Finite Group Databasebeta

← #36712377

Subgroups — #36712377

Order 22784.

458 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26033120C211yesmaximal, characteristic
2#26022539C211yesmaximal, characteristic, complemented
3#26022535C211yesmaximal, characteristic, complemented
4#26022528C211yesmaximal, characteristic, complemented
5#26022416C211yesmaximal, characteristic, complemented
6#26022416C211yesmaximal, characteristic, complemented
7#26022406C211yesmaximal, characteristic
8#26022406C211yesmaximal, characteristic
9#26022396C211yesmaximal, characteristic
10#26022394C211yesmaximal, characteristic, complemented
11#26022161C211yesmaximal, characteristic, complemented
12#26022114C211yesmaximal, characteristic
13#26022114C211yesmaximal, characteristic
14C4×#15667955C211yesmaximal, characteristic
15C89×#3383C211yesmaximal, characteristic, complemented
16#19152308—21no—
17C2×#15667955C2211yescharacteristic
18#19152003C2211yescharacteristic
19#19152000C2211yescharacteristic
20#19152000C2211yescharacteristic
21#19151899—21no—
22#19151899—21no—
23#19151889—21no—
24C4×D712C2211yescharacteristic
25C4×D712C2211yescharacteristic
26C4×D712C2211yescharacteristic
27#19151425C2211yescharacteristic
28#19151425C2211yescharacteristic
29#19151425C2211yescharacteristic
30#19151425C2211yescharacteristic
31#19151425C2211yescharacteristic
32#19151423C2211yescharacteristic
33#19151423C2211yescharacteristic
34#19151423C2211yescharacteristic
35#19151423C2211yescharacteristic
36C4⋊#13083549C2211yescharacteristic
37C4⋊#13083549C2211yescharacteristic
38C4⋊#13083549C2211yescharacteristic
39#19151402C2211yescharacteristic
40#19151402C2211yescharacteristic
41#19151402C2211yescharacteristic
42#19151398C2211yescharacteristic
43#19151398C2211yescharacteristic
44#19151398C2211yescharacteristic
45#19151374C2211yescharacteristic
46#19151374C2211yescharacteristic
47C4×#13083549C2211yescharacteristic
48#557×C89—21no—
49#500×C89C2211yescharacteristic
50#499×C89C2211yescharacteristic
51#499×C89C2211yescharacteristic
52#479×C89—21no—
53#479×C89—21no—
54#475×C89—21no—
55C8×Q8×C89C2211yescharacteristic
56C4×Q16×C89C2211yescharacteristic
57C4×Q16×C89C2211yescharacteristic
58C4×Q16×C89C2211yescharacteristic
59C2×D712C2311yescharacteristic
60#15668012—21no—
61#15668004—21no—
62#15668004—21no—
63#15667990—41no—
64#15667990—41no—
65#15667988—41no—
66C4×D356—21no—
67C4×D356—21no—
68C4×D356—21no—
69C4⋊C4×D89—41no—
70#15667955Q814yescomplemented
71C2×#13083549C2311yescharacteristic
72C2×#13083549C2311yescharacteristic
73C2×#13083549C2311yescharacteristic
74C2×#13083549C2311yescharacteristic
75#15667946—21no—
76#15667946C2311yescharacteristic
77#15667946C2311yescharacteristic
78#15667946C2311yescharacteristic
79#15667944—21no—
80#15667944—21no—
81#15667934—21no—
82#15667934—21no—
83#15667934—21no—
84C2×Q16×C89C2311yescharacteristic
85C4⋊Q8×C89—21no—
86C4⋊Q8×C89—41no—
87#125×C89—21no—
88#125×C89—21no—
89#125×C89—41no—
90#125×C89—41no—
91C4×Q8×C89—21no—
92C4×Q8×C89—21no—
93C4×Q8×C89—21no—
94C4×Q8×C89—41no—
95C4×Q8×C89D411yescharacteristic
96#107×C89—21no—
97#107×C89C2311yescharacteristic
98#107×C89C2311yescharacteristic
99#107×C89C2311yescharacteristic
100#106×C89—21no—
101#106×C89—21no—
102C4⋊C8×C89D411yescharacteristic
103C4⋊C8×C89D411yescharacteristic
104C4⋊C8×C89D411yescharacteristic
105#103×C89—21no—
106#103×C89—21no—
107#103×C89—21no—
108C4×C8×C89C2311yescharacteristic
109C4×C8×C89C2311yescharacteristic
110C4×C8×C89C2311yescharacteristic
111C2×Dic178—41no—
112C2×D356—21no—
113C2×C4×D89—41no—
114C2×C4×D89—41no—
115C2×C4×D89—41no—
116D712—21no—
117D712C2×Q812yes—
