Finite Group Databasebeta

← #36556727

Subgroups — #36556727

Order 22784.

1122 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19151566C211yesmaximal, characteristic
2C2×#19151564C211yesmaximal, characteristic
3C22×#15667922C211yesmaximal, characteristic, complemented
4#26022740C211yesmaximal, characteristic
5#26020864C218yesmaximal, complemented
6#26019949C211yesmaximal, characteristic, complemented
7#26019945C211yesmaximal, characteristic, complemented
8C2×#424×C89C211yesmaximal, characteristic, complemented
9C22×#13083574C2211yescharacteristic
10C22×#13083558C2211yescharacteristic
11C23×#10623494C2211yescharacteristic
12C22×C4×Dic89C2211yescharacteristic
13#19151566—24no—
14#19151564—24no—
15C2×#15667924C2214yes—
16C2×#15667924C2214yes—
17C2×#15667922—24no—
18C2×#15667922C2214yes—
19#19151503C412yes—
20#19151499C412yes—
21#19151475C414yes—
22C4×#13083551C2214yes—
23#19151353C2214yes—
24#19151352C2214yes—
25C22×M16×C89C2211yescharacteristic
26C23×C8×C89C2211yescharacteristic
27C2×#117×C89C2211yescharacteristic
28#424×C89—24no—
29C2×C4⋊C8×C89C2214yes—
30C2×#13083574—22no—
31C2×#13083574—22no—
32C2×#13083574—22no—
33C2×#13083574—22no—
34C2×#13083574—22no—
35C2×#13083574—22no—
36C22×Dic178C2×C411yescharacteristic
37C22×D356C2×C411yescharacteristic
38C22×C22⋊D89C2×C412yes—
39C23×Dic89C2×C412yes—
40C22×C4×D89C2312yes—
41C2×#13083558—22no—
42C2×#13083558—22no—
43C2×#13083558—22no—
44C22×#10623494—22no—
45C22×#10623494—22no—
46C22×#10623494D414yes—
47C22×#10623494C2312yes—
48C22×#10623494C2312yes—
49C2×#13083551C2×C418yes—
50C2×C4×Dic89—22no—
51C2×C4×Dic89—22no—
52C2×C4×Dic89C2×C412yes—
53C2×C4×Dic89C2312yes—
54#15667924—28no—
55#15667924—28no—
56#15667922—28no—
57#15667919C2×C418yes—
58C23×C4×C89C2311yescharacteristic
59C2×M16×C89—22no—
60C2×M16×C89—22no—
61C2×M16×C89—22no—
62C22×C8×C89—22no—
63C22×C8×C89—22no—
64C22×C8×C89D414yes—
65C22×C8×C89C2312yes—
66C22×C8×C89C2312yes—
67#117×C89—22no—
68#117×C89—22no—
69C2×C4⋊C4×C89C2×C412yes—
70C2×C22⋊C4×C89C2×C412yes—
71C2×C42×C89C2312yes—
72C4⋊C8×C89—28no—
73C23×D89C22×C412yes—
74#13083574—44no—
75#13083574—44no—
76#13083574—44no—
77#13083574—44no—
78C2×Dic178—22no—
79C2×Dic178—22no—
80C2×Dic178—22no—
81C2×D356—22no—
82C2×D356—22no—
83C2×D356—22no—
84C2×C22⋊D89—24no—
85C2×C22⋊D89—24no—
86C2×C22⋊D89—24no—
87C22×Dic89—24no—
88C22×Dic89—44no—
89C22×Dic89C22×C412yes—
90C22×Dic89C22×C414yes—
91C2×C4×D89—22no—
92C2×C4×D89—22no—
93C2×C4×D89—22no—
94C2×C4×D89—22no—
95C2×C4×D89—22no—
96C2×C4×D89—22no—
97#13083558—44no—
98C2×#10623494—28no—
99C2×#10623494—22no—
100C2×#10623494—22no—
101C2×#10623494—22no—
102C2×#10623494—22no—
103C2×#10623494—22no—
104C2×#10623494C2×D414yes—
105#13083551—48no—
106C4×Dic89—24no—
107C4×Dic89—22no—
108C4×Dic89—22no—
109C24×C89C22×C411yescharacteristic
110C22×C4×C89C22×C414yes—
111C22×C4×C89C22×C411yescharacteristic
112C22×C4×C89C2×D412yes—
113C22×C4×C89C2×D412yes—
114C22×C4×C89#4112yes—
115C22×C4×C89#4112yes—
116C22×C4×C89#4112yes—
117C22×C4×C89#4112yes—
118C22×C4×C89C2411yescharacteristic
119M16×C89—44no—
120C2×C8×C89—28no—
121C2×C8×C89—22no—
122C2×C8×C89—22no—
123C2×C8×C89—22no—
124C2×C8×C89—22no—
125C2×C8×C89—22no—
126C2×C8×C89C2×D414yes—
