Finite Group Databasebeta

← #36572023

Subgroups — #36572023

Order 22784.

704 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26028910C212yesmaximal
2#26021001C212yesmaximal, complemented
3#26020853C214yesmaximal, complemented
4#26020351C212yesmaximal
5#26020290C211yesmaximal, characteristic
6#26020133C212yesmaximal, complemented
7C8×C22⋊Dic89C211yesmaximal, characteristic
8C2×#427×C89C211yesmaximal, characteristic
9#19151659—24no—
10#19151655—22no—
11#19151651—22no—
12C2×#15667948C2212yes—
13#19151625C2214yes—
14#19151562C2212yes—
15C2×C22⋊#10623494C2211yescharacteristic
16#19151554C2214yes—
17C2×C8×Dic89C2212yes—
18C4×C22⋊Dic89C2211yescharacteristic
19#19151363C2212yes—
20#19151357C2214yes—
21#19151357C2214yes—
22#19151352C2212yes—
23C23×C8×C89C2211yescharacteristic
24C2×#117×C89C2212yes—
25#427×C89C411yescharacteristic, complemented
26#427×C89C411yescharacteristic, complemented
27#427×C89C412yescomplemented
28#427×C89C2211yescharacteristic
29#427×C89C2211yescharacteristic
30#427×C89C2212yes—
31C2×#107×C89C411yescharacteristic, complemented
32C2×#107×C89C411yescharacteristic, complemented
33C2×#106×C89C412yescomplemented
34C2×#13083558C2314yes—
35C22×#10623494C2312yes—
36C2×C22⋊Dic89—21no—
37C2×C4×Dic89C2312yes—
38#15667948—24no—
39C4⋊#10623494—24no—
40C4⋊#10623494—24no—
41C22⋊#10623494—22no—
42C22⋊#10623494—22no—
43C22⋊#10623494D411yescharacteristic
44C22⋊#10623494D411yescharacteristic
45C22⋊#10623494D412yes—
46#15667923—24no—
47C4×#10623494—24no—
48C8×Dic89—24no—
49#15667919—21no—
50C23×C4×C89C2311yescharacteristic
51C22×C8×C89—24no—
52C22×C8×C89C2×C411yescharacteristic
53C22×C8×C89C2×C411yescharacteristic
54C22×C8×C89C2×C411yescharacteristic
55C22×C8×C89C2×C411yescharacteristic
56C22×C8×C89C2311yescharacteristic
57C22×C8×C89C2311yescharacteristic
58#117×C89—22no—
59#117×C89—22no—
60#117×C89C2×C412yes—
61#117×C89C2×C412yes—
62#117×C89C2312yes—
63#117×C89C2312yes—
64C2×C4⋊C4×C89C2×C412yes—
65C2×C4⋊C4×C89C2×C412yes—
66C2×C22⋊C4×C89—22no—
67C2×C42×C89—22no—
68#107×C89C2×C412yes—
69#107×C89C2×C412yes—
70#107×C89C2×C411yescharacteristic
71#107×C89C2×C411yescharacteristic
72#107×C89C2×C411yescharacteristic
73#107×C89C2×C411yescharacteristic
74#106×C89C2×C412yes—
75#106×C89C2×C412yes—
76#106×C89C2×C412yes—
77#106×C89C2×C412yes—
78C22×Dic89—22no—
79#13083558—24no—
80#13083558—24no—
81C2×#10623494—22no—
82C2×#10623494—22no—
83C2×#10623494—24no—
84C2×#10623494C2×D412yes—
85C2×#10623494C2×D412yes—
86C22⋊Dic89—24no—
87C4×Dic89—22no—
88C4×Dic89—22no—
89C24×C89#4111yescharacteristic
90C22×C4×C89—22no—
91C22×C4×C89—22no—
92C22×C4×C89—22no—
93C22×C4×C89—22no—
94C22×C4×C89C22×C411yescharacteristic
95C22×C4×C89C22×C411yescharacteristic
96C22×C4×C89C2×D411yescharacteristic
97C22×C4×C89C2×D411yescharacteristic
98C22×C4×C89#4111yescharacteristic
99C22×C4×C89C2411yescharacteristic
100C2×C8×C89—22no—
101C2×C8×C89—22no—
102C2×C8×C89—22no—
103C2×C8×C89—22no—
104C2×C8×C89—22no—
105C2×C8×C89C22×C411yescharacteristic
106C2×C8×C89C22×C411yescharacteristic
107C2×C8×C89C22×C411yescharacteristic
108C2×C8×C89C22×C411yescharacteristic
109C2×C8×C89#4114yes—
110C4⋊C4×C89—22no—
111C4⋊C4×C89—22no—
112C4⋊C4×C89C22×C412yes—
113C4⋊C4×C89C22×C412yes—
