Finite Group Databasebeta

← #11778252

Subgroups — #11778252

Order 1152.

586 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 88 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
1C3×#5634C211yesmaximal, characteristic
2#10592749C211yesmaximal, characteristic, complemented
3#10591322C211yesmaximal, characteristic
4#10591190C211yesmaximal, characteristic, complemented
5C3×#4538C211yesmaximal, characteristic, complemented
6#10589153C211yesmaximal, characteristic, complemented
7#10589145C211yesmaximal, characteristic
8#71893—31nomaximal
9#68318—31nomaximal
10C22×#634C2211yescharacteristic
11C32×C22⋊Q8C2211yescharacteristic
12C2×#3756—21no—
13C3×#946—21no—
14C2×C3×(C2×D3)⋊C4C2211yescharacteristic
15C3×#924C2211yescharacteristic
16C3×#897—21no—
17C3×#896—21no—
18C3×#896—21no—
19C3×D3×C22⋊C4—21no—
20C3×#878—21no—
21C2×(C2×D3)⋊Dic3—21no—
22C2×#3719—21no—
23#63657C2211yescharacteristic
24#63637—21no—
25C3×C3⋊#95C2211yescharacteristic
26C3×#830C2211yescharacteristic
27#63579—21no—
28#63567—21no—
29C3×#543—31no—
30#5634D311yescharacteristic
31#5050—31no—
32#5050—31no—
33#5050—31no—
34#5050—61no—
35#4947—31no—
36#4739—31no—
37#4673—31no—
38C3×#328—31no—
39#4538—31no—
40#4538D311yescharacteristic, complemented
41#4525—31no—
42#4468—31no—
43#4452—31no—
44C23×C3×D3D411yescharacteristic
45C2×C32×Q8D411yescharacteristic
46C22×C32×C4C2311yescharacteristic
47C22×C3⋊Dic3—21no—
48C2×#634—21no—
49C2×#634—21no—
50C2×#634—42no—
51C22×C3×Dic3—21no—
52C22×C3×Dic3D411yescharacteristic
53C2×C3×C4×D3—21no—
54C2×C3×C4×D3—21no—
55C2×C3×Dic6—21no—
56C2×C3×Dic6D411yescharacteristic
57C32×C4⋊C4—21no—
58C32×C4⋊C4D411yescharacteristic
59C32×C22⋊C4—21no—
60#3756—42no—
61C2×C3⋊(C3⋊C8)—21no—
62C3×#217—21no—
63C3×(C2×D3)⋊C4—21no—
64C3×(C2×D3)⋊C4—21no—
65C3×(C2×D3)⋊C4—21no—
66C3×(C2×D3)⋊C4—21no—
67C3×C4⋊Dic3—21no—
68C3×C4⋊Dic3D411yescharacteristic
69C3×Dic3⋊C4—21no—
70C3×Dic3⋊C4—21no—
71C3×Dic3⋊C4—21no—
72C2×C3×C3⋊C8—21no—
73(C2×D3)⋊Dic3—42no—
74#3719—42no—
75#1602—91noSylow
76C22×Dic6—31no—
77C22×Dic6—31no—
78C22×Dic6—61no—
79C3×C22⋊Q8—21no—
80C3×C22⋊Q8—31no—
81C3×C22⋊Q8—61no—
82C3×C22⋊Q8—61no—
83C3×C22⋊Q8—61no—
84C3×C22⋊Q8—61no—
85C3×C22⋊Q8—61no—
86C3×C22⋊Q8C2×D311yescharacteristic
87C3×C22⋊Q8C2×D311yescharacteristic
88C2×C3×C22⋊C4—31no—
89C2×C3×C22⋊C4—61no—
90C2×#217—61no—
91#946—21no—
92C2×#216—61no—
93C2×#216—62no—
94C2×#216—61no—
95C2×#216—61no—
96C2×(C2×D3)⋊C4—61no—
97C2×(C2×D3)⋊C4C2×D311yescharacteristic
98#924C2×D311yescharacteristic
99C2×Dic12—61no—
100#897—21no—
101#896—21no—
102#896—21no—
103D3×C22⋊C4—21no—
104#878—21no—
105C3×C22⋊C8—31no—
106C3×#95—31no—
107C3⋊#95—61no—
108C3⋊#95—61no—
109C3⋊#95C2×D311yescharacteristic
