Finite Group Databasebeta

← #11778254

Subgroups — #11778254

Order 1152.

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

normal subgroups only

This group has 552 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
2C3×#5050C211yesmaximal, characteristic, complemented
3#10591304C211yesmaximal, characteristic
4#10591185C211yesmaximal, characteristic, complemented
5#10589404C211yesmaximal, characteristic, complemented
6#10589152C211yesmaximal, characteristic, complemented
7#10589145C211yesmaximal, characteristic
8#71893—31nomaximal
9#70980—31nomaximal
10C22×C3×Dic6C2211yescharacteristic
11C32×C22⋊Q8C2211yescharacteristic
12C3×#946—21no—
13C2×C3×#216—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×#3718—21no—
23#63658C2211yescharacteristic
24#63657C2211yescharacteristic
25C3×#830C2211yescharacteristic
26C3×#810—21no—
27#63579—21no—
28#63565—21no—
29C3×#543—31no—
30#5634D311yescharacteristic
31C3×#451—31no—
32#5231—31no—
33#5228—31no—
34#5050—31no—
35#5050D311yescharacteristic, complemented
36#4947—31no—
37#4538—31no—
38#4538—31no—
39#4538—31no—
40#4538—61no—
41#4525—31no—
42#4468—31no—
43#4467—31no—
44C23×C3×D3D411yescharacteristic
45C2×C32×Q8D411yescharacteristic
46C22×C32×C4C2311yescharacteristic
47C22×C3⋊Dic3—21no—
48C22×C3×Dic3—21no—
49C22×C3×Dic3D411yescharacteristic
50C2×C3×C4×D3—21no—
51C2×C3×C4×D3—21no—
52C2×C3×Dic6—21no—
53C2×C3×Dic6—21no—
54C2×C3×Dic6—21no—
55C2×C3×Dic6—42no—
56C2×C3×Dic6D411yescharacteristic
57C32×C4⋊C4—21no—
58C32×C4⋊C4D411yescharacteristic
59C32×C22⋊C4—21no—
60C2×C3⋊(C3⋊C8)—21no—
61C3×#217—21no—
62C3×#216—42no—
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#3718—42no—
75#1602—91noSylow
76C22×C3×Q8—31no—
77C22×Dic6C2×D311yescharacteristic
78C2×C3×Q16—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—21no—
93C2×#216—61no—
94C2×#216—61no—
95C2×(C2×D3)⋊C4—61no—
96C2×(C2×D3)⋊C4C2×D311yescharacteristic
97#924C2×D311yescharacteristic
98#897—21no—
99#896—21no—
100#896—21no—
101D3×C22⋊C4—21no—
102#878—21no—
103C3×#103—61no—
104C3×C22⋊C8—31no—
105#835—61no—
106C3⋊#95—31no—
107C3⋊#95—31no—
108C3⋊#95—61no—
109C3⋊#95—61no—
110C3⋊#95—61no—
111#830—31no—
112#830—31no—
113#830—61no—
114#830C2×D311yescharacteristic
115#810—21no—
116#810—61no—
117C23×C32C2×D411yescharacteristic
118C22×C3×D3—21no—
119C22×C3×D3—21no—
120C22×C3×D3—41no—
121C22×C3×D3—41no—
122C22×C3×D3Q1612yescomplemented
123C32×Q8—41no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4—41no—
127C2×C32×C4C2×D411yescharacteristic
128C2×C32×C4C2×D411yescharacteristic
129C2×C3⋊Dic3—41no—
130C2×C3⋊Dic3—41no—
131C2×C3⋊Dic3—41no—
132C2×C3×Dic3—22no—
133C2×C3×Dic3—21no—
134C2×C3×Dic3—21no—
135C2×C3×Dic3—21no—
136C2×C3×Dic3—41no—
137C2×C3×Dic3—41no—
138C2×C3×Dic3—41no—
139C3×C4×D3—81no—
140C3×C4×D3—81no—
141C3×Dic6—22no—
142C3×Dic6—22no—
143C3×Dic6—41no—
144C3×Dic6—41no—
145C3×Dic6—41no—
146C3×Dic6—81no—
147C3⋊(C3⋊C8)—41no—
148C3×C3⋊C8—41no—
149#543—31no—
150#451—31no—
151#451—91no—
152#393—91no—
153#328—91no—
154#328—91no—
155#323—91no—
156C24×C3—31no—
157C23×D3#6711yescharacteristic
158C2×C3×Q8—21no—
159C2×C3×Q8—31no—
160C2×C3×Q8—61no—
161C2×C3×Q8—61no—
162C2×C3×Q8—61no—
163C2×C3×Q8—122no—
164C2×C3×Q8#6711yescharacteristic
165C2×C3×Q8#6711yescharacteristic
166C22×C3×C4—21no—
167C22×C3×C4—31no—
