24no—
158C4⋊(C3⋊D3)—28no—
159C4⋊(C3⋊D3)—22no—
160C4⋊(C3⋊D3)C2×D414yes—
161C4×C3⋊D3—28no—
162C4×C3⋊D3—22no—
163C4×C3⋊D3—22no—
164C4×C3⋊D3—28no—
165C4×C3⋊D3—22no—
166C4×C3⋊D3C2×D414yes—
167C2×C3×Dic3—22no—
168C2×C3×Dic3—22no—
169C3×Dic6—24no—
170#627—42no—
171#624—42no—
172#624—42no—
173C3×C3⋊C8—24no—
174C22×#41—91no—
175C2×#137—92no—
176C2×C22⋊Q8—91no—
177#452—94no—
178#451—32no—
179#450—92no—
180#417—92no—
181#407—91no—
182C23×D3—32no—
183C23×D3—61no—
184C23×D3—62no—
185C23×D3—62no—
186C2×C3×Q8—28no—
187C2×C3×Q8—22no—
188C2×C3×Q8—28no—
189C2×C3×Q8—22no—
190C2×C3×Q8—22no—
191C2×C3×Q8—32no—
192C2×C3×Q8C22×D314yes—
193C22×C3×C4—22no—
194C22×C3×C4—22no—
195C22×C3×C4—21no—
196C22×C3×C4C22×D312yes—
197C22×Dic3—32no—
198C22×Dic3—61no—
199#239—316no—
200#239—68no—
201#239—68no—
202#239—68no—
203#239—68no—
204#239—68no—
205#239—616no—
206#239—616no—
207#239—616no—
208#239—616no—
209#239—68no—
210#239—68no—
211D3×Q8—68no—
212C2×D12—34no—
213C2×D12—62no—
214C2×D12—68no—
215C2×D12—64no—
216C2×D12—64no—
217C2×D12—68no—
218C2×D12—64no—
219C2×D12—64no—
220C2×D12—616no—
221C2×D12—616no—
222C2×D12—62no—
223C2×D12—62no—
224C2×C4×D3—32no—
225C2×C4×D3—32no—
226C2×C4×D3—32no—
227C2×C4×D3—32no—
228C2×C4×D3—61no—
229C2×C4×D3—61no—
230C2×C4×D3—61no—
231C2×C4×D3—61no—
232C2×C4×D3—68no—
233C2×C4×D3—64no—
234C2×C4×D3—64no—
235C2×C4×D3—68no—
236C2×C4×D3—64no—
237C2×C4×D3—64no—
238C2×C4×D3—616no—
239C2×C4×D3—616no—
240C2×C4×D3—62no—
241C2×C4×D3—62no—
242C2×Dic6—62no—
243C2×Dic6C22×D312yes—
244C3×Q16—68no—
245C2×C3×C8—34no—
246C3×C4⋊C4—32no—
247C3×C4⋊C4—62no—
248C3×C22⋊C4—62no—
249#217—22no—
250#216—28no—
251#215—68no—
252(C2×D3)⋊C4—62no—
253(C2×D3)⋊C4—62no—
254C4⋊Dic3—62no—
255C4⋊Dic3C22×D312yes—
256Dic3⋊C4—22no—
257Dic3⋊C4—62no—
258Dic3⋊C4—62no—
259C2×C3⋊C8C22×D314yes—
260#204—68no—
261ΓL(1,𝔽25)—68no—
262C22×C32—21no—
263C22×C32—21no—
264C22×C32#13712yescomplemented
265C22×C32C22×D411yescharacteristic
266C2×C3⋊D3—21no—
267C2×C3⋊D3—24no—
268C2×C3⋊D3—21no—
269C2×C3⋊D3—21no—
270C2×C3⋊D3—21no—
271C2×C3⋊D3—21no—
272C2×C3⋊D3—44no—
273C2×C3⋊D3—48no—
274C2×C3⋊D3—42no—
275C2×C3⋊D3—42no—
276C2×C3⋊D3#12012yes—
277C32×C4—24no—
278C32×C4—21no—
279C32×C4C22×D412yes—
280C3⋊Dic3—21no—
281C3⋊Dic3#12012yes—
282C3×Dic3—42no—
283C3×Dic3—42no—
284C2×#41—92no—
285C2×#41—92no—
286C2×#41—181no—
287C2×#41—181no—
288C2×#41—188no—
289C22×Q8—91no—
290C22×Q8D3211yescharacteristic
291C22×D4—91no—
292C22×D4—181no—
293C23×C4—91no—
294C23×C4—181no—
295#137—188no—
296C2×Q16—34no—
297C2×SD16—94no—
298C2×M16—92no—
299C22⋊Q8—32no—
300C22⋊Q8—92no—
301C22⋊Q8—181no—
302C22⋊Q8—181no—
303C2×C4⋊C4—91no—
304C2×C4⋊C4—181no—
305C2×C22⋊C4—181no—
