Finite Group Databasebeta

← #36709146

Subgroups — #36709146

Order 22784.

534 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26030792C211yesmaximal, characteristic, complemented
2#26029128C211yesmaximal, characteristic
3#26022414C212yesmaximal, complemented
4#26022411C212yesmaximal, complemented
5#26022397C212yesmaximal, complemented
6#26022392C212yesmaximal, complemented
7#26022041C211yesmaximal, characteristic, complemented
8C2×#19151426C211yesmaximal, characteristic, complemented
9#26022010C211yesmaximal, characteristic, complemented
10#26021964C211yesmaximal, characteristic
11C89×#2138C211yesmaximal, characteristic
12#19152226—21no—
13#19152081—21no—
14C2×#15667952C2211yescharacteristic
15#19152005—22no—
16#19152002C2212yes—
17#19151999C2212yes—
18#19151699—22no—
19Q16×Dic89C2212yes—
20D8×Dic89C2212yes—
21#19151641—21no—
22C2×#15667946C2211yescharacteristic
23C2×#15667946C2211yescharacteristic
24#19151599—21no—
25#19151553C2211yescharacteristic
26#19151426C2212yes—
27#19151426C2212yes—
28#19151425C2212yes—
29#19151423C2212yes—
30#19151416C2212yes—
31#19151416C2212yes—
32#19151402C2212yes—
33#19151399C2212yes—
34C2×#135×C89C2211yescharacteristic
35C22×C16×C89C2211yescharacteristic
36#427×C89C2211yescharacteristic
37#358×C89C2211yescharacteristic
38#358×C89C2211yescharacteristic
39#357×C89C2211yescharacteristic
40#357×C89C2211yescharacteristic
41C2×#13083558D411yescharacteristic
42#15668012—41no—
43#15668011—22no—
44#15668004—41no—
45#15668003—22no—
46#15667989—42no—
47Q8×Dic89—22no—
48D4×Dic89—22no—
49D4×Dic89—42no—
50#15667971—41no—
51#15667971D411yescharacteristic
52C2×C4⋊Dic89—21no—
53C2×C4⋊Dic89—21no—
54C2×C4⋊Dic89—41no—
55#15667952—22no—
56C2×#13083549C2312yes—
57#15667946—21no—
58#15667946—21no—
59#15667946C2312yes—
60#15667946C2312yes—
61#15667945—22no—
62#15667943—22no—
63#15667936—22no—
64#15667933—22no—
65#15667922D412yes—
66C8×Dic89C2312yes—
67C2×#41×C89—21no—
68#135×C89—22no—
69#135×C89Q814yescomplemented
70C2×Q16×C89C2311yescharacteristic
71C2×SD16×C89—21no—
72C2×D8×C89C2311yescharacteristic
73C22×C8×C89C2311yescharacteristic
74#117×C89—21no—
75C2×C16×C89—22no—
76C2×C16×C89C2311yescharacteristic
77C2×C16×C89C2311yescharacteristic
78#107×C89C2311yescharacteristic
79#107×C89C2311yescharacteristic
80#106×C89—21no—
81C2×Dic178—42no—
82C22×Dic89—41no—
83C22×Dic89—41no—
84C22×Dic89—42no—
85#13083558—22no—
86C2×#10623494C2×D412yes—
87C4⋊Dic89—22no—
88C4⋊Dic89—22no—
89C4⋊Dic89—41no—
90C4⋊Dic89—41no—
91C4⋊Dic89—42no—
92C4⋊Dic89—42no—
93C4⋊Dic89D811yescharacteristic
94C4⋊Dic89D811yescharacteristic
95#13083553—42no—
96#13083553—42no—
97C22⋊Dic89—42no—
98C22⋊Dic89D812yes—
99C4×Dic89—42no—
100C4×Dic89C2×D412yes—
101#13083549—22no—
102#41×C89—24no—
103#41×C89—42no—
104C2×Q8×C89—21no—
105C2×D4×C89—21no—
106C2×D4×C89—41no—
107C22×C4×C89—41no—