168C22×C3×C4—61no—
169C22×C3×C4—61no—
170C22×C3×C4—61no—
171C22×C3×C4C22×D311yescharacteristic
172C22×C3×C4C22×D311yescharacteristic
173C22×Dic3—21no—
174C22×Dic3—61no—
175C22×Dic3—61no—
176C22×Dic3—121no—
177C22×Dic3#6711yescharacteristic
178C2×C4×D3—21no—
179C2×C4×D3—21no—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6—21no—
183C2×Dic6—42no—
184C2×Dic6#6711yescharacteristic
185C3×Q16—122no—
186C2×C3×C8—61no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4—31no—
191C3×C4⋊C4—41no—
192C3×C4⋊C4—61no—
193C3×C4⋊C4—61no—
194C3×C4⋊C4—61no—
195C3×C4⋊C4—61no—
196C3×C4⋊C4#6711yescharacteristic
197C3×C4⋊C4#6711yescharacteristic
198C3×C22⋊C4—21no—
199C3×C22⋊C4—21no—
200C3×C22⋊C4—41no—
201C3×C22⋊C4—61no—
202C3×C22⋊C4—61no—
203C3×C22⋊C4—61no—
204C3×C22⋊C4—61no—
205C3×C22⋊C4—61no—
206#217—21no—
207#217—122no—
208#216—42no—
209#216—122no—
210#216—122no—
211(C2×D3)⋊C4—21no—
212(C2×D3)⋊C4—21no—
213(C2×D3)⋊C4—21no—
214(C2×D3)⋊C4—21no—
215(C2×D3)⋊C4—122no—
216C4⋊Dic3—21no—
217C4⋊Dic3#6711yescharacteristic
218Dic3⋊C4—21no—
219Dic3⋊C4—21no—
220Dic3⋊C4—21no—
221C2×C3⋊C8—21no—
222C2×C3⋊C8—61no—
223C2×C3⋊C8—61no—
224C2×C3⋊C8—121no—
225C22×C32—21no—
226C22×C32—21no—
227C22×C32#12011yescharacteristic
228C22×C32C2×Q1611yescharacteristic
229C22×C32#13711yescharacteristic
230C2×C3×D3—22no—
231C2×C3×D3—41no—
232C2×C3×D3—41no—
233C2×C3×D3—42no—
234C2×C3×D3—81no—
235C32×C4—21no—
236C32×C4—21no—
237C32×C4—41no—
238C32×C4—41no—
239C3⋊Dic3—81no—
240C3×Dic3—41no—
241C3×Dic3—41no—
242C3×Dic3—42no—
243C3×Dic3—41no—
244C22×Q8—31no—
245C2×Q16—61no—
246C2×Q16—181no—
247C22⋊Q8—31no—
248C22⋊Q8—61no—
249C22⋊Q8—61no—
250C22⋊Q8—61no—
251C22⋊Q8—61no—
252C22⋊Q8—61no—
253C22⋊Q8D3211yescharacteristic
254C2×C22⋊C4—31no—
255C2×C22⋊C4—61no—
256C2×C22⋊C4—181no—
257#103—61no—
258#103—181no—
259C22⋊C8—31no—
260C22⋊C8—91no—
261#95—91no—
262#95—181no—
263C23×C3—21no—
264C23×C3—32no—
265C23×C3—61no—
266C23×C3—61no—
267C23×C3—121no—
268C23×C3—121no—
269C23×C3D3×D411yescharacteristic
270C23×C3C2×#6711yescharacteristic
271C22×D3—21no—
272C22×D3—21no—
273C22×D3—41no—
274C22×D3—41no—
275C22×D3#21612yescomplemented
276C3×Q8—41no—
277C3×Q8—41no—
278C3×Q8—62no—
279C3×Q8—62no—
280C3×Q8—81no—
281C3×Q8—121no—
282C3×Q8—121no—
283C3×Q8—121no—
284C3×Q8—241no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—62no—
296C2×C3×C4—61no—
297C2×C3×C4—61no—
298C2×C3×C4—61no—
299C2×C3×C4—81no—
300C2×C3×C4—121no—
301C2×C3×C4—121no—
302C2×C3×C4—121no—
303C2×C3×C4—241no—
304C2×C3×C4—241no—
305C2×C3×C4D3×D411yescharacteristic
306C2×C3×C4C2×#6711yescharacteristic
307C2×C3×C4C2×#6711yescharacteristic
308C2×C3×C4C2×#6711yescharacteristic
309C2×Dic3—22no—
310C2×Dic3—21no—
311C2×Dic3—21no—
312C2×Dic3—21no—
313C2×Dic3—41no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316C2×Dic3—121no—
317C2×Dic3—121no—
318C2×Dic3—121no—
319C2×Dic3—121no—
320C2×Dic3—122no—
321C2×Dic3—121no—
322C2×Dic3—121no—
323C2×Dic3—241no—
324C2×Dic3—241no—
325C4×D3—81no—
326C4×D3—81no—
327Dic6—22no—
328Dic6—22no—
329Dic6—41no—
330Dic6—41no—
331Dic6—41no—
332Dic6—81no—
333C3×C8—121no—
334C3⋊C8—41no—
335C3⋊C8—121no—
336C3⋊C8—121no—
337C3⋊C8—241no—
338C2×C32—21no—
339C2×C32—21no—
340C2×C32#45111yescharacteristic
341C2×C32#45511yescharacteristic
342C2×C32B(4,𝔽2)11yescharacteristic
343C3×D3—42no—
