Finite Group Databasebeta

← #11796302

Subgroups — #11796302

Order 1152.

502 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 96 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×#5648C211yesmaximal, characteristic
2#10591528C211yesmaximal, characteristic
3#10591501C211yesmaximal, characteristic
4#10589405C211yesmaximal, characteristic, complemented
5C3×#4539C211yesmaximal, characteristic, complemented
6#10589176C211yesmaximal, characteristic, complemented
7#10589175C211yesmaximal, characteristic, complemented
8#71870—31nomaximal
9#69145—31nomaximal
10C2×C3×#237C2211yescharacteristic
11C22×C3×Dic6C411yescharacteristic, complemented
12C32×#123C2211yescharacteristic
13C3×#935C2211yescharacteristic
14C3×#929C411yescharacteristic, complemented
15C3×#896—21no—
16C3×#888—21no—
17C3×#886C411yescharacteristic, complemented
18C3×#885—21no—
19C3×#878—21no—
20C3×#877C411yescharacteristic, complemented
21#63955—21no—
22D3×#208—21no—
23#63661C2211yescharacteristic
24#63660C2211yescharacteristic
25C3×#834C2211yescharacteristic
26C3×#833C2211yescharacteristic
27#63580—21no—
28#63560—21no—
29C3×#536—31no—
30#5648D311yescharacteristic
31#5240—31no—
32#5237—31no—
33#4745—31no—
34#4740—31no—
35C3×#352—31no—
36#4539—31no—
37#4539—31no—
38#4539—31no—
39#4539—31no—
40#4539—61no—
41#4539D311yescharacteristic, complemented
42#4476—31no—
43#4472—31no—
44C2×C32×D4C2311yescharacteristic
45C22×C32×C4C2×C411yescharacteristic
46C2×C3×#67C2×C411yescharacteristic
47C2×C3×#67D411yescharacteristic
48C22×C3×Dic3C2×C411yescharacteristic
49C22×C3×Dic3C2×C411yescharacteristic
50C22×C3×Dic3D411yescharacteristic
51C3×#237—41no—
52C3×#237—41no—
53C2×C3×C4×D3D411yescharacteristic
54C2×C3×Dic6—21no—
55C2×C3×Dic6—21no—
56C2×C3×Dic6—41no—
57C2×C3×Dic6—41no—
58C2×C3×Dic6D411yescharacteristic
59C32×C4⋊C4—21no—
60C32×C22⋊C4—21no—
61C32×C22⋊C4C2×C411yescharacteristic
62#3757—21no—
63#3748—21no—
64C3×#217—21no—
65C3×#217—21no—
66C3×#217C2×C411yescharacteristic
67C3×(C2×D3)⋊C4—21no—
68C3×(C2×D3)⋊C4—21no—
69C3×C4⋊Dic3—21no—
70C3×Dic3⋊C4—21no—
71C3×Dic3⋊C4—21no—
72C3×Dic3⋊C4—21no—
73C3×C4×Dic3—21no—
74C3×#208—21no—
75Dic3⋊Dic3—41no—
76Dic32—41no—
77#3711—41no—
78D3×C3⋊C8—41no—
79#2123—91noSylow
80C2×C3×#41—31no—
81C22×C3×Q8—31no—
82C2×#237C2×D311yescharacteristic
83C22×Dic6Dic311yescharacteristic, complemented
84C3×C4⋊Q8—61no—
85C3×#124—61no—
86C3×#123—21no—
87C3×#123—31no—
88C3×#123—31no—
89C3×#123—31no—
90C3×#123C2×D311yescharacteristic
91C3×#123C2×D311yescharacteristic
92C3×C22⋊Q8—61no—
93C3×C22⋊Q8—61no—
94C3×#117—31no—
95#935C2×D311yescharacteristic
96#929Dic311yescharacteristic, complemented
97#926—61no—
98C2×#208—61no—
99D3×M16—61no—
100#896—21no—
101#888—21no—
102#886Dic311yescharacteristic, complemented
103#885—21no—
