Finite Group Databasebeta

← #44409111

Subgroups — #44409111

Order 29056.

704 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 133 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
1#28409869C211yesmaximal, characteristic
2#137×D227C212yesmaximal, complemented
3#28409607C211yesmaximal, characteristic, complemented
4#28409217C211yesmaximal, characteristic, complemented
5#28409215C211yesmaximal, characteristic, complemented
6#28409214C211yesmaximal, characteristic, complemented
7#28409211C212yesmaximal, complemented
8#28409208C212yesmaximal, complemented
9#28409047C212yesmaximal, complemented
10#28409015C211yesmaximal, characteristic, complemented
11#456×C227C211yesmaximal, characteristic, complemented
12#20300336—22no—
13#20300336C2211yescharacteristic
14#20300334—22no—
15#20300334—22no—
16#20300334C2212yes—
17C2×#16813885—22no—
18C2×#16813885C2211yescharacteristic
19#41×D227—22no—
20#41×D227—22no—
21#41×D227C2212yes—
22C2×Q8×D227C2211yescharacteristic, complemented
23#20300321C2212yescomplemented
24#20300320—22no—
25#20300319C2212yescomplemented
26#20300318C2212yescomplemented
27#20300317—22no—
28Q16×D227—22no—
29SD16×D227—22no—
30M16×D227C2212yescomplemented
31#20300286C2211yescharacteristic, complemented
32#20300277C2211yescharacteristic, complemented
33#20300276C2211yescharacteristic, complemented
34#20300273—22no—
35#20300266—22no—
36#20300245C2211yescharacteristic, complemented
37#20300216C2212yescomplemented
38#20300215C2212yescomplemented
39#20300214C2212yescomplemented
40#20300207C2211yescharacteristic, complemented
41#20300204C2212yescomplemented
422−1+4×C227C2211yescharacteristic
43#137×C227C2212yescomplemented
44C4⋊Q8×C227C2211yescharacteristic, complemented
45#104×C227C2212yescomplemented
46#101×C227C2211yescharacteristic, complemented
47#16813885—22no—
48#16813885—24no—
49#16813885—22no—
50#16813885—22no—
51#16813885—22no—
52#16813885—22no—
53#16813885D412yes—
54#16813884—24no—
55#16813884—24no—
56#16813884—22no—
57#16813884—22no—
58#16813884—22no—
59#16813884C2312yes—
60#16813883—22no—
61#16813883—22no—
62#16813883D412yes—
63C2×Dic454—21no—
64C2×Dic454C2311yescharacteristic
65Q8×D227—22no—
66Q8×D227—22no—
67Q8×D227—22no—
68Q8×D227—22no—
69Q8×D227D412yes—
70C2×D908—24no—
71C2×D908—22no—
72C2×D908D412yes—
73C2×D908C2311yescharacteristic
74D4×D227—24no—
75D4×D227—22no—
76D4×D227—24no—
77D4×D227—22no—
78D4×D227—22no—
79D4×D227D412yes—
80C2×C4×D227—24no—
81C2×C4×D227—22no—
82C2×C4×D227—21no—
83C2×C4×D227D412yes—
84C2×C4×D227C2311yescharacteristic
85Dic908—22no—
86#16813874—22no—
87#16813874—22no—
88#16813873—22no—
89#16813872—22no—
90#16813871—22no—
91#16813868C2312yes—
92#16813867—22no—
93C8×D227—22no—
94Dic227⋊C4—21no—
95Dic227⋊C4—22no—
96(C2×D227)⋊C4—21no—
97(C2×D227)⋊C4—22no—
98(C2×D227)⋊C4—22no—
99C4×Dic227C2311yescharacteristic
100#41×C227—22no—
101#41×C227—22no—
102#41×C227C2312yes—
103C2×Q8×C227—22no—
104C2×Q8×C227C2311yescharacteristic
105C2×Q8×C227C2311yescharacteristic
106Q16×C227—22no—
107SD16×C227—22no—
108M16×C227C2312yes—
109C4⋊C4×C227—22no—