114C4⋊C4×C89C22×C412yes—
115C4⋊C4×C89C22×C412yes—
116C22⋊C4×C89—22no—
117C22⋊C4×C89—22no—
118C22⋊C4×C89—22no—
119C22⋊C4×C89—22no—
120C42×C89—22no—
121C42×C89—22no—
122C42×C89—22no—
123C42×C89—22no—
124C23×C89—22no—
125C23×C89—22no—
126C23×C89—41no—
127C23×C89C4×D411yescharacteristic
128C23×C89C4×D411yescharacteristic
129C23×C89C2×#4111yescharacteristic
130C2×C4×C89—22no—
131C2×C4×C89—21no—
132C2×C4×C89—21no—
133C2×C4×C89—21no—
134C2×C4×C89—21no—
135C2×C4×C89—22no—
136C2×C4×C89—22no—
137C2×C4×C89—22no—
138C2×C4×C89—22no—
139C2×C4×C89—22no—
140C2×C4×C89—22no—
141C2×C4×C89—44no—
142C2×C4×C89—82no—
143C2×C4×C89C4×D411yescharacteristic
144C2×C4×C89C4×D411yescharacteristic
145C2×C4×C89C23×C411yescharacteristic
146C2×C4×C89C22×D411yescharacteristic
147C2×C4×C89C2×#4112yes—
148C2×C4×C892−1+412yes—
149C2×Dic89—24no—
150C2×Dic89—42no—
151C2×Dic89—42no—
152C8×C89—21no—
153C8×C89—21no—
154C8×C89#11712yes—
155C8×C89#11712yes—
156#10623494—22no—
157#10623494—22no—
158#10623494—44no—
159C22×C89—21no—
160C22×C89—21no—
161C22×C89—21no—
162C22×C89—21no—
163C22×C89—21no—
164C22×C89—21no—
165C22×C89—21no—
166C22×C89—44no—
167C22×C89#44312yes—
168C22×C89C2×C4×D411yescharacteristic
169C22×C89#55212yes—
170C4×C89—21no—
171C4×C89—21no—
172C4×C89—42no—
173C4×C89—42no—
174C4×C89—42no—
175C4×C89—42no—
176C4×C89C2×#11711yescharacteristic
177C4×C89#51911yescharacteristic
178C4×C89#54712yes—
179Dic89—44no—
180#22606—891nomaximal, Sylow
181C2×C89—22no—
182C2×C89—22no—
183C2×C89—41no—
184C2×C89#234411yescharacteristic
185C2×C89C2×#44311yescharacteristic
186C2×C89#295711yescharacteristic
187#2924—892no—
188C2×#427D8911yescharacteristic
189#1945—892no—
190#1922—894no—
191#1826—892no—
192#1810—891no—
193#1772—892no—
194C8×C22⋊C4—891no—
195C89#2260611yescharacteristic, complemented, Sylow
196C23×C8C2×D8911yescharacteristic
197C2×#117C2×D8912yes—
198#433—1784no—
199#432—1782no—
200#431—1782no—
201C2×#108—892no—
202#427Dic8911yescharacteristic, complemented
203#427Dic8911yescharacteristic, complemented
204#427Dic8912yescomplemented
205#427C2×D8912yes—
206#427C2×D8911yescharacteristic
207#427C2×D8911yescharacteristic
208C2×#107Dic8911yescharacteristic, complemented
209C2×#107Dic8911yescharacteristic, complemented
210C2×#106Dic8912yescomplemented
211#424—894no—
212#407—892no—
213C2×C22⋊C8—891no—
214#405—894no—
215C2×C4×C8—892no—
216C4×C22⋊C4—891no—
217#341—892no—
218#339—894no—
219#339—894no—
220#336—892no—
221C23×C4C22×D8911yescharacteristic
222C2×M16—894no—
223C22×C8—24no—
224C22×C8—892no—
225C22×C8C2×Dic8911yescharacteristic
226C22×C8C2×Dic8911yescharacteristic
227C22×C8C2×Dic8911yescharacteristic
228C22×C8C2×Dic8911yescharacteristic
229C22×C8C22×D8911yescharacteristic
230C22×C8C22×D8911yescharacteristic
231#117—22no—
232#117—22no—
233#117C2×Dic8912yes—
234#117C2×Dic8912yes—
235#117C22×D8912yes—
236#117C22×D8912yes—
237C2×C4⋊C4C2×Dic8912yes—
238C2×C4⋊C4C2×Dic8912yes—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—1781no—
241C2×C42—22no—
242C2×C42—892no—
243#108—1784no—
244#107C2×Dic8912yes—
245#107C2×Dic8912yes—
246#107C2×Dic8911yescharacteristic
247#107C2×Dic8911yescharacteristic
248#107C2×Dic8911yescharacteristic
249#107C2×Dic8911yescharacteristic
