Finite Group Databasebeta

← #36583496

Subgroups — #36583496

Order 22784.

874 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#15667945C211yesmaximal, characteristic
2C2×#19151625C211yesmaximal, characteristic
3#26022872C211yesmaximal, characteristic
4#26021748C214yesmaximal, complemented
5#26021743C214yesmaximal, complemented
6#26019973C211yesmaximal, characteristic
7#26019971C211yesmaximal, characteristic
8#26019968C211yesmaximal, characteristic
9C89×#2317C211yesmaximal, characteristic
10C22×#13083558C2211yescharacteristic
11C23×#10623494C2211yescharacteristic
12C2×#15667971C2211yescharacteristic
13C22×C4⋊Dic89C2211yescharacteristic
14C2×#15667945—24no—
15C2×#15667945C2212yes—
16C2×#15667945C2212yes—
17#19151625—24no—
18C2×C4⋊#10623494C2212yes—
19C2×C4⋊#10623494C2212yes—
20#19151496C2214yes—
21#19151470—22no—
22#19151467—21no—
23#19151463—21no—
24#19151360C2212yes—
25#19151360C2214yes—
26#19151360C2214yes—
27#19151360C2212yes—
28C2×#117×C89C2211yescharacteristic
29C2×#117×C89C2211yescharacteristic
30C22×C4⋊C4×C89C2211yescharacteristic
31#389×C89C2214yes—
32#385×C89—22no—
33#384×C89—21no—
34#382×C89—21no—
35C23×Dic89—21no—
36C2×#13083558—22no—
37C2×#13083558—22no—
38C2×#13083558—22no—
39C22×#10623494—22no—
40C22×#10623494—22no—
41C22×#10623494D414yes—
42C22×#10623494C2312yes—
43C22×#10623494C2311yescharacteristic
44C22×#10623494C2311yescharacteristic
45#15667971—22no—
46#15667971Q814yes—
47C2×C4⋊Dic89—22no—
48C2×C4⋊Dic89—21no—
49C2×C4⋊Dic89—22no—
50C2×C4⋊Dic89C2312yes—
51C2×C4⋊Dic89C2312yes—
52C2×#13083553—24no—
53C2×C22⋊Dic89—21no—
54C2×C4×Dic89—21no—
55#15667945—24no—
56#15667945—24no—
57C4⋊#10623494—24no—
58C4⋊#10623494—24no—
59#15667919—24no—
60C23×C4×C89—21no—
61C23×C4×C89C2311yescharacteristic
62#117×C89—22no—
63#117×C89—22no—
64#117×C89—22no—
65#117×C89Q814yes—
66C2×C4⋊C4×C89—22no—
67C2×C4⋊C4×C89—22no—
68C2×C4⋊C4×C89—21no—
69C2×C4⋊C4×C89—24no—
70C2×C4⋊C4×C89D411yescharacteristic
71C2×C4⋊C4×C89D411yescharacteristic
72C2×C4⋊C4×C89C2312yes—
73C2×C4⋊C4×C89C2312yes—
74C2×C22⋊C4×C89—21no—
75C2×C22⋊C4×C89D412yes—
76C2×C42×C89—21no—
77C2×C42×C89C2312yes—
78#95×C89—24no—
79C22×Dic89—22no—
80C22×Dic89—22no—
81C22×Dic89—42no—
82C22×Dic89—82no—
83#13083558—44no—
84C2×#10623494—28no—
85C2×#10623494—21no—
86C2×#10623494—21no—
87C2×#10623494—21no—
88C2×#10623494—21no—
89C2×#10623494—22no—
90C2×#10623494—22no—
91C2×#10623494—22no—
92C2×#10623494C2×D412yes—
93C2×#10623494C2×D412yes—
94C4⋊Dic89—22no—
95C4⋊Dic89—22no—
96C4⋊Dic89—22no—
97C4⋊Dic89—42no—
98C4⋊Dic89C2×Q814yes—
99#13083553—84no—
100C22⋊Dic89—24no—
101C4×Dic89—24no—
102C24×C89C2×D411yescharacteristic
103C22×C4×C89—22no—
104C22×C4×C89—22no—
105C22×C4×C89—42no—
106C22×C4×C89—82no—
107C22×C4×C89C2×D412yes—
108C22×C4×C89C2×D412yes—
109C22×C4×C89C2×D411yescharacteristic
110C22×C4×C89C2×D412yes—
111C22×C4×C89C2×Q812yes—
112C22×C4×C89C2×Q812yes—
113C22×C4×C89#4112yes—
114C22×C4×C89#4112yes—
115C22×C4×C89#4112yes—
116C22×C4×C89C2411yescharacteristic
117C4⋊C4×C89—22no—
118C4⋊C4×C89—22no—
119C4⋊C4×C89—22no—
120C4⋊C4×C89—22no—
121C4⋊C4×C89—22no—
