Finite Group Databasebeta

← #36574123

Subgroups — #36574123

Order 22784.

722 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#15668034C211yesmaximal, characteristic
2#26028917C212yesmaximal, complemented
3#26021140C212yesmaximal
4#26020901C214yesmaximal, complemented
5#26020424C211yesmaximal, characteristic, complemented
6#26020273C212yesmaximal, complemented
7#26020131C212yesmaximal, complemented
8C8×C4⋊C4×C89C211yesmaximal, characteristic, complemented
9C2×#15668034C2211yescharacteristic
10C4×#13083574C2212yes—
11C4×Dic356C411yescharacteristic, complemented
12C4×Dic356C411yescharacteristic, complemented
13C4×#13083563C412yescomplemented
14C4×#13083563C412yescomplemented
15C4×D712C411yescharacteristic, complemented
16C4×D712C411yescharacteristic, complemented
17#19151658—24no—
18#19151654—22no—
19#19151650—22no—
20C2×#15667948C2212yes—
21C2×#15667948C2212yes—
22#19151636C2211yescharacteristic
23#19151622C2212yes—
24C2×#15667940C2214yes—
25#19151595C2212yes—
26#19151564C2214yes—
27#19151553C2214yes—
28#19151358C2214yes—
29C2×C4⋊C8×C89C2211yescharacteristic
30C2×C4×C8×C89C2212yes—
31C2×C4×C8×C89C2211yescharacteristic
32C4×C4⋊C4×C89C2211yescharacteristic
33#336×C89C2212yes—
34C2×#13083574C2312yes—
35#15668034—22no—
36#15668034—22no—
37C2×Dic356C2×C411yescharacteristic
38C2×#13083563C2×C412yes—
39C2×D712C2×C411yescharacteristic
40C2×#13083558C2314yes—
41C4×Dic178C2×C412yes—
42C4×Dic178C2×C412yes—
43C4×D356C2×C412yes—
44C4×D356C2×C412yes—
45C4×C22⋊D89—22no—
46C4×C22⋊D89—22no—
47#15667971C2312yes—
48#15667969—22no—
49C42×D89—22no—
50#15667948—24no—
51#15667948—42no—
52#15667946C2×C411yescharacteristic
53#15667946C2×C411yescharacteristic
54#15667944C2×C412yes—
55C4⋊#10623494—22no—
56C4⋊#10623494—22no—
57#15667940—44no—
58#15667937C2×C412yes—
59#15667937C2×C412yes—
60#15667934C2×C412yes—
61#15667934C2×C412yes—
62#15667924—24no—
63#15667924—24no—
64#15667922—24no—
65C8×Dic89—24no—
66C22×C8×C89C2311yescharacteristic
67C22×C8×C89C2312yes—
68C22×C8×C89C2311yescharacteristic
69C2×C4⋊C4×C89—21no—
70C2×C42×C89C2312yes—
71C2×C42×C89C2311yescharacteristic
72C4⋊C8×C89D411yescharacteristic
73C4⋊C8×C89D411yescharacteristic
74C4⋊C8×C89D412yes—
75C4×C8×C89—24no—
76C4×C8×C89C2×C411yescharacteristic
77C4×C8×C89C2×C411yescharacteristic
78C4×C8×C89C2×C411yescharacteristic
79C4×C8×C89C2×C411yescharacteristic
80#95×C89—21no—
81#13083574—22no—
82#13083574—22no—
83#13083574—22no—
84#13083574—22no—
85C2×Dic178C22×C412yes—
86C2×D356C22×C412yes—
87C2×C22⋊D89—22no—
88C2×C4×D89—22no—
89C2×C4×D89—22no—
90C2×C4×D89—22no—
91Dic356—22no—
92#13083563—24no—
93D712—22no—
94#13083558—24no—
95#13083558—24no—
96C2×#10623494—24no—
97C4⋊Dic89C22×C412yes—
98C4⋊Dic89C22×C412yes—
99#13083553—22no—
100#13083553—22no—
101C22⋊Dic89—22no—
102#13083551—22no—
103#13083551—22no—
104C4×Dic89—22no—
105C4×Dic89—22no—
106C4×Dic89—22no—
107C22×C4×C89—22no—
108C22×C4×C89#4111yescharacteristic
109C22×C4×C89#4111yescharacteristic
110C22×C4×C89C2411yescharacteristic
111C2×C8×C89—22no—
112C2×C8×C89—21no—
113C2×C8×C89—21no—
114C2×C8×C89—22no—
115C2×C8×C89C22×C411yescharacteristic
116C2×C8×C89C22×C411yescharacteristic
117C2×C8×C89C22×C411yescharacteristic
118C2×C8×C89C22×C411yescharacteristic
119C2×C8×C89C2×D412yes—
120C2×C8×C89C2×D412yes—
121C2×C8×C89#4112yes—
122C2×C8×C89#4112yes—