104#878—21no—
105#877—61no—
106#877Dic311yescharacteristic, complemented
107C3×#100—31no—
108C3×#99—31no—
109#836—61no—
110#834—31no—
111#834—31no—
112#834—61no—
113#834—61no—
114#834C2×D311yescharacteristic
115#833—31no—
116#833—31no—
117#833—61no—
118#833C2×D311yescharacteristic
119#824—61no—
120#806—61no—
121C23×C32C22×C411yescharacteristic
122C23×C32C2×D411yescharacteristic
123C22×C3×D3C22⋊C411yescharacteristic
124C32×D4—41no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—41no—
128C2×C32×C4C2×D411yescharacteristic
129C2×C3⋊Dic3—41no—
130C3×#67—41no—
131C3×#67—41no—
132C2×C3×Dic3—21no—
133C2×C3×Dic3—21no—
134C2×C3×Dic3—21no—
135C2×C3×Dic3—21no—
136C2×C3×Dic3—21no—
137C2×C3×Dic3—21no—
138C2×C3×Dic3—41no—
139C2×C3×Dic3—41no—
140C2×C3×Dic3C22⋊C411yescharacteristic
141C2×C3×Dic3C22⋊C411yescharacteristic
142C2×C3×Dic3C22⋊C411yescharacteristic
143C3×C4×D3—21no—
144C3×C4×D3—21no—
145C3×Dic6—41no—
146C3×Dic6—42no—
147C3×Dic6—41no—
148C3×Dic6—41no—
149C3⋊(C3⋊C8)—41no—
150C3×C3⋊C8—41no—
151#536—31no—
152#413—91no—
153#410—91no—
154#352—31no—
155#352—91no—
156#352—91no—
157#352—91no—
158C3×#41—121no—
159C3×#41—121no—
160C2×C3×Q8—31no—
161C2×C3×Q8—61no—
162C2×C3×Q8—61no—
163C2×C3×Q8—121no—
164C2×C3×Q8—121no—
165C2×C3×D4—21no—
166C2×C3×D4—31no—
167C2×C3×D4—31no—
168C2×C3×D4C22×D311yescharacteristic
169C2×C3×D4C22×D311yescharacteristic
170C22×C3×C4—21no—
171C22×C3×C4—31no—
172C22×C3×C4—31no—
173C22×C3×C4—31no—
174C22×C3×C4—31no—
175C22×C3×C4C4×D311yescharacteristic
176C22×C3×C4C2×Dic311yescharacteristic
177C2×#67C2×Dic311yescharacteristic
178C2×#67#6711yescharacteristic
179C22×Dic3C2×Dic311yescharacteristic
180C22×Dic3C2×Dic311yescharacteristic
181C22×Dic3#6711yescharacteristic
182#237—41no—
183#237—41no—
184C2×C4×D3#6711yescharacteristic
185C2×Dic6—21no—
186C2×Dic6—21no—
187C2×Dic6—41no—
188C2×Dic6—41no—
189C2×Dic6#6711yescharacteristic
190C3×M16—61no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—41no—
194C3×C4⋊C4—61no—
195C3×C4⋊C4—61no—
196C3×C4⋊C4—61no—
197C3×C4⋊C4—61no—
198C3×C22⋊C4—21no—
199C3×C22⋊C4—21no—
200C3×C22⋊C4—21no—
201C3×C22⋊C4—31no—
202C3×C22⋊C4—41no—
203C3×C22⋊C4—61no—
204C3×C22⋊C4—61no—
205C3×C22⋊C4—61no—
206C3×C22⋊C4—61no—
207C3×C22⋊C4C4×D311yescharacteristic
208C3×C22⋊C4C2×Dic311yescharacteristic
209C3×C42—61no—
210#217—21no—
211#217—21no—
212#217—61no—
213#217—61no—
214#217—121no—
215#217C2×Dic311yescharacteristic
216(C2×D3)⋊C4—21no—
217(C2×D3)⋊C4—21no—
218C4⋊Dic3—21no—
219C4⋊Dic3—121no—
220Dic3⋊C4—21no—
221Dic3⋊C4—21no—
222Dic3⋊C4—21no—
223Dic3⋊C4—121no—
224C4×Dic3—21no—
225C4×Dic3—121no—
226C4×Dic3—121no—
227#208—21no—
228#208—61no—
229#208—61no—
230#208—121no—
231#208—121no—
232C2×C3⋊C8—121no—
233ΓL(1,𝔽25)—121no—
234D3×C8—121no—
235C22×C32—21no—
236C22×C32—21no—