108C22×C4×C89C2×D411yescharacteristic
109Q16×C89—21no—
110Q16×C89C2×Q812yes—
111SD16×C89—24no—
112D8×C89—21no—
113D8×C89C2×Q812yes—
114C2×C8×C89C2×D411yescharacteristic
115C2×C8×C89C2×Q812yes—
116C2×C8×C89#4112yes—
117C2×C8×C89C2411yescharacteristic
118C4⋊C4×C89—21no—
119C4⋊C4×C89—21no—
120C22⋊C4×C89—41no—
121C42×C89—41no—
122C16×C89—21no—
123C16×C89—21no—
124C23×C89—41no—
125C23×C89C2×D811yescharacteristic
126Q8×C89—22no—
127Q8×C89—41no—
128D4×C89—22no—
129D4×C89—44no—
130D4×C89—41no—
131C2×C4×C89—41no—
132C2×C4×C89—44no—
133C2×C4×C89—41no—
134C2×C4×C89—41no—
135C2×C4×C89C22⋊Q812yes—
136C2×C4×C89C2×D811yescharacteristic
137C2×C4×C89#13712yes—
138C2×C4×C89C22×D411yescharacteristic
139C2×Dic89—42no—
140C2×Dic89—42no—
141C2×Dic89—162no—
142C2×Dic89—161no—
143C2×Dic89—161no—
144C2×Dic89C2×D812yes—
145C2×Dic89C2×D812yes—
146Dic178—162no—
147C8×C89C22⋊Q812yes—
148C8×C89C22×Q811yescharacteristic
149C8×C89C2×#4111yescharacteristic
150#10623494—22no—
151C22×C89—21no—
152C22×C89—21no—
153C22×C89—42no—
154C22×C89—81no—
155C22×C89—81no—
156C22×C89#47412yes—
157C22×C89C22×D811yescharacteristic
158C4×C89—42no—
159C4×C89—81no—
160C4×C89—81no—
161C4×C89#47412yes—
162C4×C89C2×C22⋊Q811yescharacteristic
163C4×C89C2×#13711yescharacteristic
164Dic89—22no—
165Dic89—22no—
166Dic89—82no—
167Dic89—82no—
168#33634—891nomaximal, Sylow
169C2×C89—22no—
170C2×C89—82no—
171C2×C89C2×#47411yescharacteristic
172C2×C89#341611yescharacteristic
173C2×C89#341611yescharacteristic
174#3164—891no—
175#2951—891no—
176#2229—892no—
177#2228—892no—
178#2226—892no—
179#2225—892no—
180#2159—891no—
181C2×#367—891no—
182#2151—891no—
183#2139—891no—
184#2138D8911yescharacteristic
185C89#3363411yescharacteristic, complemented, Sylow
186C2×#135C2×D8911yescharacteristic
187#544—1781no—
188#520—1781no—
189C2×M32—891no—
190C22×C16C2×D8911yescharacteristic
191#501—1782no—
192#500—892no—
193#499—892no—
194#440—1782no—
195C4×Q16—892no—
196C4×D8—892no—
197#428—1781no—
198#427C2×D8911yescharacteristic
199C2×#107—891no—
200C2×#107—891no—
201#417—1781no—
202#405—891no—
203#367—892no—
204#367—892no—
205#367—892no—
206#366—892no—
207C4⋊C16—892no—
208C4⋊C16—892no—
209#358—892no—
210#358C2×D8911yescharacteristic
211#358C2×D8911yescharacteristic
212#357—892no—
213#357C2×D8911yescharacteristic
214#357C2×D8911yescharacteristic
215C2×#41—21no—
216#135—22no—
217#135Dic17814yescomplemented
218C2×Q16C22×D8911yescharacteristic
219C2×SD16—21no—
220C2×D8C22×D8911yescharacteristic
221C2×M16—891no—
222C22×C8C22×D8911yescharacteristic
223C4⋊Q8—3561no—
224C4⋊Q8—1782no—
225#125—3561no—
226#125—1782no—
227C22⋊Q8—3562no—
228C4×Q8—1782no—
229C4×D4—3562no—
230C4×D4—1782no—
231#117—21no—
232#117—891no—
233#117—3561no—
234C2×C4⋊C4—3561no—
235C2×C4⋊C4—1781no—
236C2×C4⋊C4—1781no—
237M32—1782no—
238C2×C16—22no—
239C2×C16—892no—