127C4⋊C4×C89—24no—
128C22⋊C4×C89—24no—
129C42×C89—22no—
130C42×C89—22no—
131C23×C89—24no—
132C23×C89#11712yes—
133C23×C89C4×D412yes—
134C23×C89C4×D412yes—
135C23×C89C23×C411yescharacteristic
136C22×D89—22no—
137C22×D89—22no—
138C22×D89—22no—
139C22×D89—28no—
140C2×C4×C89—24no—
141C2×C4×C89—28no—
142C2×C4×C89#11712yes—
143C2×C4×C89C4×D412yes—
144C2×C4×C89C4×D412yes—
145C2×C4×C89#12114yes—
146C2×C4×C89#12314yes—
147C2×C4×C89#12414yes—
148C2×C4×C89#12614yes—
149C2×C4×C89C22×D411yescharacteristic
150C2×C4×C89C22×D411yescharacteristic
151C2×C4×C89C2×#4111yescharacteristic
152C2×C4×C89C2×#4111yescharacteristic
153C2×C4×C89C2×#4111yescharacteristic
154C2×C4×C89C2×#4111yescharacteristic
155C22⋊D89—48no—
156C2×Dic89—22no—
157C2×Dic89—24no—
158C2×Dic89—22no—
159C2×Dic89—22no—
160C2×Dic89—24no—
161C2×Dic89—24no—
162C2×Dic89—48no—
163D356—44no—
164C4×D89—42no—
165C4×D89—42no—
166C4×D89—42no—
167C4×D89—42no—
168Dic178—44no—
169C8×C89—24no—
170C8×C89—42no—
171#10623494—24no—
172#10623494—42no—
173C22×C89—24no—
174C22×C89—24no—
175C22×C89—24no—
176C22×C89#38814yes—
177C22×C89C2×#11711yescharacteristic
178C22×C89C2×C4×D411yescharacteristic
179C22×C89C2×C4×D411yescharacteristic
180C22×C89#56814yes—
181C2×D89—44no—
182C2×D89—44no—
183C2×D89—44no—
184C2×D89—42no—
185C4×C89—44no—
186C4×C89#38814yes—
187C4×C89C2×#12111yescharacteristic
188C4×C89C2×#12311yescharacteristic
189C4×C89C2×#12411yescharacteristic
190C4×C89C2×#12611yescharacteristic
191Dic89—44no—
192Dic89—42no—
193#21047—891nomaximal, Sylow
194C2×C89—24no—
195C2×C89C2×#38811yescharacteristic
196C2×C89#292312yes—
197C2×C89#293012yes—
198C2×C89#293212yes—
199D89—82no—
200C2×#424D8911yescharacteristic, complemented
201C2×#408—891no—
202C2×#408—891no—
203C22×#97—891no—
204#2297—891no—
205#1924—898no—
206#1729—891no—
207#1728—891no—
208C89#2104711yescharacteristic, complemented, Sylow
209C22×#41—891no—
210C22×M16—891no—
211C22×M16C2×D8911yescharacteristic
212C23×C8—891no—
213C23×C8C2×D8911yescharacteristic
214C2×#117C2×D8911yescharacteristic
215C22×C42—891no—
216#424—24no—
217C2×C4⋊C8C2×D8914yes—
218#408—1784no—
219#408—1784no—
220C2×C22⋊C8—894no—
221C2×C22⋊C8—894no—
222C2×#97—894no—
223C2×#97—1784no—
224#391—892no—
225#390—892no—
226#386—894no—
227C4×C22⋊C4—894no—
228#336—894no—
229#336—894no—
230C2×#41—1782no—
231C2×#41—1782no—
232C2×#41—1782no—
233C2×#41—1782no—
234C2×#41—1782no—
235C2×#41—1782no—
236C22×Q8—891no—
237C22×D4—892no—
238C22×D4—891no—
239C23×C4—892no—
240C23×C4—892no—
241C23×C4C22×D8911yescharacteristic
242C2×M16—22no—
243C2×M16—22no—
244C2×M16—22no—
245C2×M16—1782no—
246C2×M16—1782no—
247C2×M16—1782no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—894no—
251C22×C8—892no—
252C22×C8—892no—
253C22×C8—1782no—
254C22×C8—1782no—
255C22×C8C22⋊D8914yes—
256C22×C8C22×D8912yes—
257C22×C8C22×D8912yes—
258#117—22no—
259#117—22no—
260C2×C4⋊C4C4×D8912yes—
261C2×C22⋊C4—898no—
262C2×C22⋊C4C4×D8912yes—
263C2×C42—892no—
264C2×C42—892no—
265C2×C42—1782no—
266C2×C42—1782no—
267C2×C42C22×D8912yes—
268C4⋊C8—28no—
269C22⋊C8—1788no—
270C22⋊C8—1788no—
271#97—1788no—
272#95—898no—
273C24—892no—
274C24C2×C4×D8911yescharacteristic
275#41—3564no—
276#41—3564no—