237C22×C32—41no—
238C22×C32—41no—
239C22×C32C2×C22⋊C411yescharacteristic
240C2×C3×D3—21no—
241C2×C3×D3—41no—
242C32×C4—21no—
243C32×C4—41no—
244C32×C4—41no—
245C3⋊Dic3—81no—
246C3×Dic3—21no—
247C3×Dic3—21no—
248C3×Dic3—41no—
249C3×Dic3—41no—
250C3×Dic3—81no—
251C3×Dic3#9911yescharacteristic
252C3×Dic3#9911yescharacteristic
253C2×#41—31no—
254C22×Q8—31no—
255C2×M16—181no—
256C4⋊Q8—61no—
257#124—61no—
258#123—31no—
259#123—31no—
260#123—31no—
261#123D3211yescharacteristic
262C22⋊Q8—61no—
263C22⋊Q8—61no—
264#117—31no—
265#117—181no—
266#101—181no—
267#100—31no—
268#100—91no—
269#99—31no—
270#99—91no—
271#99—181no—
272C23×C3—21no—
273C23×C3—21no—
274C23×C3—31no—
275C23×C3C2×C4×D311yescharacteristic
276C23×C3D3×D411yescharacteristic
277C23×C3C22×Dic311yescharacteristic
278C23×C3C2×#6711yescharacteristic
279C22×D3#21711yescharacteristic
280C3×Q8—121no—
281C3×Q8—121no—
282C3×Q8—121no—
283C3×Q8—122no—
284C3×D4—41no—
285C3×D4—41no—
286C3×D4—81no—
287C3×D4—121no—
288C3×D4—121no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—31no—
295C2×C3×C4—31no—
296C2×C3×C4—31no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—61no—
302C2×C3×C4—61no—
303C2×C3×C4—61no—
304C2×C3×C4—61no—
305C2×C3×C4—61no—
306C2×C3×C4—61no—
307C2×C3×C4—61no—
308C2×C3×C4—61no—
309C2×C3×C4—81no—
310C2×C3×C4—121no—
311C2×C3×C4—121no—
312C2×C3×C4D3×D411yescharacteristic
313C2×C3×C4C2×#6711yescharacteristic
314#67—41no—
315#67—41no—
316C2×Dic3—21no—
317C2×Dic3—21no—
318C2×Dic3—21no—
319C2×Dic3—21no—
320C2×Dic3—21no—
321C2×Dic3—21no—
322C2×Dic3—41no—
323C2×Dic3—41no—
324C2×Dic3—121no—
325C2×Dic3—121no—
326C2×Dic3—241no—
327C2×Dic3#21711yescharacteristic
328C2×Dic3#21711yescharacteristic
329C2×Dic3#21711yescharacteristic
330C4×D3—21no—
331C4×D3—21no—
332Dic6—41no—
333Dic6—42no—
334Dic6—41no—
335Dic6—41no—
336C3×C8—121no—
337C3⋊C8—41no—
338C3⋊C8—121no—
339C3⋊C8—121no—
340C3⋊C8—241no—
341C2×C32—21no—
342C2×C32—41no—
343C2×C32—41no—
344C2×C32C2×#9911yescharacteristic
345C3×D3—41no—
346#41—121no—
347#41—121no—
348C2×Q8—31no—
349C2×Q8—61no—
350C2×Q8—61no—
351C2×Q8—121no—
352C2×Q8—121no—
353C2×D4—31no—
354C2×D4—31no—
355C2×D4C2×D3211yescharacteristic
356C22×C4—31no—
357C22×C4—31no—
358C22×C4—31no—
359C22×C4—31no—
360C22×C4D3×Dic311yescharacteristic
361M16—61no—
362M16—181no—
363M16—361no—
364C2×C8—361no—
365C4⋊C4—21no—
366C4⋊C4—61no—
367C4⋊C4—61no—
368C4⋊C4—61no—
369C4⋊C4—61no—
370C4⋊C4—361no—
371C22⋊C4—21no—
372C22⋊C4—31no—
373C22⋊C4—61no—
374C22⋊C4—61no—
375C22⋊C4—61no—
376C22⋊C4—61no—
377C22⋊C4—181no—
378C22⋊C4D3×Dic311yescharacteristic
379C42—61no—
380C42—361no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—41no—
387C22×C3—41no—
388C22×C3—41no—
389C22×C3—41no—
390C22×C3—41no—
391C22×C3—41no—
392C22×C3—61no—