122C4⋊C4×C89—42no—
123C4⋊C4×C89—84no—
124C4⋊C4×C89C2×Q814yes—
125C22⋊C4×C89—24no—
126C22⋊C4×C89—24no—
127C42×C89—24no—
128C42×C89—22no—
129C42×C89—22no—
130C23×C89—22no—
131C23×C89—22no—
132C23×C89#12112yes—
133C23×C89C22⋊Q812yes—
134C23×C89C22⋊Q812yes—
135C23×C89C22×D411yescharacteristic
136C2×C4×C89—22no—
137C2×C4×C89—22no—
138C2×C4×C89—24no—
139C2×C4×C89—24no—
140C2×C4×C89—24no—
141C2×C4×C89—42no—
142C2×C4×C89—42no—
143C2×C4×C89—42no—
144C2×C4×C89—44no—
145C2×C4×C89—84no—
146C2×C4×C89#12112yes—
147C2×C4×C89C22⋊Q812yes—
148C2×C4×C89C22⋊Q812yes—
149C2×C4×C89#12314yes—
150C2×C4×C89#12514yes—
151C2×C4×C89#12514yes—
152C2×C4×C89C4⋊Q814yes—
153C2×C4×C89C22×D411yescharacteristic
154C2×C4×C89C22×Q811yescharacteristic
155C2×C4×C89C22×Q811yescharacteristic
156C2×C4×C89C2×#4111yescharacteristic
157C2×C4×C89C2×#4111yescharacteristic
158C2×C4×C89C2×#4111yescharacteristic
159C2×Dic89—24no—
160C2×Dic89—42no—
161C2×Dic89—42no—
162C2×Dic89—42no—
163C2×Dic89—44no—
164C2×Dic89—84no—
165#10623494—22no—
166#10623494—22no—
167#10623494—42no—
168C22×C89—24no—
169C22×C89—24no—
170C22×C89—22no—
171C22×C89—22no—
172C22×C89#39814yes—
173C22×C89C2×#12111yescharacteristic
174C22×C89C2×C22⋊Q811yescharacteristic
175C22×C89C2×C22⋊Q811yescharacteristic
176C22×C89#57612yes—
177C22×C89#57812yes—
178C4×C89—44no—
179C4×C89—42no—
180C4×C89—42no—
181C4×C89—82no—
182C4×C89#39814yes—
183C4×C89C2×#12311yescharacteristic
184C4×C89C2×#12511yescharacteristic
185C4×C89C2×#12511yescharacteristic
186C4×C89C2×C4⋊Q811yescharacteristic
187Dic89—44no—
188Dic89—82no—
189#23667—891nomaximal, Sylow
190C2×C89—24no—
191C2×C89C2×#39811yescharacteristic
192C2×C89#305811yescharacteristic
193C2×C89#305911yescharacteristic
194C2×C89#308312yes—
195C2×C89#308312yes—
196C22×#107—891no—
197C2×#424—891no—
198#2317—891no—
199#2317D8911yescharacteristic
200#2087—894no—
201#2086—894no—
202#1736—891no—
203#1736—891no—
204#1735—891no—
205C89#2366711yescharacteristic, complemented, Sylow
206C22×M16—891no—
207C23×C8—891no—
208C2×#117—891no—
209C2×#117C2×D8911yescharacteristic
210C2×#117C2×D8911yescharacteristic
211C22×C4⋊C4—891no—
212C22×C4⋊C4C2×D8911yescharacteristic
213C2×#107—892no—
214C2×#107—892no—
215C2×#107—1784no—
216#424—1784no—
217C2×C4⋊C8—892no—
218C2×C4⋊C8—892no—
219#389—894no—
220#389C2×D8914yes—
221#385—22no—
222#385—1782no—
223#384—21no—
224#384—1781no—
225#382—21no—
226#382—1781no—
227#340—892no—
228#340—894no—
229#340—894no—
230#340—892no—
231C23×C4—21no—
232C23×C4—1781no—
233C23×C4C22×D8911yescharacteristic
234C2×M16—1782no—
235C2×M16—1782no—
236C2×M16—1782no—
237C22×C8—894no—
238C22×C8—891no—
239C22×C8—891no—
240C22×C8—892no—
241C22×C8—1782no—
242C22×C8—1782no—
243#117—22no—
244#117—22no—
245#117—22no—
246#117—894no—
247#117—1782no—
248#117Dic17814yes—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—892no—
254C2×C4⋊C4—892no—
255C2×C4⋊C4—1781no—
256C2×C4⋊C4—1782no—
257C2×C4⋊C4—1782no—
258C2×C4⋊C4—1784no—
259C2×C4⋊C4C22⋊D8911yescharacteristic
260C2×C4⋊C4C22⋊D8911yescharacteristic
261C2×C4⋊C4C22×D8912yes—
262C2×C4⋊C4C22×D8912yes—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—1781no—
