Finite Group Databasebeta

← #36363186

Subgroups — #36363186

Order 22144.

496 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×#19149708C211yesmaximal, characteristic, complemented
2C22×#14583809C211yesmaximal, characteristic, complemented
3#26012001C211yesmaximal, characteristic
4C22⋊Q8×D173C211yesmaximal, characteristic, complemented
5#26011750C211yesmaximal, characteristic, complemented
6#26011748C211yesmaximal, characteristic, complemented
7#26011736C211yesmaximal, characteristic, complemented
8#26011734C211yesmaximal, characteristic, complemented
9#26011732C211yesmaximal, characteristic, complemented
10#26011730C211yesmaximal, characteristic, complemented
11#26011718C211yesmaximal, characteristic, complemented
12#26011530C211yesmaximal, characteristic, complemented
13#26011498C211yesmaximal, characteristic, complemented
14#26011489C211yesmaximal, characteristic, complemented
15#465×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×Q8×D173C2211yescharacteristic
18C22×C4×D173C2211yescharacteristic
19#19149708—24no—
20C2×Dic692C2211yescharacteristic, complemented
21C2×#14583816C2211yescharacteristic, complemented
22C2×#14583814C2211yescharacteristic, complemented
23C2×#14583809—24no—
24C2×#14583809C2211yescharacteristic
25C2×#14583809C2211yescharacteristic
26C2×#14583809C2211yescharacteristic
27C2×#14583809C2211yescharacteristic
28C22×#13063436C2211yescharacteristic
29#19149661—21no—
30#19149659C2211yescharacteristic
31#19149659C2211yescharacteristic
32#19149658C2211yescharacteristic
33#19149658C2211yescharacteristic
34#19149657—21no—
35#19149656—21no—
36#19149655C2211yescharacteristic
37#19149650C2211yescharacteristic
38#19149647—21no—
39D4×Dic173—21no—
40#19149639—21no—
41#19149637C2211yescharacteristic
42C4⋊C4×D173—21no—
43C4⋊C4×D173C2211yescharacteristic
44C22⋊C4×D173—21no—
45#19149613C2211yescharacteristic, complemented
46#19149612C2211yescharacteristic, complemented
47#19149610C2211yescharacteristic, complemented
48#19149604C2211yescharacteristic, complemented
49#19149604C2211yescharacteristic, complemented
50#19149603C2211yescharacteristic, complemented
51#19149602C2211yescharacteristic, complemented
52#19149602C2211yescharacteristic, complemented
53#19149600C2211yescharacteristic, complemented
54C2×SD16×C173C2211yescharacteristic, complemented
55C22×C8×C173C2211yescharacteristic
56C22⋊Q8×C173C2211yescharacteristic
57#121×C173C2211yescharacteristic
58#106×C173C2211yescharacteristic, complemented
59#103×C173C2211yescharacteristic, complemented
60#102×C173C2211yescharacteristic, complemented
61C23×D173D411yescharacteristic
62#14583825—22no—
63#14583825—22no—
64C2×Dic346—21no—
65C2×Dic346C2311yescharacteristic
66C2×Dic346C2311yescharacteristic
67Q8×D173—22no—
68Q8×D173—22no—
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C22×Dic173—21no—
72C22×Dic173D411yescharacteristic
73C2×C4×D173—22no—
74C2×C4×D173—22no—