118C4⋊Dic89—21no—
119C4⋊Dic89—21no—
120C4⋊Dic89—21no—
121C4⋊Dic89—41no—
122#13083553—41no—
123#13083553—41no—
124#13083553—41no—
125#13083553—41no—
126#13083551—41no—
127#13083551—41no—
128#13083551—41no—
129#13083549—21no—
130#13083549—21no—
131#13083549—21no—
132#13083549C2×Q812yes—
133C2×Q8×C89—21no—
134C2×Q8×C89—41no—
135C2×Q8×C89D811yescharacteristic
136Q16×C89—21no—
137Q16×C89C2×Q812yes—
138C2×C8×C89C2×D411yescharacteristic
139C2×C8×C89C2×D411yescharacteristic
140C2×C8×C89C2×D411yescharacteristic
141C2×C8×C89C2411yescharacteristic
142C4⋊C4×C89—21no—
143C4⋊C4×C89—21no—
144C4⋊C4×C89—21no—
145C4⋊C4×C89—41no—
146C4⋊C4×C89—41no—
147C4⋊C4×C89—41no—
148C4⋊C4×C89—41no—
149C4⋊C4×C89—41no—
150C4⋊C4×C89—41no—
151C4⋊C4×C89—41no—
152C4⋊C4×C89—41no—
153C4⋊C4×C89D811yescharacteristic
154C4⋊C4×C89D811yescharacteristic
155C4⋊C4×C89D811yescharacteristic
156C42×C89—41no—
157C42×C89—41no—
158C42×C89—41no—
159C42×C89C2×D411yescharacteristic
160C42×C89C2×D411yescharacteristic
161C42×C89C2×D411yescharacteristic
162C22×D89—41no—
163Q8×C89—22no—
164Q8×C89—41no—
165Q8×C89—41no—
166Q8×C89—82no—
167C2×C4×C89—41no—
168C2×C4×C89—41no—
169C2×C4×C89—41no—
170C2×C4×C89—41no—
171C2×C4×C89C2×D811yescharacteristic
172C2×C4×C89C2×D811yescharacteristic
173C2×C4×C89C2×D811yescharacteristic
174C2×C4×C89C2×D811yescharacteristic
175C2×C4×C89C2×D811yescharacteristic
176C2×C4×C89C2×D811yescharacteristic
177C2×C4×C89C22×D411yescharacteristic
178C2×Dic89—41no—
179C2×Dic89—41no—
180C2×Dic89—41no—
181D356—22no—
182D356—41no—
183C4×D89—82no—
184C4×D89—161no—
185C4×D89—161no—
186Dic178—161no—
187C8×C89—21no—
188C8×C89—21no—
189C8×C89—21no—
190C8×C89C22×Q811yescharacteristic
191C8×C892−1+411yescharacteristic
192C22×C89C22×D811yescharacteristic
193C2×D89—42no—
194C2×D89—81no—
195C2×D89—81no—
196C4×C89—21no—
197C4×C89—21no—
198C4×C89—21no—
199C4×C89—21no—
200C4×C89—21no—
201C4×C89—21no—
202C4×C89—42no—
203C4×C89—81no—
204C4×C89—81no—
205C4×C89—81no—
206C4×C89D4×Q811yescharacteristic
207C4×C89#57811yescharacteristic
208Dic89—81no—
209Dic89—81no—
210Dic89—81no—
211#33955—891nomaximal, Sylow
212C2×C89Q8×D811yescharacteristic
213C2×C89#341411yescharacteristic
214C2×C89#341611yescharacteristic
215D89—82no—
216#3383D8911yescharacteristic, complemented
217#3380—891no—
218#2255—891no—
219#2255—891no—
220#2253—891no—
221#2229—891no—
222#2229—891no—
223#2228—891no—
224#2228—891no—
225#2226—891no—
226#2225—891no—
227#2183—891no—
228#2175—891no—
229#2175—891no—
230C4×SD32—891no—
231C89#3395511yescharacteristic, complemented, Sylow
232#557—21no—
233#554—1781no—
234C2×SD32—891no—
235#500—891no—
236#500C2×D8911yescharacteristic
237#499—891no—
238#499—891no—
239#499C2×D8911yescharacteristic
240#499C2×D8911yescharacteristic
241#479—21no—
242#479—21no—
243#478—1781no—
244#478—1781no—
245#476—1781no—
246#475—21no—
247C8×Q8C2×D8911yescharacteristic
248C4×Q16C2×D8911yescharacteristic
249C4×Q16C2×D8911yescharacteristic
250C4×Q16C2×D8911yescharacteristic
251C4×D8—891no—
252C4×D8—891no—
253C4×D8—891no—
254#367—891no—
255#367—891no—
256#367—891no—
257#367—891no—
258#367—891no—
259#366—891no—
260#366—891no—
261#366—891no—
262#366—891no—
263C4⋊C16—891no—
264C4⋊C16—891no—
265C4⋊C16—891no—
266#358—891no—
267#358—891no—
268#358—891no—
269#357—891no—
270#357—891no—
271#357—891no—
272#346—891no—
273#346—891no—
274C4×C16—891no—
275C2×Q16C22×D8911yescharacteristic
276C2×D8—891no—
277C4⋊Q8—21no—
278C4⋊Q8—41no—
279C4⋊Q8—1781no—
280#125—21no—
281#125—21no—
282#125—41no—
283#125—41no—
284#125—1781no—
285#125—1781no—
286C22⋊Q8—3561no—
287C22⋊Q8—3561no—
288C22⋊Q8—3561no—
289C4×Q8—21no—
