Finite Group Databasebeta

← #44408556

Subgroups — #44408556

Order 29056.

484 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20300317C211yesmaximal, characteristic
2C4×#16813883C211yesmaximal, characteristic
3C4×Q8×D227C211yesmaximal, characteristic, complemented
4#28409140C211yesmaximal, characteristic, complemented
5#28409137C211yesmaximal, characteristic, complemented
6#28409132C211yesmaximal, characteristic, complemented
7#28409126C211yesmaximal, characteristic, complemented
8#28409124C211yesmaximal, characteristic, complemented
9#28409123C211yesmaximal, characteristic, complemented
10SD16×Dic227C211yesmaximal, characteristic, complemented
11#28409073C211yesmaximal, characteristic, complemented
12#28409039C211yesmaximal, characteristic, complemented
13#28409030C211yesmaximal, characteristic
14#28408984C211yesmaximal, characteristic, complemented
15#440×C227C211yesmaximal, characteristic, complemented
16C2×#16813883C2211yescharacteristic
17C2×Q8×D227C2211yescharacteristic
18#20300317C418yescomplemented
19C2×Dic908C2211yescharacteristic
20C2×#16813874C2211yescharacteristic
21C2×#16813872C2211yescharacteristic
22C2×#16813867C2211yescharacteristic
23C2×#16813867C2211yescharacteristic
24Q8×Dic227C2211yescharacteristic
25C4×Dic454—21no—
26C4×Dic454C2211yescharacteristic
27C4×Dic454C2211yescharacteristic
28#20300253—21no—
29#20300253C2211yescharacteristic
30#20300253C2211yescharacteristic
31D4×Dic227C2211yescharacteristic
32C4×C22⋊D227—21no—
33(C22⋊D227)⋊C4—21no—
34#20300246C2211yescharacteristic
35#20300244—21no—
36C4⋊C4×D227—21no—
37C4⋊C4×D227C2211yescharacteristic
38C2×C4×Dic227—21no—
39C42×D227—21no—
40C42×D227C2211yescharacteristic
41#20300221C2211yescharacteristic
42#20300220C2211yescharacteristic
43#20300219C2211yescharacteristic
44#20300213C2211yescharacteristic
45#20300213C2211yescharacteristic
46#20300212C2211yescharacteristic
47#20300211C2211yescharacteristic
48#20300211C2211yescharacteristic
49#20300209C2211yescharacteristic
50#20300199C2211yescharacteristic
51C8×Dic227C2211yescharacteristic
52C8×Dic227C2211yescharacteristic
53C2×SD16×C227C2211yescharacteristic
54C4×Q8×C227C2211yescharacteristic
55C4×D4×C227C2211yescharacteristic
56#107×C227C2211yescharacteristic
57#103×C227C2211yescharacteristic
58#102×C227C2211yescharacteristic
59#97×C227C2211yescharacteristic
60#16813883—22no—
61#16813883C2×C414yes—
62C2×Dic454—21no—
63C2×Dic454C2311yescharacteristic
64C2×Dic454C2311yescharacteristic
65Q8×D227—22no—
66Q8×D227C2×C414yescomplemented
67C2×C22⋊D227—21no—
68C22×Dic227—21no—
69C22×Dic227—21no—
70C2×C4×D227—21no—
71C2×C4×D227—21no—
72C2×C4×D227D411yescharacteristic
73C2×C4×D227D411yescharacteristic
74C2×C4×D227C2311yescharacteristic
75Dic908C2×C414yescomplemented
76#16813874C2×C414yescomplemented
77#16813872C2×C414yescomplemented
78#16813867—22no—
79#16813867C2×C414yescomplemented
80C2×C227⋊C8C2311yescharacteristic
81C2×C227⋊C8C2311yescharacteristic
82C4⋊Dic227—21no—
83C4⋊Dic227C2311yescharacteristic
84C4⋊Dic227C2311yescharacteristic
85Dic227⋊C4—21no—
86Dic227⋊C4—21no—
87Dic227⋊C4—21no—
88Dic227⋊C4—21no—
89Dic227⋊C4—21no—
90C22⋊Dic227—21no—
91(C2×D227)⋊C4—21no—
92C4×Dic227—21no—
93C4×Dic227—21no—
94C4×Dic227—21no—
95C4×Dic227—21no—
96C4×Dic227—21no—
97C4×Dic227—21no—