240C2×C16C22×D8911yescharacteristic
241C2×C16C22×D8911yescharacteristic
242#107—892no—
243#107—892no—
244#107—1782no—
245#107—1781no—
246#107—1781no—
247#107C22×D8911yescharacteristic
248#107C22×D8911yescharacteristic
249#106—21no—
250#106—1782no—
251#103—1782no—
252#102—1782no—
253#97—892no—
254C4×C8—892no—
255#41—24no—
256#41—42no—
257C2×Q8—21no—
258C2×Q8—3562no—
259C2×D4—21no—
260C2×D4—41no—
261C22×C4—41no—
262C22×C4—3562no—
263C22×C4—3561no—
264C22×C4—3561no—
265C22×C4D4×D8911yescharacteristic
266Q16—21no—
267Q16C2×Dic17812yes—
268SD16—24no—
269D8—21no—
270D8C2×Dic17812yes—
271M16—1782no—
272C2×C8—892no—
273C2×C8D4×D8911yescharacteristic
274C2×C8C2×Dic17812yes—
275C2×C8#1308357312yes—
276C2×C8C23×D8911yescharacteristic
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—891no—
280C4⋊C4—891no—
281C4⋊C4—3562no—
282C4⋊C4—3562no—
283C4⋊C4—3562no—
284C4⋊C4—3562no—
285C4⋊C4—3561no—
286C4⋊C4—3561no—
287C4⋊C4—1782no—
288C4⋊C4—1782no—
289C22⋊C4—41no—
290C22⋊C4—892no—
291C22⋊C4—3562no—
292C42—41no—
293C42—892no—
294C42—3562no—
295C16—21no—
296C16—21no—
297C16—1782no—
298C23—41no—
299C23D8×D8911yescharacteristic
300Q8—22no—
301Q8—41no—
302Q8—14242no—
303D4—22no—
304D4—44no—
305D4—41no—
306C2×C4—41no—
307C2×C4—44no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—892no—
311C2×C4—892no—
312C2×C4—3562no—
313C2×C4—3562no—
314C2×C4—14242no—
315C2×C4—14241no—
316C2×C4—14241no—
317C2×C4#1566798912yes—
318C2×C4D8×D8911yescharacteristic
319C2×C4#1566804112yes—
320C2×C4C2×D4×D8911yescharacteristic
321C8—1782no—
322C8#1566798912yes—
323C8C22×Dic17811yescharacteristic
324C8C2×#1308357311yescharacteristic
325C22—21no—
326C22—21no—
327C22—42no—
328C22—81no—
329C22—81no—
330C22#1915187612yes—
331C22C2×D8×D8911yescharacteristic
332C4—42no—
333C4—81no—
334C4—81no—
335C4—1782no—
336C4—1782no—
337C4—7122no—
338C4—7122no—
339C4#1915187612yes—
340C4C2×#1566798911yescharacteristic
341C4C2×#1566804111yescharacteristic
342C2—22no—
343C2—82no—
344C2C2×#1915187611yescharacteristic
345C2#2603339511yescharacteristic
346C2#2603339511yescharacteristic
subgroupG/H
centerC22C2×D8×D89
commutator, cosocleC2×C8×C89C24
FrattiniC2×C8C23×D89
FittingC89×#2138C2
socleC22×C89C22×D8
prime radicalC2×#135×C89C22
hypercenter#2138D89
hypocenterC89#33634
LCSG ▷ C2×C8×C89 ▷ C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
1. #26030792C2semidirect
3. #26022414C2semidirect
4. #26022411C2semidirect
5. #26022397C2semidirect
6. #26022392C2semidirect
7. #26022041C2semidirect
8. C2×#19151426C2semidirect
9. #26022010C2semidirect
69. #135×C89Q8semidirect
185. C89#33634semidirect
217. #135Dic178semidirect

Maximal subgroups

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

subgroupindex
#260307922
#260291282
#260224142
#260224112
#260223972
#260223922
#260220412
C2×#191514262
#260220102
#260219642
C89×#21382
#3363489