277#41—3564no—
278#41—3564no—
279C2×Q8—1782no—
280C2×Q8—1782no—
281C2×Q8—1782no—
282C2×D4—1784no—
283C2×D4—1784no—
284C2×D4—1784no—
285C2×D4—1782no—
286C2×D4—1782no—
287C2×D4—1782no—
288C22×C4—892no—
289C22×C4—894no—
290C22×C4—3564no—
291C22×C4—1784no—
292C22×C4—1782no—
293C22×C4—1782no—
294C22×C4—1782no—
295C22×C4—1782no—
296C22×C4—1782no—
297C22×C4—1782no—
298C22×C4C2×C4×D8914yes—
299C22×C4C2×C4×D8911yescharacteristic
300C22×C4D4×D8912yes—
301C22×C4C2×C22⋊D8912yes—
302C22×C4#1308357312yes—
303C22×C4#1308357412yes—
304C22×C4#1308357512yes—
305C22×C4#1308357512yes—
306C22×C4C23×D8911yescharacteristic
307M16—44no—
308M16—3564no—
309C2×C8—28no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—894no—
316C2×C8—1782no—
317C2×C8—1782no—
318C2×C8—1782no—
319C2×C8—1788no—
320C2×C8—1782no—
321C2×C8—1782no—
322C2×C8C2×C22⋊D8914yes—
323C4⋊C4—24no—
324C22⋊C4—24no—
325C22⋊C4—3568no—
326C42—22no—
327C42—22no—
328C42—1784no—
329C42—1782no—
330C42—1782no—
331C23—24no—
332C23—1782no—
333C23—1782no—
334C23—1782no—
335C23—1788no—
336C23#1566797012yes—
337C23C4×C22⋊D8912yes—
338C23#1566797412yes—
339C23C22×C4×D8911yescharacteristic
340Q8—3564no—
341D4—3568no—
342D4—3564no—
343C2×C4—24no—
344C2×C4—28no—
345C2×C4—3568no—
346C2×C4—3562no—
347C2×C4—3562no—
348C2×C4—3562no—
349C2×C4—3562no—
350C2×C4—1782no—
351C2×C4—1782no—
352C2×C4—1784no—
353C2×C4—1782no—
354C2×C4—1784no—
355C2×C4—1784no—
356C2×C4#1566797012yes—
357C2×C4C4×C22⋊D8912yes—
358C2×C4#1566797412yes—
359C2×C4#1566798414yes—
360C2×C4#1566799714yes—
361C2×C4#1566800114yes—
362C2×C4#1566800714yes—
363C2×C4C2×D4×D8911yescharacteristic
364C2×C4C22×C22⋊D8911yescharacteristic
365C2×C4C2×#1308357311yescharacteristic
366C2×C4C2×#1308357411yescharacteristic
367C2×C4C2×#1308357511yescharacteristic
368C2×C4C2×#1308357511yescharacteristic
369C8—24no—
370C8—42no—
371C8—3562no—
372C8—1784no—
373C22—24no—
374C22—24no—
375C22—24no—
376C22—3562no—
377C22—3564no—
378C22—3564no—
379C22—3564no—
380C22#1915148814yes—
381C22C2×#1566797011yescharacteristic
382C22C2×C4×C22⋊D8911yescharacteristic
383C22C2×#1566797411yescharacteristic
384C22#1915236314yes—
385C4—44no—
386C4—3562no—
387C4—3564no—
388C4#1915148814yes—
389C4C2×#1566798411yescharacteristic
390C4C2×#1566799711yescharacteristic
391C4C2×#1566800111yescharacteristic
392C4C2×#1566800711yescharacteristic
393C2—24no—
394C2—7122no—
395C2C2×#1915148811yescharacteristic
396C2#2602890212yes—
397C2#2602895412yes—
398C2#2602896812yes—
subgroupG/H
center, FrattiniC22×C4C23×D89
commutator, socleC23×C89C23×C4
FittingC2×#424×C89C2
prime radicalC22×#13083574C22
cosocleC22×C4×C89C24
hypercenterC2×#424D89
hypocenterC89#21047
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C22×C4 ◁ C2×#424

Complements

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

NH ≅ G/Ntype
3. C22×#15667922C2semidirect
5. #26020864C2semidirect
6. #26019949C2semidirect
7. #26019945C2semidirect
8. C2×#424×C89C2semidirect
200. C2×#424D89semidirect
208. C89#21047semidirect

Maximal subgroups

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

subgroupindex
C2×#191515662
C2×#191515642
C22×#156679222
#260227402
#260208642
#260199492
#260199452
C2×#424×C892
#2104789