Finite Group Databasebeta

← #36452628

Subgroups — #36452628

Order 22784.

444 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26021045C211yesmaximal, characteristic
2#26020122C211yesmaximal, characteristic, complemented
3C4×#15667934C211yesmaximal, characteristic
4C2×#19151325C211yesmaximal, characteristic
5#26018091C211yesmaximal, characteristic, complemented
6#26018079C211yesmaximal, characteristic, complemented
7C2×#335×C89C211yesmaximal, characteristic, complemented
8C2×C4×D356C2211yescharacteristic
9C2×#15667934C411yescharacteristic
10C2×#15667934C411yescharacteristic
11C2×#15667924—21no—
12C2×#15667924—21no—
13C2×#15667922—21no—
14C2×C4×#10623494C2211yescharacteristic
15C2×C4×#10623494C2211yescharacteristic
16C4×C4⋊Dic89C2211yescharacteristic
17#19151361C411yescharacteristic
18#19151361C411yescharacteristic
19#19151352—21no—
20#19151325—22no—
21C2×C4⋊C8×C89C2211yescharacteristic
22C2×C4⋊C8×C89C2211yescharacteristic
23C2×C4×C8×C89C2211yescharacteristic
24#335×C89—22no—
25C22×D356C2×C411yescharacteristic
26C22×C4×D89—21no—
27C22×#10623494—21no—
28C22×#10623494D411yescharacteristic
29C22×#10623494D411yescharacteristic
30C4×D356—21no—
31C4×D356—21no—
32C4×D356—21no—
33C4×D356—21no—
34C2×C4⋊Dic89C2×C411yescharacteristic
35C2×C4⋊Dic89C2×C411yescharacteristic
36C2×C4⋊Dic89C2×C411yescharacteristic
37C2×#13083551—21no—
38C2×C4×Dic89—21no—
39#15667934—22no—
40#15667934—22no—
41#15667924—42no—
42#15667924—42no—
43#15667922—42no—
44C4×#10623494—21no—
45C4×#10623494—21no—
46C4×#10623494—21no—
47C4×#10623494—21no—
48#15667919—21no—
49C22×C8×C89—21no—
50C22×C8×C89—21no—
51C22×C8×C89C2×C411yescharacteristic
52C22×C8×C89C2×C411yescharacteristic
53C2×C42×C89C2311yescharacteristic
54C4⋊C8×C89—21no—
55C4⋊C8×C89—21no—
56C4⋊C8×C89—21no—
57C4⋊C8×C89D412yes—
58C4×C8×C89—21no—
59C4×C8×C89—21no—
60C23×D89—21no—
61C2×D356—21no—
62C2×D356—21no—
63C2×D356—21no—
64C2×D356—21no—
65C2×D356—21no—
66C2×D356M1612yes—
67C22×Dic89—21no—
68C22×Dic89—21no—
69C22×Dic89—21no—
70C2×C4×D89—41no—
71C2×C4×D89—41no—
72C2×C4×D89—41no—
73C2×C4×D89—41no—
74C2×C4×D89—41no—
75C2×C4×D89—41no—
76C2×#10623494—21no—
77C2×#10623494—21no—
78C2×#10623494—22no—
79C2×#10623494—21no—
80C2×#10623494—21no—
81C2×#10623494—21no—
82C2×#10623494—41no—
83C2×#10623494—41no—
84C2×#10623494D812yes—
85C4⋊Dic89—21no—
86C4⋊Dic89—21no—
87C4⋊Dic89—21no—
88C4⋊Dic89—21no—
89C4⋊Dic89—21no—
90C4⋊Dic89—21no—
91#13083551—41no—
92#13083551—41no—
93C4×Dic89—41no—
94C4×Dic89—41no—
95C22×C4×C89C22×C411yescharacteristic
96C22×C4×C89C2×D411yescharacteristic
97C22×C4×C89#4111yescharacteristic
98C2×C8×C89—21no—
99C2×C8×C89—21no—
100C2×C8×C89—21no—
101C2×C8×C89—22no—
102C2×C8×C89—21no—
103C2×C8×C89—21no—
104C2×C8×C89—41no—
105C2×C8×C89—41no—
106C2×C8×C89—41no—
107C2×C8×C89—41no—
108C2×C8×C89—41no—
109C2×C8×C89M1612yes—
110C42×C89C2×D411yescharacteristic
111C42×C89#4111yescharacteristic
112C42×C89#4111yescharacteristic
113C42×C89#4111yescharacteristic
114C23×C89C4×D411yescharacteristic
115C22×D89—22no—
116C22×D89—41no—
117C22×D89—41no—
118C22×D89—41no—
119C22×D89—41no—
120C22×D89—41no—
121C22×D89—41no—
122C2×C4×C89—21no—
123C2×C4×C89—21no—
124C2×C4×C89#11711yescharacteristic
125C2×C4×C89C4×D411yescharacteristic
126C2×C4×C89#12111yescharacteristic
127C2×C4×C89#12311yescharacteristic
128C2×C4×C89#12411yescharacteristic