123C4⋊C4×C89—24no—
124C42×C89—22no—
125C42×C89—22no—
126C42×C89C22×C411yescharacteristic
127C42×C89C22×C411yescharacteristic
128C42×C89C2×D411yescharacteristic
129C42×C89C2×D411yescharacteristic
130C23×C89C2×#4111yescharacteristic
131C22×D89—22no—
132C2×C4×C89—22no—
133C2×C4×C89—24no—
134C2×C4×C89—41no—
135C2×C4×C89—42no—
136C2×C4×C89—42no—
137C2×C4×C89C4×D411yescharacteristic
138C2×C4×C89C4×D411yescharacteristic
139C2×C4×C89C4×D411yescharacteristic
140C2×C4×C89C4×D411yescharacteristic
141C2×C4×C89C23×C411yescharacteristic
142C2×C4×C89C22×D411yescharacteristic
143C2×C4×C89C2×#4111yescharacteristic
144C2×C4×C89C2×#4111yescharacteristic
145C2×C4×C89H(5,𝔽2)11yescharacteristic
146C2×C4×C892−1+411yescharacteristic
147C22⋊D89—42no—
148C22⋊D89—42no—
149C2×Dic89—22no—
150C2×Dic89—22no—
151C2×Dic89—22no—
152D356—22no—
153D356—22no—
154C4×D89—42no—
155C4×D89—42no—
156C4×D89—44no—
157C4×D89—82no—
158Dic178—22no—
159Dic178—22no—
160C8×C89—21no—
161C8×C89—21no—
162C8×C89—22no—
163C8×C89—22no—
164C8×C89#11712yes—
165C8×C89#11712yes—
166#10623494—44no—
167C22×C89—21no—
168C22×C89—21no—
169C22×C89C2×C4×D411yescharacteristic
170C22×C89#55511yescharacteristic
171C22×C89#55611yescharacteristic
172C2×D89—42no—
173C2×D89—42no—
174C2×D89—42no—
175C4×C89—21no—
176C4×C89—21no—
177C4×C89—21no—
178C4×C89—44no—
179C4×C89#44312yes—
180C4×C89C2×#11711yescharacteristic
181C4×C89#52011yescharacteristic
182C4×C89#54711yescharacteristic
183C4×C89#55011yescharacteristic
184Dic89—42no—
185Dic89—42no—
186Dic89—42no—
187#22802—891nomaximal, Sylow
188C2×C89—21no—
189C2×C89—21no—
190C2×C89#235411yescharacteristic
191C2×C89C2×#44311yescharacteristic
192C2×C89#295711yescharacteristic
193D89—82no—
194C4×#135—891no—
195#2925—892no—
196#1974—892no—
197#1929—894no—
198#1848—891no—
199#1806—892no—
200#1772—892no—
201C8×C4⋊C4D8911yescharacteristic, complemented
202C89#2280211yescharacteristic, complemented, Sylow
203C2×#135—891no—
204C4×#41—892no—
205C4×Q16—891no—
206C4×Q16—891no—
207C4×SD16—892no—
208C4×SD16—892no—
209C4×D8—891no—
210C4×D8—891no—
211#433—1784no—
212#432—1782no—
213#431—1782no—
214C2×#108—892no—
215C2×#108—892no—
216#427—891no—
217#423—892no—
218C2×C4⋊C8C2×D8911yescharacteristic
219C2×#104—894no—
220#416—892no—
221#408—894no—
222#405—894no—
223C2×C4×C8C2×D8912yes—
224C2×C4×C8C2×D8911yescharacteristic
225C4×C4⋊C4C2×D8911yescharacteristic
226#339—894no—
227#336C2×D8912yes—
228C2×#41—892no—
229#135—1782no—
230#135—1782no—
231C2×Q16—891no—
232C2×SD16—892no—
233C2×D8—891no—
234C2×M16—894no—
235C22×C8C22×D8911yescharacteristic
236C22×C8C22×D8912yes—
237C22×C8C22×D8911yescharacteristic
238C4×Q8—892no—
239C4×Q8—892no—
240C4×D4—892no—
241C4×D4—892no—
242C4×D4—1782no—
243C4×D4—1782no—
244#117—892no—
245#117—1782no—
246C2×C4⋊C4—21no—
247C2×C42—1782no—
248C2×C42C22×D8912yes—
249C2×C42C22×D8911yescharacteristic
250#108—3562no—
251#108—1784no—
252#107—891no—
253#107—891no—
254#106—892no—
255C4⋊C8—1782no—
256C4⋊C8—1782no—
257C4⋊C8D35611yescharacteristic
258C4⋊C8D35611yescharacteristic
259C4⋊C8D35612yes—
260#104—3564no—
261#103—892no—
262#103—892no—
263#102—892no—
264#102—892no—
265C22⋊C8—1784no—
266C22⋊C8—1784no—
267#97—1784no—
268C4×C8—24no—
269C4×C8—1784no—
270C4×C8C4×D8911yescharacteristic
271C4×C8C4×D8911yescharacteristic
272C4×C8C4×D8911yescharacteristic