250#106C2×Dic8912yes—
251#106C2×Dic8912yes—
252#106C2×Dic8912yes—
253#106C2×Dic8912yes—
254C4⋊C8—1784no—
255C4⋊C8—1784no—
256C22⋊C8—891no—
257C22⋊C8—891no—
258C22⋊C8—892no—
259C22⋊C8—1782no—
260C22⋊C8—1782no—
261#97—1784no—
262C4×C8—1784no—
263C4×C8—1784no—
264#95—1781no—
265C24#1308357311yescharacteristic
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—1782no—
271C22×C4C22×Dic8911yescharacteristic
272C22×C4C22×Dic8911yescharacteristic
273C22×C4D4×D8911yescharacteristic
274C22×C4D4×D8911yescharacteristic
275C22×C4#1308357311yescharacteristic
276C22×C4C23×D8911yescharacteristic
277M16—1784no—
278M16—1784no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—892no—
285C2×C8—892no—
286C2×C8—1784no—
287C2×C8—1782no—
288C2×C8—1782no—
289C2×C8C22×Dic8911yescharacteristic
290C2×C8C22×Dic8911yescharacteristic
291C2×C8C22×Dic8911yescharacteristic
292C2×C8C22×Dic8911yescharacteristic
293C2×C8#1308357414yes—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4C22×Dic8912yes—
297C4⋊C4C22×Dic8912yes—
298C4⋊C4C22×Dic8912yes—
299C4⋊C4C22×Dic8912yes—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—1784no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—1782no—
310C42—1782no—
311C23—22no—
312C23—22no—
313C23—41no—
314C23D4×Dic8911yescharacteristic
315C23D4×Dic8911yescharacteristic
316C23C2×#1308357311yescharacteristic
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—44no—
329C2×C4—82no—
330C2×C4—3562no—
331C2×C4—3562no—
332C2×C4—1784no—
333C2×C4D4×Dic8911yescharacteristic
334C2×C4D4×Dic8911yescharacteristic
335C2×C4C23×Dic8911yescharacteristic
336C2×C4C2×D4×D8911yescharacteristic
337C2×C4C2×#1308357412yes—
338C2×C4#1566806012yes—
339C8—21no—
340C8—21no—
341C8—3564no—
342C8—1782no—
343C8—1782no—
344C8#1566797112yes—
345C8#1566797112yes—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—44no—
354C22#1915171512yes—
355C22C2×D4×Dic8911yescharacteristic
356C22#1915228312yes—
357C4—21no—
358C4—21no—
359C4—42no—
360C4—42no—
361C4—42no—
362C4—42no—
363C4—3564no—
364C4C2×#1566797111yescharacteristic
365C4#1915207611yescharacteristic
366C4#1915224912yes—
367C2—22no—
368C2—22no—
369C2—41no—
370C2#2602312111yescharacteristic
371C2C2×#1915171511yescharacteristic
372C2#2602916911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC22×C4C23×D89
FittingC2×#427×C89C2
socleC22×C89C2×C4×D4
prime radicalC24×C89#41
cosocleC22×C4×C89C24
hypercenterC2×#427D89
hypocenterC89#22606
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#427

Complements

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

NH ≅ G/Ntype
2. #26021001C2semidirect
3. #26020853C2semidirect
6. #26020133C2semidirect
25. #427×C89C4semidirect
26. #427×C89C4semidirect
27. #427×C89C4semidirect
31. C2×#107×C89C4semidirect
32. C2×#107×C89C4semidirect
33. C2×#106×C89C4semidirect
195. C89#22606semidirect
202. #427Dic89semidirect
203. #427Dic89semidirect
204. #427Dic89semidirect
208. C2×#107Dic89semidirect
209. C2×#107Dic89semidirect
210. C2×#106Dic89semidirect

Maximal subgroups

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

subgroupindex
#260289102
#260210012
#260208532
#260203512
#260202902
#260201332
C8×C22⋊Dic892
C2×#427×C892
#2260689