129C2×C4×C89#12611yescharacteristic
130C2×C4×C89C2×M1611yescharacteristic
131C2×C4×C89C2×M1611yescharacteristic
132C2×C4×C89#13111yescharacteristic
133C2×C4×C89#13111yescharacteristic
134C2×C4×C89C2×D811yescharacteristic
135C2×C4×C89#13511yescharacteristic
136C2×C4×C89AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C89#13711yescharacteristic
138C2×Dic89—41no—
139C2×Dic89—41no—
140C2×Dic89—41no—
141C2×Dic89—41no—
142C2×Dic89—41no—
143C2×Dic89—41no—
144C2×Dic89—41no—
145C2×Dic89—41no—
146C2×Dic89—41no—
147D356—22no—
148D356—22no—
149D356—41no—
150D356—41no—
151C4×D89—81no—
152C4×D89—81no—
153C4×D89—81no—
154C4×D89—81no—
155C8×C89—22no—
156C8×C89—41no—
157C8×C89—42no—
158C8×C89—81no—
159#10623494—22no—
160#10623494—42no—
161#10623494—41no—
162C22×C89#38811yescharacteristic
163C22×C89#43511yescharacteristic
164C22×C89#43611yescharacteristic
165C22×C89C4×D811yescharacteristic
166C22×C89#44011yescharacteristic
167C22×C89#44311yescharacteristic
168C22×C89#44411yescharacteristic
169C2×D89—42no—
170C2×D89—42no—
171C2×D89—42no—
172C2×D89—81no—
173C2×D89—81no—
174C2×D89—81no—
175C2×D89—81no—
176C4×C89—21no—
177C4×C89—21no—
178C4×C89#43211yescharacteristic
179C4×C89#43311yescharacteristic
180C4×C89#43511yescharacteristic
181C4×C89#43511yescharacteristic
182C4×C89#45911yescharacteristic
183C4×C89#46411yescharacteristic
184C4×C89#48011yescharacteristic
185C4×C89#48211yescharacteristic
186Dic89—81no—
187Dic89—81no—
188Dic89—81no—
189#9545—891nomaximal, Sylow
190C2×C89#158111yescharacteristic
191C2×C89#158511yescharacteristic
192C2×C89#158911yescharacteristic
193C2×C89#159211yescharacteristic
194C2×C89#192411yescharacteristic
195C2×C89#192511yescharacteristic
196C2×C89#192911yescharacteristic
197D89—82no—
198#1954—891no—
199#1770—891no—
200C4×#102—891no—
201C2×#335D8911yescharacteristic, complemented
202C2×D4⋊C8—891no—
203#1279—891no—
204#1273—891no—
205C89#954511yescharacteristic, complemented, Sylow
206C2×C4×D4—891no—
207C2×C4⋊C8C2×D8911yescharacteristic
208C2×C4⋊C8C2×D8911yescharacteristic
209C2×#102—891no—
210C2×#102—891no—
211C2×C22⋊C8—1781no—
212C2×C22⋊C8—1781no—
213C2×#97—1781no—
214C2×C4×C8—891no—
215C2×C4×C8—891no—
216C2×C4×C8C2×D8911yescharacteristic
217C4×C4⋊C4—891no—
218#340—891no—
219#340—891no—
220#336—1781no—
221#335—22no—
222D4⋊C8—1782no—
223C22×D4—891no—
224C23×C4—1781no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—891no—
228C22×C8—891no—
229C22×C8—1781no—
230C22×C8C4×D8911yescharacteristic
231C22×C8C4×D8911yescharacteristic
232C4×D4—1781no—
233C4×D4—1781no—
234C4×D4—1781no—
235C4×D4—1781no—
236C2×C4⋊C4—891no—
237C2×C4⋊C4—891no—
238C2×C4⋊C4—891no—
239C2×C22⋊C4—1781no—
240C2×C42—1781no—
241C2×C42C22×D8911yescharacteristic
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8C22⋊D8912yes—
246#102—1782no—
247#102—1782no—
248C22⋊C8—3562no—
249C22⋊C8—3562no—
250#97—3562no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—1781no—
254C4×C8—1781no—
255C4×C8—1781no—
256C4×C8—1781no—
257#95—1781no—
258C24—1781no—
259C2×D4—892no—
260C2×D4—1781no—
261C2×D4—1781no—
262C2×D4—1781no—
263C2×D4—1781no—
264C2×D4—1781no—
265C22×C4—3561no—
266C22×C4—3561no—
267C22×C4—3561no—
268C22×C4—3561no—
269C22×C4—3561no—
270C22×C4—3561no—
271C22×C4—1781no—
272C22×C4—1781no—
273C22×C4—1781no—
274C22×C4C2×C4×D8911yescharacteristic
275C22×C4D4×D8911yescharacteristic