273C4×C8C4×D8911yescharacteristic
274#95—21no—
275#41—1782no—
276#41—1782no—
277#41—1782no—
278#41—1782no—
279C2×Q8—892no—
280C2×D4—892no—
281C2×D4—1782no—
282C22×C4—22no—
283C22×C4—1782no—
284C22×C4—1782no—
285C22×C4—1782no—
286C22×C4#1308357411yescharacteristic
287C22×C4#1308357411yescharacteristic
288C22×C4C23×D8911yescharacteristic
289Q16—1782no—
290SD16—1784no—
291D8—1782no—
292M16—1784no—
293M16—1784no—
294C2×C8—22no—
295C2×C8—21no—
296C2×C8—22no—
297C2×C8—21no—
298C2×C8—1784no—
299C2×C8C2×C4×D8911yescharacteristic
300C2×C8C2×C4×D8911yescharacteristic
301C2×C8C2×C4×D8911yescharacteristic
302C2×C8C2×C4×D8911yescharacteristic
303C2×C8C2×D35612yes—
304C2×C8C2×D35612yes—
305C2×C8#1308357312yes—
306C2×C8#1308357512yes—
307C4⋊C4—24no—
308C4⋊C4—892no—
309C4⋊C4—892no—
310C4⋊C4—1782no—
311C4⋊C4—1782no—
312C22⋊C4—1782no—
313C22⋊C4—1782no—
314C22⋊C4—1782no—
315C42—22no—
316C42—22no—
317C42—1782no—
318C42—1782no—
319C42—1782no—
320C42C2×C4×D8911yescharacteristic
321C42C2×C4×D8911yescharacteristic
322C42C2×D35611yescharacteristic
323C42C2×D35611yescharacteristic
324C23—1782no—
325C23C2×#1308357411yescharacteristic
326Q8—1782no—
327Q8—1782no—
328D4—3562no—
329D4—3562no—
330D4—1782no—
331D4—1782no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—3564no—
338C2×C4—3562no—
339C2×C4—3562no—
340C2×C4—1782no—
341C2×C4—1782no—
342C2×C4—1782no—
343C2×C4—7122no—
344C2×C4C4×D35611yescharacteristic
345C2×C4C4×D35611yescharacteristic
346C2×C4C4×D35611yescharacteristic
347C2×C4C4×D35611yescharacteristic
348C2×C4C22×C4×D8911yescharacteristic
349C2×C4C22×D35611yescharacteristic
350C2×C4C2×#1308357311yescharacteristic
351C2×C4C2×#1308357511yescharacteristic
352C2×C4#1566805811yescharacteristic
353C2×C4#1566805911yescharacteristic
354C8—22no—
355C8—22no—
356C8—21no—
357C8—21no—
358C8—3564no—
359C8#1566797012yes—
360C8#1566797012yes—
361C22—21no—
362C22—21no—
363C22—3562no—
364C22—3562no—
365C22—3562no—
366C22C2×C4×D35611yescharacteristic
367C22#1915231311yescharacteristic
368C22#1915232111yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—44no—
373C4—3562no—
374C4—3562no—
375C4—3562no—
376C4#1915171612yes—
377C4C2×#1566797011yescharacteristic
378C4#1915207811yescharacteristic
379C4#1915225211yescharacteristic
380C4#1915227011yescharacteristic
381C2—21no—
382C2—21no—
383C2—7122no—
384C2#2602319911yescharacteristic
385C2C2×#1915171611yescharacteristic
386C2#2602917011yescharacteristic
subgroupG/H
centerC2×C4C22×D356
commutatorC2×C4×C89C23×C4
FrattiniC22×C4C23×D89
FittingC8×C4⋊C4×C89C2
socleC22×C89C2×C4×D4
prime radicalC2×#15668034C22
cosocleC22×C4×C89C24
hypercenterC8×C4⋊C4D89
hypocenterC89#22802
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C2×C4⋊C8 ◁ C8×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #26028917C2semidirect
4. #26020901C2semidirect
5. #26020424C2semidirect
6. #26020273C2semidirect
7. #26020131C2semidirect
8. C8×C4⋊C4×C89C2semidirect
11. C4×Dic356C4semidirect
12. C4×Dic356C4semidirect
13. C4×#13083563C4semidirect
14. C4×#13083563C4semidirect
15. C4×D712C4semidirect
16. C4×D712C4semidirect
201. C8×C4⋊C4D89semidirect
202. C89#22802semidirect

Maximal subgroups

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

subgroupindex
C4×#156680342
#260289172
#260211402
#260209012
#260204242
#260202732
#260201312
C8×C4⋊C4×C892
#2280289