Finite Group Databasebeta

← #36526217

Subgroups — #36526217

Order 22784.

600 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26030822C211yesmaximal, characteristic
2#26029256C212yesmaximal, complemented
3#26019150C211yesmaximal, characteristic, complemented
4#26019138C211yesmaximal, characteristic, complemented
5#26019097C212yesmaximal, complemented
6#26019090C212yesmaximal, complemented
7C8×Dic356C211yesmaximal, characteristic, complemented
8C8×D712C211yesmaximal, characteristic, complemented
9#26019000C211yesmaximal, characteristic, complemented
10#26018661C212yesmaximal, complemented
11C8×M16×C89C211yesmaximal, characteristic, complemented
12#19152132C2212yes—
13#19152000C411yescharacteristic
14#19151982C411yescharacteristic
15#19151953C412yes—
16#19151839C411yescharacteristic
17#19151834C411yescharacteristic
18#19151827C412yes—
19#19151728C2212yes—
20C8×Dic178C2212yes—
21C4×Dic356C2211yescharacteristic
22C4×#13083563C2212yes—
23C4×D712C2211yescharacteristic
24#19151677C2212yes—
25#19151674—22no—
26#19151667—22no—
27C8×D356C2212yes—
28#19151660—22no—
29#19151651C2212yes—
30#19151555—22no—
31#19151351C2211yescharacteristic
32#19151351C2211yescharacteristic
33#19151349C2212yes—
34#19151330C2212yes—
35#19151330C2212yes—
36#19151327C2212yes—
37#19151327C2212yes—
38#431×C89C2211yescharacteristic
39C4×M16×C89C2211yescharacteristic
40C2×C4×C8×C89C2211yescharacteristic
41#322×C89C2211yescharacteristic
42#322×C89C2211yescharacteristic
43C82×C89C2211yescharacteristic
44C82×C89C2211yescharacteristic
45C2×Dic356C2×C411yescharacteristic
46C2×#13083563C2×C412yes—
47C2×D712C2×C411yescharacteristic
48C2×#13083557—22no—
49C2×C8×D89—22no—
50#15668006—22no—
51#15668004C2×C412yes—
52#15668000C2×C412yes—
53#15667995—22no—
54#15667991C2×C412yes—
55#15667986C2×C412yes—
56C4×Dic178C2312yes—
57C4×D356C2312yes—
58C4×C22⋊D89—22no—
59#15667969—22no—
60#15667946C2×C411yescharacteristic
61#15667946C2×C411yescharacteristic
62#15667944C2×C412yes—
63C4⋊#10623494C2312yes—
64C4⋊#10623494C2312yes—
65#15667941—22no—
66#15667941—22no—
67#15667937C2×C412yes—
68#15667937C2×C412yes—
69#15667934C2×C412yes—
70#15667934C2×C412yes—
71C22⋊#10623494—22no—
72#15667924—22no—
73#15667924—22no—
74#15667922—22no—
75C4×#10623494—22no—
76C8×Dic89—22no—
77C2×M16×C89D411yescharacteristic
78C2×M16×C89D411yescharacteristic
79C22×C8×C89C2×C411yescharacteristic
80C22×C8×C89C2×C411yescharacteristic
81C2×C42×C89C2311yescharacteristic
82C4⋊C8×C89—21no—
83C22⋊C8×C89—21no—
84#97×C89D411yescharacteristic
85#97×C89D411yescharacteristic
86C4×C8×C89C2×C411yescharacteristic
87C4×C8×C89C2×C411yescharacteristic
88C4×C8×C89C2311yescharacteristic
89C4×C8×C89C2311yescharacteristic
90C4×C8×C89C2311yescharacteristic
91C4×C8×C89C2311yescharacteristic
92C4×C8×C89C2311yescharacteristic
93C4×C8×C89C2311yescharacteristic
94C2×Dic178C22×C412yes—
95C2×D356C22×C412yes—
96C2×C22⋊D89—22no—
97C2×C4×D89—22no—
98Dic356—22no—
99#13083563—42no—
100D712—22no—
101#13083557—44no—
102C2×#10623494—22no—
103C2×#10623494—22no—
104C8×D89—44no—
105C4⋊Dic89C22×C412yes—
106C4⋊Dic89C22×C412yes—
107#13083553—22no—
108#13083553—22no—
109#13083553—22no—
110#13083553—22no—
111C22⋊Dic89—22no—
112#13083551—22no—
113#13083551—22no—
114#13083551—22no—
115#13083551—22no—
116C4×Dic89—22no—
117C22×C4×C89C22×C411yescharacteristic
118C22×C4×C89C2×D411yescharacteristic
119C22×C4×C89#4111yescharacteristic
120M16×C89—22no—
121M16×C89—22no—