393C22×C3—81no—
394C22×C3—81no—
395C22×C3—121no—
396C22×C3D3×C22⋊C411yescharacteristic
397C22×C3C2×#21711yescharacteristic
398C2×D3—21no—
399C2×D3—41no—
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—31no—
403C3×C4—31no—
404C3×C4—41no—
405C3×C4—41no—
406C3×C4—41no—
407C3×C4—41no—
408C3×C4—41no—
409C3×C4—61no—
410C3×C4—61no—
411C3×C4—81no—
412C3×C4—81no—
413C3×C4—121no—
414C3×C4—121no—
415C3×C4—241no—
416Dic3—21no—
417Dic3—21no—
418Dic3—41no—
419Dic3—41no—
420Dic3—81no—
421Dic3—241no—
422Dic3—241no—
423Dic3—481no—
424Dic3#83411yescharacteristic
425Dic3#83411yescharacteristic
426C32#212311yescharacteristic, complemented, Sylow
427C23—31no—
428C23C2×D3×Dic311yescharacteristic
429C23D3×#6711yescharacteristic
430Q8—121no—
431Q8—121no—
432Q8—121no—
433Q8—122no—
434D4—41no—
435D4—121no—
436D4—121no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—31no—
440C2×C4—31no—
441C2×C4—31no—
442C2×C4—41no—
443C2×C4—61no—
444C2×C4—61no—
445C2×C4—61no—
446C2×C4—61no—
447C2×C4—61no—
448C2×C4—61no—
449C2×C4—61no—
450C2×C4—61no—
451C2×C4—121no—
452C2×C4—121no—
453C2×C4—361no—
454C2×C4D3×#6711yescharacteristic
455C8—121no—
456C8—361no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—41no—
461C2×C3—41no—
462C2×C3—41no—
463C2×C3—41no—
464C2×C3—41no—
465C2×C3—81no—
466C2×C3—81no—
467C2×C3—121no—
468C2×C3D3×#9911yescharacteristic
469C2×C3C2×#83411yescharacteristic
470D3—41no—
471C22—21no—
472C22—21no—
473C22—41no—
474C22—41no—
475C22—61no—
476C22—121no—
477C22D3×#21711yescharacteristic
478C4—21no—
479C4—31no—
480C4—31no—
481C4—41no—
482C4—41no—
483C4—61no—
484C4—61no—
485C4—121no—
486C4—121no—
487C4—241no—
488C4—721no—
489C3—21no—
490C3#6914511yescharacteristic, complemented
491C3#7187011yescharacteristic, complemented
492C2—21no—
493C2—41no—
494C2—41no—
495C2—121no—
496C2D3×#83411yescharacteristic
subgroupG/H
centerC2D3×#834
commutatorC23×C32C22×C4
FrattiniC2×D4C2×D32
FittingC32×#123C22
socleC2×C32C2×#99
prime radicalC2×C3×#237C22
cosocleC2×C32×D4C23
hypercenter#123D32
hypocenterC32#2123
LCSG ▷ C23×C32 ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #123

Complements

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

NH ≅ G/Ntype
4. #10589405C2semidirect
5. C3×#4539C2semidirect
6. #10589176C2semidirect
7. #10589175C2semidirect
11. C22×C3×Dic6C4semidirect
14. C3×#929C4semidirect
17. C3×#886C4semidirect
20. C3×#877C4semidirect
41. #4539D3semidirect
83. C22×Dic6Dic3semidirect
96. #929Dic3semidirect
102. #886Dic3semidirect
106. #877Dic3semidirect
426. C32#2123semidirect
490. C3#69145semidirect
491. C3#71870semidirect

Maximal subgroups

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

subgroupindex
C3×#56482
#105915282
#105915012
#105894052
C3×#45392
#105891762
#105891752
#718703
#691453