110C42×C227C2311yescharacteristic
111C22×D227—24no—
112C22×D227—22no—
113C22×D227C2×D412yes—
114C22×D227C2×D411yescharacteristic
115Dic454—22no—
116Dic454—22no—
117Dic454—22no—
118Dic454—41no—
119Dic454—41no—
120Dic454C2×D412yes—
121D908—24no—
122D908—22no—
123D908—22no—
124D908—24no—
125D908—22no—
126D908C2×D412yes—
127C22⋊D227—24no—
128C22⋊D227—22no—
129C22⋊D227—22no—
130C22⋊D227—22no—
131C2×Dic227—22no—
132C2×Dic227—21no—
133C2×Dic227—21no—
134C2×Dic227C2×D411yescharacteristic
135C4×D227—24no—
136C4×D227—24no—
137C4×D227—22no—
138C4×D227—22no—
139C4×D227—22no—
140C4×D227—22no—
141C4×D227—22no—
142C4×D227—42no—
143C4×D227C2×D412yes—
144C4×D227C2×D412yes—
145C227⋊C8—22no—
146Q8×C227—22no—
147Q8×C227—22no—
148Q8×C227—22no—
149Q8×C227C2×D412yes—
150D4×C227—22no—
151D4×C227—22no—
152D4×C227C2×D412yes—
153C2×C4×C227—24no—
154C2×C4×C227—22no—
155C2×C4×C227—21no—
156C2×C4×C227—21no—
157C2×C4×C227C2×D412yes—
158C2×C4×C227C2411yescharacteristic
159C8×C227—22no—
160C22×C227—22no—
161C22×C227C22×D411yescharacteristic
162C2×D227—22no—
163C2×D227—22no—
164C2×D227—41no—
165C2×D227—42no—
166C2×D227—82no—
167C2×D227—84no—
168C2×D227#12012yes—
169C4×C227—22no—
170C4×C227—22no—
171C4×C227—41no—
172C4×C227—41no—
173C4×C227C22×D412yes—
174Dic227—22no—
175Dic227—41no—
176Dic227—41no—
177Dic227#12011yescharacteristic
178Dic227#12011yescharacteristic
179C2×C227—21no—
180C2×C227—42no—
181C2×C227C2×#12011yescharacteristic
182D227—22no—
183D227—42no—
184D227—42no—
185C227#302211yescharacteristic, complemented, Sylow
186#3022—2271nomaximal, Sylow
187#585—2271no—
188C2×#137—2272no—
189#536—2271no—
190#456—2271no—
191#456—2271no—
192#456—2271no—
193#456D22711yescharacteristic, complemented
194#455—2272no—
195#455—2272no—
196#421—2272no—
197#413—2271no—
1982−1+4—4542no—
1992−1+4—2271no—
2002−1+4C2×D22711yescharacteristic
201H(5,𝔽2)—4542no—
202H(5,𝔽2)—4542no—
203H(5,𝔽2)—2272no—
204C2×#41—4542no—
205C2×#41—4542no—
206C2×#41—4542no—
207C2×#41—2272no—
208C2×#41—2271no—
209C22×Q8—2271no—
210#137—4542no—
211#137—4542no—
212#137—2272no—
213#137—2272no—
214#137—2272no—
215#137C2×D22712yescomplemented
216C2×Q16—4542no—
217C2×SD16—4542no—
218C2×M16—2272no—
219C4⋊Q8—2271no—
220C4⋊Q8C2×D22711yescharacteristic, complemented
221#124—2271no—
222#124—2271no—
223#123—4542no—
224C22⋊Q8—4542no—
225#117—2271no—
226#104—2272no—
227#104—2272no—
228#104—2272no—
229#104C2×D22712yescomplemented
230#101—2271no—
231#101C2×D22711yescharacteristic, complemented
232#100—2272no—
233#41—22no—
234#41—22no—
235#41—4544no—
236#41—4544no—
237#41—4542no—
238#41—4542no—
239#41—4542no—
240#41—4542no—
241#41—4544no—
242#41—4542no—
243#41—4542no—
244#41—4542no—
245#41—4542no—
246#41—4542no—
247#41—4542no—
248#41—2272no—
249#41—2272no—
250#41—2272no—
251#41C22×D22712yes—
252C2×Q8—22no—
253C2×Q8—4541no—
254C2×Q8—4542no—
255C2×Q8—4542no—
256C2×Q8—4542no—
257C2×Q8—4542no—
258C2×Q8—2272no—
259C2×Q8—2271no—
260C2×Q8C22×D22711yescharacteristic
261C2×Q8C22×D22711yescharacteristic
262C2×D4—4544no—
263C2×D4—4542no—
264C2×D4—4544no—
265C2×D4—4542no—
266C2×D4—4544no—