122C2×C8×C89—22no—
123C2×C8×C89—21no—
124C2×C8×C89—21no—
125C2×C8×C89—41no—
126C2×C8×C89C22×C411yescharacteristic
127C2×C8×C89C22×C411yescharacteristic
128C2×C8×C89C22×C411yescharacteristic
129C2×C8×C89C22×C411yescharacteristic
130C2×C8×C89C2×D411yescharacteristic
131C2×C8×C89C2×D411yescharacteristic
132C2×C8×C89C2×D411yescharacteristic
133C2×C8×C89C2×D411yescharacteristic
134C42×C89C22×C411yescharacteristic
135C42×C89C2×D411yescharacteristic
136C42×C89#4111yescharacteristic
137C42×C89C2411yescharacteristic
138C23×C89C4×D411yescharacteristic
139C22×D89—22no—
140C2×C4×C89—21no—
141C2×C4×C89—21no—
142C2×C4×C89—21no—
143C2×C4×C89—21no—
144C2×C4×C89—41no—
145C2×C4×C89C4×D411yescharacteristic
146C2×C4×C89C4×D411yescharacteristic
147C2×C4×C89C4×D411yescharacteristic
148C2×C4×C89C23×C411yescharacteristic
149C2×C4×C89C22×D411yescharacteristic
150C2×C4×C89C2×#4111yescharacteristic
151C22⋊D89—82no—
152C2×Dic89—22no—
153C2×Dic89—22no—
154C2×Dic89—22no—
155D356—22no—
156D356—22no—
157C4×D89—42no—
158C4×D89—42no—
159Dic178—22no—
160Dic178—22no—
161C8×C89—21no—
162C8×C89—21no—
163C8×C89—21no—
164C8×C89—21no—
165C8×C89—21no—
166C8×C89—41no—
167C8×C89—41no—
168C8×C89#13112yes—
169C8×C89#13512yes—
170C8×C89#13512yes—
171#10623494—42no—
172#10623494—42no—
173C22×C89—21no—
174C22×C89—21no—
175C22×C89—21no—
176C22×C89C2×C4×D411yescharacteristic
177C2×D89—42no—
178C2×D89—42no—
179C2×D89—42no—
180C4×C89—21no—
181C4×C89—21no—
182C4×C89—21no—
183C4×C89—21no—
184C4×C89C2×#13111yescharacteristic
185C4×C89#56811yescharacteristic
186C4×C89C2×#13511yescharacteristic
187C4×C89C2×#13511yescharacteristic
188Dic89—42no—
189Dic89—42no—
190Dic89—42no—
191#18069—891nomaximal, Sylow
192C2×C89—41no—
193C2×C89#293511yescharacteristic
194C2×C89C4×#13511yescharacteristic
195C2×C89#295711yescharacteristic
196D89—82no—
197#3168—891no—
198#2969—892no—
199#1591—891no—
200#1589—891no—
201#1583—892no—
202#1582—892no—
203C8×Q16—891no—
204C8×D8—891no—
205#1564—891no—
206#1482—892no—
207C8×M16D8911yescharacteristic, complemented
208C89#1806911yescharacteristic, complemented, Sylow
209#529—892no—
210#499—891no—
211#495—891no—
212#488—892no—
213#467—891no—
214#466—891no—
215#465—892no—
216#446—892no—
217C8×Q8—892no—
218C4×Q16—891no—
219C4×SD16—892no—
220C4×D8—891no—
221#436—892no—
222#436—1782no—
223#435—1782no—
224C8×D4—892no—
225#433—1782no—
226#431—892no—
227#431C2×D8911yescharacteristic
228#405—1782no—
229C4×M16C2×D8911yescharacteristic
230C2×C4×C8C2×D8911yescharacteristic
231#335—891no—
232#335—891no—
233#334—892no—
234Q8⋊C8—892no—
235Q8⋊C8—892no—
236D4⋊C8—892no—
237D4⋊C8—892no—
238#322C2×D8911yescharacteristic
239#322C2×D8911yescharacteristic
240C82C2×D8911yescharacteristic
241C82C2×D8911yescharacteristic
242C2×Q16—891no—
243C2×SD16—892no—
244C2×D8—891no—
245C2×M16—1782no—
246C2×M16D35611yescharacteristic
247C2×M16D35611yescharacteristic
248C22×C8—1782no—
249C22×C8C4×D8911yescharacteristic
250C22×C8C4×D8911yescharacteristic
251#126—1782no—
252#125—892no—
253#124—892no—
254#123—1782no—
255C22⋊Q8—892no—
256#121—892no—
257C4×Q8—892no—
258C4×D4—892no—
259C4×D4—1782no—
260#117—1782no—
261C2×C42C22×D8911yescharacteristic
262#107—891no—
263#107—891no—
264#106—892no—
265C4⋊C8—21no—
266C4⋊C8—892no—
267C4⋊C8—892no—
268C4⋊C8—1782no—
269C4⋊C8—1782no—
270#103—892no—
271#103—892no—
272#102—892no—
273#102—892no—
274C22⋊C8—21no—
275C22⋊C8—1782no—
276C22⋊C8—1782no—
277C22⋊C8—1782no—
278#97—1782no—