110#830—31no—
111#830—31no—
112#830—61no—
113#830C2×D311yescharacteristic
114#816—61no—
115#810—61no—
116#810—62no—
117#810—61no—
118#810—61no—
119C23×C32C2×D411yescharacteristic
120C22×C3×D3—21no—
121C22×C3×D3—21no—
122C22×C3×D3—41no—
123C22×C3×D3—41no—
124C22×C3×D3Q1612yescomplemented
125C32×Q8—41no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—41no—
129C2×C32×C4C2×D411yescharacteristic
130C2×C32×C4C2×D411yescharacteristic
131C2×C3⋊Dic3—22no—
132C2×C3⋊Dic3—41no—
133C2×C3⋊Dic3—41no—
134#634—22no—
135#634—22no—
136#634—41no—
137#634—41no—
138C2×C3×Dic3—21no—
139C2×C3×Dic3—21no—
140C2×C3×Dic3—21no—
141C2×C3×Dic3—41no—
142C2×C3×Dic3—41no—
143C2×C3×Dic3—41no—
144C2×C3×Dic3—41no—
145C3×C4×D3—81no—
146C3×C4×D3—81no—
147C3×Dic6—41no—
148C3×Dic6—81no—
149C3⋊(C3⋊C8)—41no—
150C3×C3⋊C8—41no—
151#543—31no—
152#451—91no—
153#451—91no—
154#393—91no—
155#328—31no—
156#328—91no—
157#323—91no—
158C24×C3—31no—
159C23×D3#6711yescharacteristic
160C2×C3×Q8—21no—
161C2×C3×Q8—31no—
162C2×C3×Q8—61no—
163C2×C3×Q8D1211yescharacteristic
164C2×C3×Q8#6711yescharacteristic
165C22×C3×C4—21no—
166C22×C3×C4—31no—
167C22×C3×C4—61no—
168C22×C3×C4—61no—
169C22×C3×C4—61no—
170C22×C3×C4C22×D311yescharacteristic
171C22×C3×C4C22×D311yescharacteristic
172C22×Dic3—21no—
173C22×Dic3—61no—
174C22×Dic3—61no—
175C22×Dic3—121no—
176C22×Dic3#6711yescharacteristic
177C2×C4×D3—21no—
178C2×C4×D3—21no—
179C2×Dic6—21no—
180C2×Dic6—62no—
181C2×Dic6—61no—
182C2×Dic6—61no—
183C2×Dic6—61no—
184C2×Dic6—61no—
185C2×Dic6—121no—
186C2×Dic6—122no—
187C2×Dic6—122no—
188C2×Dic6—124no—
189C2×Dic6#6711yescharacteristic
190C2×C3×C8—61no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—31no—
195C3×C4⋊C4—41no—
196C3×C4⋊C4—61no—
197C3×C4⋊C4—61no—
198C3×C4⋊C4—61no—
199C3×C4⋊C4—61no—
200C3×C4⋊C4D1211yescharacteristic
201C3×C4⋊C4#6711yescharacteristic
202C3×C22⋊C4—21no—
203C3×C22⋊C4—21no—
204C3×C22⋊C4—41no—
205C3×C22⋊C4—61no—
206C3×C22⋊C4—61no—
207C3×C22⋊C4—61no—
208C3×C22⋊C4—61no—
209C3×C22⋊C4—61no—
210#217—21no—
211#217—122no—
212#216—122no—
213#216—122no—
214#216—122no—
215#216—124no—
216(C2×D3)⋊C4—21no—
217(C2×D3)⋊C4—21no—
218(C2×D3)⋊C4—21no—
219(C2×D3)⋊C4—21no—
220(C2×D3)⋊C4—122no—
221C4⋊Dic3—21no—
222C4⋊Dic3#6711yescharacteristic
223Dic3⋊C4—21no—
224Dic3⋊C4—21no—
225Dic3⋊C4—21no—
226C2×C3⋊C8—21no—
227C2×C3⋊C8—62no—
228C2×C3⋊C8—61no—
229C2×C3⋊C8—61no—
230Dic12—122no—
231C22×C32—21no—
232C22×C32—21no—
233C22×C32#12011yescharacteristic
234C22×C32C2×Q1611yescharacteristic
235C22×C32#13711yescharacteristic
236C2×C3×D3—22no—
237C2×C3×D3—41no—
238C2×C3×D3—41no—
239C2×C3×D3—42no—
240C2×C3×D3—81no—
241C32×C4—21no—
242C32×C4—21no—
243C32×C4—41no—