265C2×C22⋊C4C22⋊D8912yes—
266C2×C42—21no—
267C2×C42—1781no—
268C2×C42C22×D8912yes—
269#107—1784no—
270#107—1784no—
271C4⋊C8—1784no—
272C4⋊C8—1784no—
273#95—24no—
274#95—1784no—
275C24C2×C22⋊D8911yescharacteristic
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—42no—
279C22×C4—82no—
280C22×C4—3562no—
281C22×C4—1782no—
282C22×C4—1782no—
283C22×C4—7122no—
284C22×C4D4×D8912yes—
285C22×C4C2×C22⋊D8912yes—
286C22×C4C2×C22⋊D8912yes—
287C22×C4C2×C22⋊D8911yescharacteristic
288C22×C4Q8×D8912yes—
289C22×C4C2×Dic17812yes—
290C22×C4#1308357312yes—
291C22×C4#1308357412yes—
292C22×C4#1308357512yes—
293C22×C4C23×D8911yescharacteristic
294M16—3564no—
295C2×C8—892no—
296C2×C8—892no—
297C2×C8—1782no—
298C2×C8—1782no—
299C2×C8—1782no—
300C2×C8—1788no—
301C2×C8—1781no—
302C2×C8—1781no—
303C2×C8—1781no—
304C2×C8—1781no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—42no—
311C4⋊C4—84no—
312C4⋊C4—894no—
313C4⋊C4—3562no—
314C4⋊C4—1782no—
315C4⋊C4—1782no—
316C4⋊C4—1782no—
317C4⋊C4—7124no—
318C4⋊C4C2×Dic17814yes—
319C22⋊C4—24no—
320C22⋊C4—24no—
321C22⋊C4—1784no—
322C42—24no—
323C42—22no—
324C42—22no—
325C42—1784no—
326C23—22no—
327C23—22no—
328C23#1566798412yes—
329C23#1566798812yes—
330C23#1566799112yes—
331C23C22×C22⋊D8911yescharacteristic
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—84no—
342C2×C4—3562no—
343C2×C4—3562no—
344C2×C4—3564no—
345C2×C4—3562no—
346C2×C4—1784no—
347C2×C4—7124no—
348C2×C4#1566798412yes—
349C2×C4#1566798812yes—
350C2×C4#1566799112yes—
351C2×C4#1566799714yes—
352C2×C4#1566800214yes—
353C2×C4#1566800314yes—
354C2×C4#1566801114yes—
355C2×C4C2×D4×D8911yescharacteristic
356C2×C4C2×Q8×D8911yescharacteristic
357C2×C4C22×Dic17811yescharacteristic
358C2×C4C2×#1308357311yescharacteristic
359C2×C4C2×#1308357411yescharacteristic
360C2×C4C2×#1308357511yescharacteristic
361C8—3562no—
362C8—1782no—
363C8—1782no—
364C22—24no—
365C22—24no—
366C22—22no—
367C22—22no—
368C22#1915153114yes—
369C22C2×#1566798411yescharacteristic
370C22C2×#1566798811yescharacteristic
371C22C2×#1566799111yescharacteristic
372C22#1915240112yes—
373C22#1915241312yes—
374C4—44no—
375C4—42no—
376C4—42no—
377C4—82no—
378C4—3564no—
379C4—7122no—
380C4#1915153114yes—
381C4C2×#1566799711yescharacteristic
382C4C2×#1566800211yescharacteristic
383C4C2×#1566800311yescharacteristic
384C4C2×#1566801111yescharacteristic
385C2—24no—
386C2C2×#1915153111yescharacteristic
387C2#2602989711yescharacteristic
388C2#2602990411yescharacteristic
389C2#2603009912yes—
390C2#2603010312yes—
subgroupG/H
centerC23C22×C22⋊D89
commutator, cosocleC22×C4×C89C24
FrattiniC22×C4C23×D89
FittingC89×#2317C2
socleC23×C89C22×D4
prime radicalC24×C89C2×D4
hypercenter#2317D89
hypocenterC89#23667
LCSG ▷ C22×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
4. #26021748C2semidirect
5. #26021743C2semidirect
205. C89#23667semidirect

Maximal subgroups

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

subgroupindex
C22×#156679452
C2×#191516252
#260228722
#260217482
#260217432
#260199732
#260199712
#260199682
C89×#23172
#2366789