279#97D35611yescharacteristic
280#97D35611yescharacteristic
281C4×C8—1782no—
282C4×C8—1782no—
283C4×C8C4×D8911yescharacteristic
284C4×C8C4×D8911yescharacteristic
285C4×C8C22×D8911yescharacteristic
286C4×C8C22×D8911yescharacteristic
287C4×C8C22×D8911yescharacteristic
288C4×C8C22×D8911yescharacteristic
289C4×C8C22×D8911yescharacteristic
290C4×C8C22×D8911yescharacteristic
291C2×Q8—892no—
292C2×D4—892no—
293C2×D4—1782no—
294C22×C4—1782no—
295C22×C4C2×C4×D8911yescharacteristic
296C22×C4C2×D35611yescharacteristic
297C22×C4#1308357411yescharacteristic
298Q16—1782no—
299SD16—3562no—
300D8—1782no—
301M16—22no—
302M16—22no—
303M16—3564no—
304C2×C8—22no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—41no—
308C2×C8—3564no—
309C2×C8—1782no—
310C2×C8—1782no—
311C2×C8C2×C4×D8911yescharacteristic
312C2×C8C2×C4×D8911yescharacteristic
313C2×C8C2×C4×D8911yescharacteristic
314C2×C8C2×C4×D8911yescharacteristic
315C2×C8C2×D35611yescharacteristic
316C2×C8C2×D35611yescharacteristic
317C2×C8C2×D35611yescharacteristic
318C2×C8C2×D35611yescharacteristic
319C4⋊C4—892no—
320C4⋊C4—892no—
321C4⋊C4—1782no—
322C4⋊C4—1782no—
323C4⋊C4—1782no—
324C4⋊C4—1782no—
325C22⋊C4—1782no—
326C22⋊C4—1782no—
327C22⋊C4—1782no—
328C22⋊C4—1782no—
329C22⋊C4—1782no—
330C42—1782no—
331C42C2×C4×D8911yescharacteristic
332C42C2×D35611yescharacteristic
333C42#1308357411yescharacteristic
334C42C23×D8911yescharacteristic
335C23—1782no—
336C23C4×D35611yescharacteristic
337Q8—1782no—
338Q8—1782no—
339D4—1782no—
340D4—1782no—
341D4—7122no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—41no—
347C2×C4—3562no—
348C2×C4—3562no—
349C2×C4—1782no—
350C2×C4—1782no—
351C2×C4—1782no—
352C2×C4C4×D35611yescharacteristic
353C2×C4C4×D35611yescharacteristic
354C2×C4C4×D35611yescharacteristic
355C2×C4C22×C4×D8911yescharacteristic
356C2×C4C22×D35611yescharacteristic
357C2×C4C2×#1308357411yescharacteristic
358C8—21no—
359C8—21no—
360C8—21no—
361C8—21no—
362C8—21no—
363C8—41no—
364C8—41no—
365C8—3562no—
366C8—3562no—
367C8#1566802012yes—
368C8#1566803412yes—
369C8#1566803412yes—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—3562no—
374C22—3562no—
375C22—3562no—
376C22C2×C4×D35611yescharacteristic
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—3562no—
382C4—3562no—
383C4—3562no—
384C4C2×#1566802011yescharacteristic
385C4#1915236411yescharacteristic
386C4C2×#1566803411yescharacteristic
387C4C2×#1566803411yescharacteristic
388C2—41no—
389C2—7122no—
390C2#2602899811yescharacteristic
391C2C4×#1566803411yescharacteristic
392C2#2602917011yescharacteristic
subgroupG/H
centerC2×C4C22×D356
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC8×M16×C89C2
socleC22×C89C2×C4×D4
prime radical#19151834C4
cosocleC42×C89C24
hypercenterC8×M16D89
hypocenterC89#18069
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ C8×M16

Complements

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

NH ≅ G/Ntype
2. #26029256C2semidirect
3. #26019150C2semidirect
4. #26019138C2semidirect
5. #26019097C2semidirect
6. #26019090C2semidirect
7. C8×Dic356C2semidirect
8. C8×D712C2semidirect
9. #26019000C2semidirect
10. #26018661C2semidirect
11. C8×M16×C89C2semidirect
207. C8×M16D89semidirect
208. C89#18069semidirect

Maximal subgroups

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

subgroupindex
#260308222
#260292562
#260191502
#260191382
#260190972
#260190902
C8×Dic3562
C8×D7122
#260190002
#260186612
C8×M16×C892
#1806989