306#103—34no—
307#102—94no—
308C22⋊C8—32no—
309C22⋊C8—92no—
310C23×C3—21no—
311C23×C3D3×D412yes—
312C22×D3—32no—
313C22×D3—32no—
314C22×D3—61no—
315C22×D3—61no—
316C22×D3—64no—
317C22×D3—64no—
318C22×D3—68no—
319C22×D3—68no—
320C22×D3—64no—
321C22×D3—64no—
322C22×D3—64no—
323C22×D3—122no—
324C22×D3—122no—
325C22×D3—122no—
326C22×D3—124no—
327C22×D3—124no—
328C22×D3—124no—
329C22×D3—124no—
330C3×Q8—28no—
331C3×Q8—28no—
332C3×Q8—24no—
333C3×Q8—24no—
334C3×Q8—24no—
335C3×Q8—64no—
336C3×Q8D3×D418yes—
337C2×C3×C4—28no—
338C2×C3×C4—28no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—24no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—62no—
346C2×C3×C4—62no—
347C2×C3×C4D3×D412yes—
348C2×C3×C4C23×D312yes—
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—32no—
352C2×Dic3—32no—
353C2×Dic3—61no—
354C2×Dic3—61no—
355C2×Dic3—64no—
356C2×Dic3—64no—
357D12—38no—
358D12—64no—
359D12—616no—
360D12—616no—
361D12—616no—
362D12—68no—
363D12—616no—
364D12—68no—
365D12—64no—
366D12—64no—
367C4×D3—38no—
368C4×D3—64no—
369C4×D3—64no—
370C4×D3—616no—
371C4×D3—64no—
372C4×D3—64no—
373C4×D3—616no—
374C4×D3—64no—
375C4×D3—616no—
376C4×D3—616no—
377C4×D3—64no—
378C4×D3—64no—
379C4×D3—124no—
380C4×D3—124no—
381Dic6—24no—
382Dic6—124no—
383C3×C8—64no—
384C3⋊C8—24no—
385C2×C32—22no—
386C2×C32C2×#12011yescharacteristic
387C2×C32C2×#13711yescharacteristic
388C2×C32#57511yescharacteristic
389C3⋊D3—22no—
390C3⋊D3—41no—
391C3⋊D3—44no—
392C24—91no—
393C24—181no—
394#41—98no—
395#41—184no—
396#41—184no—
397#41—188no—
398#41—188no—
399#41—184no—
400C2×Q8—24no—
401C2×Q8—21no—
402C2×Q8—32no—
403C2×Q8—181no—
404C2×Q8—184no—
405C2×Q8C2×D3212yes—
406C2×D4—92no—
407C2×D4—184no—
408C2×D4—182no—
409C2×D4—182no—
410C2×D4—188no—
411C2×D4—181no—
412C22×C4—21no—
413C22×C4—91no—
414C22×C4—91no—
415C22×C4—91no—
416C22×C4—91no—
417C22×C4—91no—
418C22×C4—184no—
419C22×C4—182no—
420C22×C4—182no—
421C22×C4—188no—
422C22×C4—181no—
423C22×C4—181no—
424C22×C4C2×D3211yescharacteristic
425Q16—68no—
426SD16—188no—
427M16—184no—
428C2×C8—34no—
429C4⋊C4—32no—
430C4⋊C4—62no—
431C4⋊C4—181no—
432C4⋊C4—182no—
433C22⋊C4—62no—
434C22⋊C4—182no—
435C22×C3—22no—
436C22×C3—22no—
437C22×C3—22no—
438C22×C3—22no—
439C22×C3—21no—
440C22×C3—22no—
441C22×C3#91814yescomplemented
442C22×C3C2×D3×D412yes—
443C2×D3—34no—
444C2×D3—62no—
445C2×D3—62no—
446C2×D3—62no—
447C2×D3—68no—
448C2×D3—62no—
449C2×D3—62no—
450C2×D3—62no—
451C2×D3—68no—
452C2×D3—62no—
453C2×D3—121no—
454C2×D3—121no—
455C2×D3—121no—
456C2×D3—121no—
457C2×D3—128no—
458C2×D3—128no—
459C2×D3—1216no—
460C2×D3—1216no—
461C2×D3—124no—
462C2×D3—124no—
463C2×D3—124no—
464C2×D3—124no—
465C3×C4—28no—