267C2×D4—4542no—
268C2×D4—4542no—
269C2×D4—2272no—
270C2×D4—2272no—
271C2×D4—2271no—
272C22×C4—4541no—
273C22×C4—4544no—
274C22×C4—4542no—
275C22×C4—2272no—
276C22×C4—2271no—
277Q16—22no—
278Q16—4542no—
279Q16—4542no—
280Q16—4542no—
281SD16—22no—
282SD16—4542no—
283SD16—4542no—
284SD16—4542no—
285M16—4542no—
286M16—2272no—
287M16C22×D22712yes—
288C2×C8—4542no—
289C4⋊C4—22no—
290C4⋊C4—4541no—
291C4⋊C4—4542no—
292C22⋊C4—4541no—
293C22⋊C4—4542no—
294C22⋊C4—4542no—
295C42—2271no—
296C42C22×D22711yescharacteristic
297C23—4544no—
298C23—4542no—
299C23—2272no—
300C23—2271no—
301Q8—22no—
302Q8—22no—
303Q8—22no—
304Q8—9081no—
305Q8—9081no—
306Q8—4542no—
307Q8—4542no—
308Q8—4542no—
309Q8—2272no—
310Q8D4×D22712yes—
311D4—22no—
312D4—22no—
313D4—4544no—
314D4—4542no—
315D4—4542no—
316D4—4544no—
317D4—4544no—
318D4—4542no—
319D4—4542no—
320D4—4542no—
321D4—4542no—
322D4—2272no—
323D4D4×D22712yes—
324C2×C4—24no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—9082no—
329C2×C4—4544no—
330C2×C4—4544no—
331C2×C4—4542no—
332C2×C4—4542no—
333C2×C4—4542no—
334C2×C4—4542no—
335C2×C4—4541no—
336C2×C4—4541no—
337C2×C4—4542no—
338C2×C4—4542no—
339C2×C4—2271no—
340C2×C4—2272no—
341C2×C4—2272no—
342C2×C4D4×D22712yes—
343C2×C4C23×D22711yescharacteristic
344C8—22no—
345C8—4542no—
346C22—22no—
347C22—18164no—
348C22—18162no—
349C22—9082no—
350C22—9081no—
351C22—4542no—
352C22—4542no—
353C22—2272no—
354C22C2×D4×D22711yescharacteristic
355C4—22no—
356C4—22no—
357C4—41no—
358C4—41no—
359C4—9081no—
360C4—9081no—
361C4—4542no—
362C4—2271no—
363C4—2271no—
364C4C2×D4×D22712yes—
365C2—21no—
366C2—42no—
367C2—9082no—
368C2—9082no—
369C2—4542no—
370C2#120×D22711yescharacteristic
subgroupG/H
centerC2#120×D227
commutator, cosocleC2×C4×C227C24
FrattiniC2×C4C23×D227
Fitting#456×C227C2
socleC2×C227C2×#120
prime radical#28409869C2
hypercenter#456D227
hypocenterC227#3022
LCSG ▷ C2×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
2. #137×D227C2semidirect
3. #28409607C2semidirect
4. #28409217C2semidirect
5. #28409215C2semidirect
6. #28409214C2semidirect
7. #28409211C2semidirect
8. #28409208C2semidirect
9. #28409047C2semidirect
10. #28409015C2semidirect
11. #456×C227C2semidirect
22. C2×Q8×D227C22semidirect
23. #20300321C22semidirect
25. #20300319C22semidirect
26. #20300318C22semidirect
30. M16×D227C22semidirect
31. #20300286C22semidirect
32. #20300277C22semidirect
33. #20300276C22semidirect
36. #20300245C22semidirect
37. #20300216C22semidirect
38. #20300215C22semidirect
39. #20300214C22semidirect
40. #20300207C22semidirect
41. #20300204C22semidirect
43. #137×C227C22semidirect
44. C4⋊Q8×C227C22semidirect
45. #104×C227C22semidirect
46. #101×C227C22semidirect
185. C227#3022semidirect
193. #456D227semidirect
215. #137C2×D227semidirect
220. C4⋊Q8C2×D227semidirect
229. #104C2×D227semidirect
231. #101C2×D227semidirect

Maximal subgroups

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

subgroupindex
#284098692
#137×D2272
#284096072
#284092172
#284092152
#284092142
#284092112
#284092082
#284090472
#284090152
#456×C2272
#3022227