276C22×C4#1308357511yescharacteristic
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—22no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—892no—
289C2×C8—3561no—
290C2×C8—3561no—
291C2×C8—1782no—
292C2×C8—1781no—
293C2×C8—1781no—
294C2×C8—1781no—
295C2×C8—1781no—
296C2×C8—1781no—
297C2×C8#1308355712yes—
298C4⋊C4—1781no—
299C4⋊C4—1781no—
300C4⋊C4—1781no—
301C4⋊C4—1781no—
302C4⋊C4—1781no—
303C4⋊C4—1781no—
304C22⋊C4—3561no—
305C22⋊C4—3561no—
306C42—3561no—
307C42—3561no—
308C42C2×C22⋊D8911yescharacteristic
309C42#1308357311yescharacteristic
310C42#1308357411yescharacteristic
311C42#1308357511yescharacteristic
312C23—3561no—
313C23—3561no—
314C23—3561no—
315C23—3561no—
316C23—3561no—
317C23—3561no—
318C23—1782no—
319C23#1566797411yescharacteristic
320D4—3561no—
321D4—3561no—
322D4—1782no—
323D4—1782no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—3561no—
327C2×C4—3561no—
328C2×C4—3561no—
329C2×C4—3561no—
330C2×C4—3561no—
331C2×C4—3561no—
332C2×C4—3561no—
333C2×C4—3561no—
334C2×C4—3561no—
335C2×C4—7121no—
336C2×C4—7121no—
337C2×C4—7121no—
338C2×C4—7121no—
339C2×C4#1566797011yescharacteristic
340C2×C4C4×C22⋊D8911yescharacteristic
341C2×C4#1566798411yescharacteristic
342C2×C4#1566799711yescharacteristic
343C2×C4#1566800111yescharacteristic
344C2×C4#1566800711yescharacteristic
345C2×C4M16×D8911yescharacteristic
346C2×C4C2×#1308355711yescharacteristic
347C2×C4#1566801811yescharacteristic
348C2×C4#1566802011yescharacteristic
349C2×C4D8×D8911yescharacteristic
350C2×C4#1566803511yescharacteristic
351C2×C4#1566803611yescharacteristic
352C2×C4#1566804411yescharacteristic
353C8—22no—
354C8—41no—
355C8—42no—
356C8—81no—
357C8—3562no—
358C8—3561no—
359C8—1782no—
360C22—3562no—
361C22—3562no—
362C22—3562no—
363C22—7121no—
364C22—7121no—
365C22—7121no—
366C22—7121no—
367C22#1915148811yescharacteristic
368C22#1915166811yescharacteristic
369C22#1915167511yescharacteristic
370C22#1915168011yescharacteristic
371C22#1915169711yescharacteristic
372C22#1915171411yescharacteristic
373C22#1915171911yescharacteristic
374C4—21no—
375C4—21no—
376C4—7121no—
377C4—7121no—
378C4—7121no—
379C4#1915165411yescharacteristic
380C4#1915165811yescharacteristic
381C4#1915166711yescharacteristic
382C4#1915166911yescharacteristic
383C4#1915179111yescharacteristic
384C4#1915182211yescharacteristic
385C4#1915191311yescharacteristic
386C4#1915192511yescharacteristic
387C2—7122no—
388C2#2601908011yescharacteristic
389C2#2601910811yescharacteristic
390C2#2601913611yescharacteristic
391C2#2601915311yescharacteristic
392C2#2602086411yescharacteristic
393C2#2602087111yescharacteristic
394C2#2602089911yescharacteristic
subgroupG/H
centerC22×C4D4×D89
commutatorC22×C4×C89C22×C4
FrattiniC2×C42C22×D89
FittingC2×#335×C89C2
socleC23×C89C4×D4
prime radicalC22×D356C2×C4
cosocleC2×C42×C89C23
hypercenterC2×#335D89
hypocenterC89#9545
LCSG ▷ C22×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×#335

Complements

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

NH ≅ G/Ntype
2. #26020122C2semidirect
5. #26018091C2semidirect
6. #26018079C2semidirect
7. C2×#335×C89C2semidirect
201. C2×#335D89semidirect
205. C89#9545semidirect

Maximal subgroups

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

subgroupindex
#260210452
#260201222
C4×#156679342
C2×#191513252
#260180912
#260180792
C2×#335×C892
#954589