290C4×Q8—21no—
291C4×Q8—21no—
292C4×Q8—41no—
293C4×Q8C22⋊D8911yescharacteristic
294C4×D4—1781no—
295C4×D4—1781no—
296C4×D4—1781no—
297C2×C4⋊C4—3561no—
298SD32—894no—
299C2×C16—891no—
300C2×C16—891no—
301C2×C16—891no—
302C2×C16—891no—
303#107—21no—
304#107—891no—
305#107—891no—
306#107—891no—
307#107—1781no—
308#107C22×D8911yescharacteristic
309#107C22×D8911yescharacteristic
310#107C22×D8911yescharacteristic
311#106—21no—
312#106—21no—
313#106—1781no—
314#106—1781no—
315C4⋊C8C22⋊D8911yescharacteristic
316C4⋊C8C22⋊D8911yescharacteristic
317C4⋊C8C22⋊D8911yescharacteristic
318#103—21no—
319#103—21no—
320#103—21no—
321#102—1781no—
322#102—1781no—
323#102—1781no—
324C4×C8C22×D8911yescharacteristic
325C4×C8C22×D8911yescharacteristic
326C4×C8C22×D8911yescharacteristic
327C2×Q8—21no—
328C2×Q8—41no—
329C2×Q8—3561no—
330C2×Q8#1308355911yescharacteristic
331C2×D4—1781no—
332C22×C4—3561no—
333C22×C4—3561no—
334C22×C4—3561no—
335Q16—21no—
336Q16Q8×D8912yes—
337D8—892no—
338D8—1781no—
339C2×C8C2×C22⋊D8911yescharacteristic
340C2×C8C2×C22⋊D8911yescharacteristic
341C2×C8C2×C22⋊D8911yescharacteristic
342C2×C8C23×D8911yescharacteristic
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—3561no—
355C4⋊C4—3561no—
356C4⋊C4—3561no—
357C4⋊C4—3561no—
358C4⋊C4—3561no—
359C4⋊C4—1781no—
360C4⋊C4—1781no—
361C4⋊C4—1781no—
362C4⋊C4#1308355911yescharacteristic
363C4⋊C4#1308355911yescharacteristic
364C4⋊C4#1308355911yescharacteristic
365C22⋊C4—3561no—
366C22⋊C4—3561no—
367C22⋊C4—3561no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42C2×C22⋊D8911yescharacteristic
372C42C2×C22⋊D8911yescharacteristic
373C42C2×C22⋊D8911yescharacteristic
374C16—892no—
375C16—1781no—
376C16—1781no—
377C16—1781no—
378C23—3561no—
379Q8—22no—
380Q8—41no—
381Q8—41no—
382Q8—82no—
383Q8—14241no—
384D4—3561no—
385D4—1782no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—3561no—
391C2×C4—3561no—
392C2×C4—3561no—
393C2×C4—14241no—
394C2×C4—14241no—
395C2×C4—7122no—
396C2×C4C2×#1308355911yescharacteristic
397C2×C4C2×#1308355911yescharacteristic
398C2×C4C2×#1308355911yescharacteristic
399C2×C4C2×#1308355911yescharacteristic
400C2×C4C2×#1308355911yescharacteristic
401C2×C4C2×#1308355911yescharacteristic
402C2×C4C22×C22⋊D8911yescharacteristic
403C8—21no—
404C8—21no—
405C8—21no—
406C8C2×Q8×D8911yescharacteristic
407C8#1566806011yescharacteristic
408C22—3562no—
409C22—7121no—
410C22—7121no—
411C22C22×#1308355911yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—42no—
419C4—81no—
420C4—81no—
421C4—81no—
422C4—7121no—
423C4—7121no—
424C4—7121no—
425C4Q8×C22⋊D8911yescharacteristic
426C4#1915241311yescharacteristic
427C2—7122no—
428C2Q8×#1308355911yescharacteristic
429C2#2603338611yescharacteristic
430C2#2603339611yescharacteristic
subgroupG/H
centerC22C22×#13083559
commutator, cosocleC2×C8×C89C24
FrattiniC2×C8C23×D89
FittingC89×#3383C2
socleC22×C89C22×D8
prime radicalC2×D712C23
hypercenter#3383D89
hypocenterC89#33955
LCSG ▷ C2×C8×C89 ▷ C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ #3383

Complements

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

NH ≅ G/Ntype
2. #26022539C2semidirect
3. #26022535C2semidirect
4. #26022528C2semidirect
5. #26022416C2semidirect
6. #26022416C2semidirect
10. #26022394C2semidirect
11. #26022161C2semidirect
15. C89×#3383C2semidirect
70. #15667955Q8semidirect
216. #3383D89semidirect
231. C89#33955semidirect

Maximal subgroups

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

subgroupindex
#260331202
#260225392
#260225352
#260225282
#260224162
#260224062
#260223962
#260223942
#260221612
#260221142
C4×#156679552
C89×#33832
#3395589