75C2×C4×D173—21no—
76C2×C4×D173—21no—
77C2×C4×D173D411yescharacteristic
78C2×C4×D173D411yescharacteristic
79C2×C4×D173C2311yescharacteristic
80C2×C4×D173C2311yescharacteristic
81Dic692—22no—
82#14583816—22no—
83#14583814—22no—
84#14583809—22no—
85#14583809—22no—
86#14583809—22no—
87#14583809D414yescomplemented
88C2×#13063436—22no—
89C2×#13063436C2311yescharacteristic
90C2×#13063436C2311yescharacteristic
91C4⋊Dic173—21no—
92C4⋊Dic173—21no—
93C4⋊Dic173C2311yescharacteristic
94C4⋊Dic173C2311yescharacteristic
95#14583805—21no—
96#14583805—21no—
97#14583805—21no—
98#14583805—21no—
99C22⋊Dic173—21no—
100C22⋊Dic173—21no—
101C22⋊Dic173—21no—
102#14583803—21no—
103#14583803—21no—
104#14583803—21no—
105#14583803—21no—
106C4×Dic173—21no—
107C2×Q8×C173C2311yescharacteristic
108C2×D4×C173—21no—
109C2×D4×C173C2311yescharacteristic
110C22×C4×C173C2311yescharacteristic
111SD16×C173—22no—
112C2×C8×C173—22no—
113C2×C8×C173C2311yescharacteristic
114C2×C8×C173C2311yescharacteristic
115C4⋊C4×C173—21no—
116C4⋊C4×C173C2311yescharacteristic
117C4⋊C4×C173C2311yescharacteristic
118C22⋊C4×C173—21no—
119C22⋊C4×C173—21no—
120C22×D173—22no—
121C22×D173—22no—
122C22×D173—41no—
123C22×D173C2×D411yescharacteristic
124C22×D173C2×D411yescharacteristic
125Dic346—21no—
126Dic346—21no—
127Dic346—21no—
128Dic346—21no—
129Dic346—42no—
130C22⋊D173—42no—
131C22⋊D173—42no—
132C2×Dic173—22no—
133C2×Dic173—21no—
134C2×Dic173—21no—
135C2×Dic173—21no—
136C2×Dic173—21no—
137C2×Dic173—42no—
138C2×Dic173C2×D411yescharacteristic
139C2×Dic173C2×D411yescharacteristic
140C4×D173—22no—
141C4×D173—21no—
142C4×D173—21no—
143C4×D173—22no—
144C4×D173—42no—
145C4×D173—42no—
146C4×D173C2×D412yes—
147C4×D173#4112yes—
148#13063436—21no—
149#13063436C2×D412yes—
150C23×C173—21no—
151C23×C173C2×D411yescharacteristic
152Q8×C173—21no—
153Q8×C173—21no—
154D4×C173—21no—
155D4×C173—21no—
156D4×C173—42no—
157C2×C4×C173—21no—
158C2×C4×C173—21no—
159C2×C4×C173—21no—
160C2×C4×C173—21no—
161C2×C4×C173—21no—
162C2×C4×C173C2×D411yescharacteristic
163C2×C4×C173C2411yescharacteristic
164C8×C173—21no—
165C8×C173C2×D412yes—
166C22×C173—21no—
167C22×C173—21no—
168C22×C173—41no—
169C22×C173—41no—
170C22×C173—41no—
171C22×C173#13712yescomplemented
172C22×C173C22×D411yescharacteristic
173C2×D173—21no—
174C2×D173—21no—
175C2×D173—21no—
176C2×D173—21no—
177C2×D173—21no—
178C2×D173—44no—
179C2×D173#12112yes—
180C4×C173—21no—
181C4×C173—41no—
182C4×C173—41no—
183C4×C173—41no—
184C4×C173C22×D411yescharacteristic
185C4×C173C2×#4111yescharacteristic
186Dic173—21no—
187Dic173—41no—
188Dic173—41no—
189Dic173—41no—
190Dic173—41no—
191Dic173#12112yes—
192C2×C173—22no—
193C2×C173—81no—
194C2×C173C2×#12111yescharacteristic
195C2×C173C2×#13711yescharacteristic
196C2×C173#57511yescharacteristic
197D173—22no—
198D173—41no—
199C173#305611yescharacteristic, complemented, Sylow
200#3056—1731nomaximal, Sylow
201C2×#137—1731no—
202C22×M16—1731no—