466C3×C4—22no—
467C3×C4—28no—
468C3×C4—22no—
469C3×C4—22no—
470C3×C4—122no—
471C3×C4—122no—
472C3×C4C2×D3×D414yes—
473Dic3—34no—
474Dic3—42no—
475Dic3—42no—
476Dic3—62no—
477Dic3—62no—
478Dic3—62no—
479C32#300711yescharacteristic, complemented, Sylow
480C23—91no—
481C23—91no—
482C23—182no—
483C23—184no—
484C23—182no—
485C23—182no—
486C23—361no—
487C23—362no—
488C23—362no—
489C23#381111yescharacteristic
490Q8—24no—
491Q8—22no—
492Q8—64no—
493Q8—362no—
494Q8#381114yes—
495D4—94no—
496D4—188no—
497D4—188no—
498D4—184no—
499D4—182no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—62no—
504C2×C4—62no—
505C2×C4—91no—
506C2×C4—94no—
507C2×C4—91no—
508C2×C4—182no—
509C2×C4—188no—
510C2×C4—182no—
511C2×C4—188no—
512C2×C4—182no—
513C2×C4—182no—
514C2×C4—362no—
515C2×C4—362no—
516C2×C4#381111yescharacteristic
517C2×C4C22×D3211yescharacteristic
518C8—64no—
519C2×C3—24no—
520C2×C3—24no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3D3×#12012yes—
525C2×C3C2×#91812yes—
526C2×C3#577712yes—
527D3—64no—
528D3—122no—
529D3—122no—
530D3—128no—
531D3—122no—
532D3—128no—
533C22—21no—
534C22—21no—
535C22—92no—
536C22—181no—
537C22—181no—
538C22—181no—
539C22—184no—
540C22—181no—
541C22—181no—
542C22—364no—
543C22—368no—
544C22—362no—
545C22—362no—
546C22#6394512yescomplemented
547C22C2×#381111yescharacteristic
548C4—24no—
549C4—21no—
550C4—92no—
551C4—122no—
552C4—122no—
553C4—181no—
554C4C2×#381112yes—
555C3—21no—
556C3#8018412yescomplemented
557C2—22no—
558C2—182no—
559C2—361no—
560C2—364no—
561C2#1059532311yescharacteristic
562C2C2×#6394511yescharacteristic
563C2#1059552711yescharacteristic
subgroupG/H
centerC22C2×#3811
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC22×C32×Q8C22
socleC22×C32C22×D4
prime radicalC22×#3832C2
hypercenterC22×Q8D32
hypocenterC32#3007
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#63945C2semidirect
3. #10594925C2semidirect
4. C3×#5228C2semidirect
5. #10591602C2semidirect
6. #10591554C2semidirect
7. #10591183C2semidirect
8. #10591159C2semidirect
20. C2×#3797C22semidirect
21. C2×C3×#216C22semidirect
26. #63900C22semidirect
28. #63871C22semidirect
30. C2×#3717C22semidirect
31. C2×#3712C22semidirect
32. C3×#835C22semidirect
34. #63564C22semidirect
35. #63557C22semidirect
41. #5228D3semidirect
116. C2×#216C2×D3semidirect
130. #835C2×D3semidirect
264. C22×C32#137semidirect
441. C22×C3#918semidirect
479. C32#3007semidirect
546. C22#63945semidirect
556. C3#80184semidirect

Maximal subgroups

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

subgroupindex
C22×#38322
C2×#639452
#105949252
C3×#52282
#105916022
#105915542
#105911832
#105911592
#801843