244C32×C4—41no—
245C3⋊Dic3—42no—
246C3×Dic3—41no—
247C3×Dic3—41no—
248C3×Dic3—41no—
249C3×Dic3—81no—
250C22×Q8—91no—
251C2×Q16—181no—
252C2×Q16—181no—
253C22⋊Q8—31no—
254C22⋊Q8—61no—
255C22⋊Q8—61no—
256C22⋊Q8—61no—
257C22⋊Q8—61no—
258C22⋊Q8—61no—
259C22⋊Q8D3211yescharacteristic
260C2×C22⋊C4—31no—
261C2×C22⋊C4—61no—
262C2×C22⋊C4—181no—
263#103—181no—
264#103—181no—
265C22⋊C8—31no—
266C22⋊C8—91no—
267#95—31no—
268#95—181no—
269C23×C3—21no—
270C23×C3—32no—
271C23×C3—61no—
272C23×C3—61no—
273C23×C3—121no—
274C23×C3—121no—
275C23×C3D3×D411yescharacteristic
276C23×C3C2×#6711yescharacteristic
277C22×D3—21no—
278C22×D3—21no—
279C22×D3—41no—
280C22×D3—41no—
281C22×D3#21612yescomplemented
282C3×Q8—41no—
283C3×Q8—41no—
284C3×Q8—42no—
285C3×Q8—121no—
286C3×Q8—241no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—42no—
298C2×C3×C4—61no—
299C2×C3×C4—61no—
300C2×C3×C4—61no—
301C2×C3×C4—121no—
302C2×C3×C4—121no—
303C2×C3×C4—121no—
304C2×C3×C4—121no—
305C2×C3×C4—241no—
306C2×C3×C4—241no—
307C2×C3×C4C2×D1211yescharacteristic
308C2×C3×C4D3×D411yescharacteristic
309C2×C3×C4C2×#6711yescharacteristic
310C2×C3×C4C2×#6711yescharacteristic
311C2×Dic3—21no—
312C2×Dic3—21no—
313C2×Dic3—21no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316C2×Dic3—41no—
317C2×Dic3—41no—
318C2×Dic3—62no—
319C2×Dic3—62no—
320C2×Dic3—122no—
321C2×Dic3—122no—
322C2×Dic3—121no—
323C2×Dic3—121no—
324C2×Dic3—122no—
325C2×Dic3—121no—
326C2×Dic3—121no—
327C4×D3—81no—
328C4×D3—81no—
329Dic6—41no—
330Dic6—62no—
331Dic6—64no—
332Dic6—62no—
333Dic6—62no—
334Dic6—62no—
335Dic6—81no—
336Dic6—122no—
337Dic6—122no—
338Dic6—121no—
339Dic6—121no—
340Dic6—121no—
341Dic6—121no—
342Dic6—122no—
343C3×C8—121no—
344C3⋊C8—41no—
345C3⋊C8—122no—
346C3⋊C8—121no—
347C3⋊C8—121no—
348C2×C32—21no—
349C2×C32—21no—
350C2×C32#45111yescharacteristic
351C2×C32#45511yescharacteristic
352C2×C32B(4,𝔽2)11yescharacteristic
353C3×D3—42no—
354C24—31no—
355C2×Q8—31no—
356C2×Q8—61no—
357C2×Q8—181no—
358C2×Q8—181no—
359C2×Q8—362no—
360C2×Q8#62611yescharacteristic
361C22×C4—31no—
362C22×C4—61no—
363C22×C4—61no—
364C22×C4—61no—
365C22×C4—181no—
366C22×C4C2×D3211yescharacteristic
367Q16—362no—
368Q16—362no—
369C2×C8—61no—
370C2×C8—181no—
371C4⋊C4—21no—
372C4⋊C4—31no—
373C4⋊C4—61no—
374C4⋊C4—61no—
375C4⋊C4—61no—
376C4⋊C4—61no—
377C4⋊C4#62611yescharacteristic
378C22⋊C4—21no—
379C22⋊C4—61no—
380C22⋊C4—61no—
381C22⋊C4—61no—
382C22⋊C4—61no—
383C22⋊C4—61no—
384C22⋊C4—362no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—41no—
393C22×C3—41no—
394C22×C3—62no—
395C22×C3—122no—
396C22×C3—121no—
397C22×C3—121no—
398C22×C3—241no—
399C22×C3#88211yescharacteristic