98C4×Dic227D411yescharacteristic
99C4×Dic227D411yescharacteristic
100C4×Dic227C2311yescharacteristic
101C2×Q8×C227C2311yescharacteristic
102C2×D4×C227C2311yescharacteristic
103C22×C4×C227—21no—
104SD16×C227C2×C414yescomplemented
105C2×C8×C227C2311yescharacteristic
106C2×C8×C227C2311yescharacteristic
107C4⋊C4×C227—21no—
108C4⋊C4×C227C2311yescharacteristic
109C4⋊C4×C227C2311yescharacteristic
110C22⋊C4×C227—21no—
111C42×C227—21no—
112C42×C227C2311yescharacteristic
113C22×D227C2×D411yescharacteristic
114Dic454—21no—
115Dic454—21no—
116Dic454—24no—
117Dic454C22×C412yes—
118Dic454C22×C412yes—
119C22⋊D227—24no—
120C2×Dic227—21no—
121C2×Dic227—21no—
122C2×Dic227—24no—
123C2×Dic227—21no—
124C2×Dic227—21no—
125C2×Dic227—42no—
126C2×Dic227C2×D411yescharacteristic
127C2×Dic227C2×D411yescharacteristic
128C2×Dic227C2×D411yescharacteristic
129C4×D227—24no—
130C4×D227—22no—
131C4×D227—41no—
132C4×D227—42no—
133C4×D227C22×C412yes—
134C4×D227#4112yes—
135C227⋊C8—21no—
136C227⋊C8C22×C412yes—
137C23×C227—21no—
138Q8×C227—21no—
139Q8×C227C22×C412yes—
140D4×C227—21no—
141D4×C227C22×C412yes—
142C2×C4×C227—21no—
143C2×C4×C227—21no—
144C2×C4×C227—21no—
145C2×C4×C227—42no—
146C2×C4×C227C2×D411yescharacteristic
147C2×C4×C227C2×D411yescharacteristic
148C2×C4×C227C2411yescharacteristic
149C8×C227—21no—
150C8×C227C22×C412yes—
151C22×C227—22no—
152C22×C227—41no—
153C22×C227—41no—
154C22×C227C22×D411yescharacteristic
155C2×D227—21no—
156C2×D227—21no—
157C2×D227C4×D412yes—
158C4×C227—22no—
159C4×C227—21no—
160C4×C227—21no—
161C4×C227—41no—
162C4×C227—41no—
163C4×C227C23×C411yescharacteristic
164C4×C227C2×#4111yescharacteristic
165Dic227—22no—
166Dic227—22no—
167Dic227—21no—
168Dic227—41no—
169Dic227—41no—
170Dic227C4×D412yes—
171Dic227#13712yescomplemented
172C2×C227—42no—
173C2×C227C2×C4×D411yescharacteristic
174C2×C227C2×#13711yescharacteristic
175C2×C227#57511yescharacteristic
176D227—22no—
177C227C4×#13711yescharacteristic, complemented, Sylow
178C4×#137—2271nomaximal, Sylow
179C2×#137—2271no—
180C4×#41—2271no—
181C2×C4×Q8—2271no—
182#441—2271no—
183#441—2271no—
184#440—2271no—
185#440D22711yescharacteristic, complemented
186C4×Q16—2271no—
187C4×Q16—2271no—
188C4×SD16—2271no—
189C4×SD16—2271no—
190#428—2271no—
191#419—2271no—
192#417—2271no—
193C4×M16—2271no—
194C2×#41—2271no—
195C22×Q8—2271no—
196#137—2278no—
197C2×Q16—2271no—
198C2×Q16—2271no—
199C2×SD16—2271no—
200C2×SD16C2×D22711yescharacteristic
201C2×M16—2271no—
202C2×M16—2271no—
203C4×Q8—4541no—
204C4×Q8—4541no—
205C4×Q8—2271no—
206C4×Q8—2271no—
207C4×Q8—2271no—
208C4×Q8—2271no—
209C4×Q8—2271no—
210C4×Q8C2×D22711yescharacteristic
211C4×D4—4541no—
212C4×D4—4541no—
213C4×D4—2271no—
214C4×D4C2×D22711yescharacteristic
215#117—4541no—
216#117—2271no—
217C2×C4⋊C4—4541no—
218C2×C4⋊C4—2271no—
219C2×C42—4541no—
220C2×C42—4541no—
221C2×C42—2271no—
222#107—2271no—
223#107C2×D22711yescharacteristic
224#106—2271no—
225#106—2271no—
226#103—2271no—
227#103—2271no—
228#103—2271no—
229#103—2271no—
230#103—2271no—
231#103C2×D22711yescharacteristic
232#102—2271no—
233#102C2×D22711yescharacteristic
234#97—2271no—
235#97C2×D22711yescharacteristic
236C4×C8—2271no—
237C4×C8—2271no—
238#41—4542no—
239#41—2274no—
240C2×Q8—4541no—
241C2×Q8—4542no—
242C2×Q8—2274no—