344C24—31no—
345C2×Q8—31no—
346C2×Q8—61no—
347C2×Q8—61no—
348C2×Q8—61no—
349C2×Q8—122no—
350C2×Q8#62511yescharacteristic
351C22×C4—31no—
352C22×C4—61no—
353C22×C4—61no—
354C22×C4—61no—
355C22×C4—181no—
356C22×C4C2×D3211yescharacteristic
357Q16—122no—
358Q16—362no—
359C2×C8—61no—
360C2×C8—181no—
361C4⋊C4—21no—
362C4⋊C4—31no—
363C4⋊C4—61no—
364C4⋊C4—61no—
365C4⋊C4—61no—
366C4⋊C4—61no—
367C4⋊C4#62511yescharacteristic
368C22⋊C4—21no—
369C22⋊C4—61no—
370C22⋊C4—61no—
371C22⋊C4—61no—
372C22⋊C4—61no—
373C22⋊C4—61no—
374C22⋊C4—362no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—41no—
383C22×C3—41no—
384C22×C3—62no—
385C22×C3—122no—
386C22×C3—121no—
387C22×C3—121no—
388C22×C3—241no—
389C22×C3D3×Q1611yescharacteristic
390C22×C3#91811yescharacteristic
391C22×C3#93711yescharacteristic
392C22×C3#94211yescharacteristic
393C22×C3C2×#21611yescharacteristic
394C22×C3#95311yescharacteristic
395C2×D3—22no—
396C2×D3—41no—
397C2×D3—41no—
398C2×D3—42no—
399C2×D3—81no—
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—21no—
403C3×C4—21no—
404C3×C4—41no—
405C3×C4—41no—
406C3×C4—41no—
407C3×C4—41no—
408C3×C4—41no—
409C3×C4—41no—
410C3×C4—81no—
411C3×C4—81no—
412C3×C4—121no—
413C3×C4—121no—
414C3×C4—122no—
415C3×C4—121no—
416Dic3—41no—
417Dic3—41no—
418Dic3—41no—
419Dic3—42no—
420Dic3—241no—
421Dic3—241no—
422Dic3—242no—
423C32#160211yescharacteristic, complemented, Sylow
424C23—32no—
425C23—61no—
426C23—61no—
427C23—121no—
428C23—121no—
429C23D3×#6711yescharacteristic
430Q8—41no—
431Q8—62no—
432Q8—62no—
433Q8—121no—
434Q8—121no—
435Q8—121no—
436Q8—241no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—41no—
440C2×C4—62no—
441C2×C4—61no—
442C2×C4—61no—
443C2×C4—61no—
444C2×C4—121no—
445C2×C4—121no—
446C2×C4—121no—
447C2×C4—241no—
448C2×C4—241no—
449C2×C4—361no—
450C2×C4—361no—
451C2×C4—361no—
452C2×C4C2×#62511yescharacteristic
453C2×C4D3×#6711yescharacteristic
454C8—121no—
455C8—361no—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—41no—
464C2×C3—41no—
465C2×C3—122no—
466C2×C3#503211yescharacteristic
467C2×C3#518611yescharacteristic
468C2×C3#520211yescharacteristic
469C2×C3#522811yescharacteristic
470C2×C3#524211yescharacteristic
471C2×C3#524411yescharacteristic
472D3—42no—
473C22—21no—
474C22—21no—
475C22—62no—
476C22—122no—
477C22—121no—
478C22—121no—
479C22—241no—
480C22D3×#21611yescharacteristic
481C22#6394811yescharacteristic
482C22#6397611yescharacteristic
483C4—21no—
484C4—21no—
485C4—41no—
486C4—41no—
487C4—122no—
488C4—121no—
489C4—121no—
490C4—121no—
491C4—721no—
492C3—21no—
493C3#7098011yescharacteristic, complemented
494C3#7189311yescharacteristic, complemented
495C2—21no—
496C2—21no—
497C2—122no—
498C2#1059153011yescharacteristic
499C2#1059159411yescharacteristic
500C2#1059166411yescharacteristic
subgroupG/H
centerC22#63976
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. C3×#5050C2semidirect
4. #10591185C2semidirect
5. #10589404C2semidirect
6. #10589152C2semidirect
35. #5050D3semidirect
122. C22×C3×D3Q16semidirect
275. C22×D3#216semidirect
423. C32#1602semidirect
493. C3#70980semidirect
494. C3#71893semidirect

Maximal subgroups

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

subgroupindex
C3×#56342
C3×#50502
#105913042
#105911852
#105894042
#105891522
#105891452
#718933
#709803