400C22×C3#91511yescharacteristic
401C22×C3D3×Q1611yescharacteristic
402C22×C3#94211yescharacteristic
403C22×C3C2×#21611yescharacteristic
404C22×C3#95311yescharacteristic
405C2×D3—22no—
406C2×D3—41no—
407C2×D3—41no—
408C2×D3—42no—
409C2×D3—81no—
410C3×C4—22no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—41no—
416C3×C4—41no—
417C3×C4—41no—
418C3×C4—41no—
419C3×C4—41no—
420C3×C4—81no—
421C3×C4—81no—
422C3×C4—121no—
423C3×C4—121no—
424C3×C4—121no—
425C3×C4—241no—
426Dic3—41no—
427Dic3—41no—
428Dic3—41no—
429Dic3—81no—
430Dic3—124no—
431Dic3—122no—
432Dic3—122no—
433C32#160211yescharacteristic, complemented, Sylow
434C23—32no—
435C23—61no—
436C23—61no—
437C23—121no—
438C23—121no—
439C23D3×#6711yescharacteristic
440Q8—41no—
441Q8—121no—
442Q8—182no—
443Q8—182no—
444Q8—241no—
445Q8—361no—
446Q8—361no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—41no—
450C2×C4—61no—
451C2×C4—61no—
452C2×C4—61no—
453C2×C4—121no—
454C2×C4—121no—
455C2×C4—121no—
456C2×C4—121no—
457C2×C4—182no—
458C2×C4—241no—
459C2×C4—241no—
460C2×C4—361no—
461C2×C4—361no—
462C2×C4C2×#62611yescharacteristic
463C2×C4D3×#6711yescharacteristic
464C8—121no—
465C8—361no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—41no—
474C2×C3—41no—
475C2×C3—122no—
476C2×C3#474111yescharacteristic
477C2×C3#480911yescharacteristic
478C2×C3#482511yescharacteristic
479C2×C3#522811yescharacteristic
480C2×C3#524211yescharacteristic
481C2×C3#524411yescharacteristic
482D3—42no—
483C22—21no—
484C22—21no—
485C22—62no—
486C22—122no—
487C22—121no—
488C22—121no—
489C22—241no—
490C22D3×#21611yescharacteristic
491C22#6394611yescharacteristic
492C22#6397711yescharacteristic
493C4—21no—
494C4—21no—
495C4—41no—
496C4—41no—
497C4—121no—
498C4—121no—
499C4—121no—
500C4—241no—
501C4—362no—
502C3—21no—
503C3#6831811yescharacteristic, complemented
504C3#7189311yescharacteristic, complemented
505C2—21no—
506C2—21no—
507C2—122no—
508C2#1059153111yescharacteristic
509C2#1059159511yescharacteristic
510C2#1059168811yescharacteristic
subgroupG/H
centerC22#63977
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC32×C22⋊Q8C22
socleC22×C32#120
prime radicalC23×C3×D3D4
hypercenterC22⋊Q8D32
hypocenterC32#1602
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #10592749C2semidirect
4. #10591190C2semidirect
5. C3×#4538C2semidirect
6. #10589153C2semidirect
40. #4538D3semidirect
124. C22×C3×D3Q16semidirect
281. C22×D3#216semidirect
433. C32#1602semidirect
503. C3#68318semidirect
504. C3#71893semidirect

Maximal subgroups

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

subgroupindex
C3×#56342
#105927492
#105913222
#105911902
C3×#45382
#105891532
#105891452
#718933
#683183