243C2×Q8—2271no—
244C2×Q8—2271no—
245C2×Q8C22×D22711yescharacteristic
246C2×D4—4541no—
247C2×D4C22×D22711yescharacteristic
248C22×C4—21no—
249C22×C4—4541no—
250C22×C4—4541no—
251C22×C4—4541no—
252C22×C4—4541no—
253C22×C4—2271no—
254C22×C4—2271no—
255C22×C4—2271no—
256Q16—2274no—
257Q16—2274no—
258SD16—2274no—
259SD16C4×D22714yescomplemented
260M16—4542no—
261M16—2274no—
262C2×C8—2271no—
263C2×C8—2271no—
264C2×C8C22×D22711yescharacteristic
265C2×C8C22×D22711yescharacteristic
266C4⋊C4—21no—
267C4⋊C4—4541no—
268C4⋊C4—4541no—
269C4⋊C4—4541no—
270C4⋊C4—4541no—
271C4⋊C4—4541no—
272C4⋊C4—4541no—
273C4⋊C4—2271no—
274C4⋊C4—2271no—
275C4⋊C4C22×D22711yescharacteristic
276C4⋊C4C22×D22711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—4541no—
279C22⋊C4—4541no—
280C42—21no—
281C42—4541no—
282C42—4541no—
283C42—4541no—
284C42—4541no—
285C42—4541no—
286C42—4541no—
287C42—2271no—
288C42—2271no—
289C42—2271no—
290C42C22×D22711yescharacteristic
291C23—21no—
292C23—2271no—
293Q8—21no—
294Q8—4544no—
295Q8—4541no—
296Q8—4541no—
297Q8—2272no—
298Q8—2272no—
299Q8C2×C4×D22712yes—
300D4—21no—
301D4—4544no—
302D4C2×C4×D22712yes—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—42no—
307C2×C4—9081no—
308C2×C4—9082no—
309C2×C4—9082no—
310C2×C4—4544no—
311C2×C4—4542no—
312C2×C4—4541no—
313C2×C4—4541no—
314C2×C4—4541no—
315C2×C4—4541no—
316C2×C4—4544no—
317C2×C4—2271no—
318C2×C4—2271no—
319C2×C4—2271no—
320C2×C4—2272no—
321C2×C4—2272no—
322C2×C4D4×D22711yescharacteristic
323C2×C4D4×D22711yescharacteristic
324C2×C4C23×D22711yescharacteristic
325C8—21no—
326C8—4541no—
327C8—2272no—
328C8C2×C4×D22712yes—
329C22—22no—
330C22—41no—
331C22—41no—
332C22—4541no—
333C22—4541no—
334C22—2272no—
335C22C2×D4×D22711yescharacteristic
336C4—22no—
337C4—21no—
338C4—21no—
339C4—41no—
340C4—41no—
341C4—9081no—
342C4—9081no—
343C4—4542no—
344C4—4542no—
345C4—4541no—
346C4—2272no—
347C4—2272no—
348C4C22×C4×D22711yescharacteristic
349C4#41×D22711yescharacteristic
350C2—42no—
351C2—4542no—
352C2C4×D4×D22711yescharacteristic
353C2#137×D22711yescharacteristic
354C2#2840982811yescharacteristic
subgroupG/H
centerC22C2×D4×D227
commutatorC4×C227C23×C4
FrattiniC2×C4C23×D227
Fitting#440×C227C2
socleC22×C227C22×D4
prime radicalC2×#16813883C22
cosocleC2×C4×C227C24
hypercenter#440D227
hypocenterC227C4×#137
LCSG ▷ C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
3. C4×Q8×D227C2semidirect
4. #28409140C2semidirect
5. #28409137C2semidirect
6. #28409132C2semidirect
7. #28409126C2semidirect
8. #28409124C2semidirect
9. #28409123C2semidirect
10. SD16×Dic227C2semidirect
11. #28409073C2semidirect
12. #28409039C2semidirect
14. #28408984C2semidirect
15. #440×C227C2semidirect
18. #20300317C4semidirect
66. Q8×D227C2×C4semidirect
75. Dic908C2×C4semidirect
76. #16813874C2×C4semidirect
77. #16813872C2×C4semidirect
79. #16813867C2×C4semidirect
104. SD16×C227C2×C4semidirect
171. Dic227#137semidirect
177. C227C4×#137semidirect
185. #440D227semidirect
259. SD16C4×D227semidirect

Maximal subgroups

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

subgroupindex
C2×#203003172
C4×#168138832
C4×Q8×D2272
#284091402
#284091372
#284091322
#284091262
#284091242
#284091232
SD16×Dic2272
#284090732
#284090392
#284090302
#284089842
#440×C2272
C4×#137227