203#525—1731no—
204C2×C22⋊Q8—1731no—
205#470—1731no—
206#470—1731no—
207#468—1731no—
208#468—1731no—
209#467—1731no—
210#467—1731no—
211#465—1731no—
212#465D17311yescharacteristic, complemented
213#428—1731no—
214#419—1731no—
215#417—1731no—
216C2×#41—1731no—
217C22×Q8—1731no—
218C23×C4—1731no—
219#137—3464no—
220C2×Q16—1731no—
221C2×Q16—1731no—
222C2×SD16—1731no—
223C2×SD16C2×D17311yescharacteristic, complemented
224C2×M16—3464no—
225C2×M16—1731no—
226C2×M16—1731no—
227C2×M16—1731no—
228C2×M16—1731no—
229C22×C8—1731no—
230C22×C8C2×D17311yescharacteristic
231#123—3461no—
232C22⋊Q8—3461no—
233C22⋊Q8—3461no—
234C22⋊Q8—1731no—
235C22⋊Q8—1731no—
236C22⋊Q8—1731no—
237C22⋊Q8—1731no—
238C22⋊Q8—1731no—
239C22⋊Q8C2×D17311yescharacteristic
240#121—1731no—
241#121C2×D17311yescharacteristic
242#120—3461no—
243C4×D4—3461no—
244C4×D4—3461no—
245#117—1731no—
246C2×C4⋊C4—3461no—
247C2×C4⋊C4—1731no—
248C2×C22⋊C4—3461no—
249#107—1731no—
250#107—1731no—
251#106—1731no—
252#106C2×D17311yescharacteristic, complemented
253#103—1731no—
254#103—1731no—
255#103—1731no—
256#103—1731no—
257#103—1731no—
258#103C2×D17311yescharacteristic, complemented
259#102—1731no—
260#102C2×D17311yescharacteristic, complemented
261C24—1731no—
262#41—3462no—
263#41—3462no—
264C2×Q8—3461no—
265C2×Q8—3462no—
266C2×Q8—3462no—
267C2×Q8—1731no—
268C2×Q8—1731no—
269C2×Q8C22×D17311yescharacteristic
270C2×D4—21no—
271C2×D4—3461no—
272C2×D4—3461no—
273C2×D4C22×D17311yescharacteristic
274C22×C4—3462no—
275C22×C4—3462no—
276C22×C4—3461no—
277C22×C4—3461no—
278C22×C4—3461no—
279C22×C4—1731no—
280C22×C4—1731no—
281C22×C4—1731no—
282C22×C4—1731no—
283C22×C4—1731no—
284C22×C4C22×D17311yescharacteristic
285Q16—3462no—
286Q16—3462no—
287SD16—22no—
288SD16—3462no—
289M16—3462no—
290M16—3462no—
291M16—3462no—
292M16—1734no—
293C2×C8—22no—
294C2×C8—3462no—
295C2×C8—1731no—
296C2×C8—1731no—
297C2×C8C22×D17311yescharacteristic
298C2×C8C22×D17311yescharacteristic
299C4⋊C4—21no—
300C4⋊C4—3461no—
301C4⋊C4—3461no—
302C4⋊C4—3461no—
303C4⋊C4—3461no—
304C4⋊C4—3461no—
305C4⋊C4—3461no—
306C4⋊C4—1731no—
307C4⋊C4—1731no—
308C4⋊C4C22×D17311yescharacteristic
309C4⋊C4C22×D17311yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—3461no—
313C22⋊C4—3461no—
314C22⋊C4—3461no—
315C22⋊C4—3461no—
316C22⋊C4—3461no—
317C22⋊C4—3461no—
318C22⋊C4—3461no—
319C42—3461no—
320C23—21no—
321C23—3462no—
322C23—3462no—
323C23—1731no—
324C23—1731no—
325C23—6921no—
326C23D4×D17311yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—3461no—
330Q8—3461no—
331Q8—3461no—
332Q8—3461no—
333Q8—6922no—
334D4—21no—
335D4—21no—
336D4—42no—
337D4—6922no—
338D4—6922no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—3462no—
345C2×C4—3462no—
346C2×C4—3462no—
347C2×C4—3461no—
348C2×C4—3461no—
349C2×C4—3461no—
350C2×C4—3461no—
351C2×C4—3461no—
352C2×C4—3461no—
353C2×C4—1731no—
354C2×C4—1731no—
355C2×C4—1732no—
356C2×C4—1732no—
357C2×C4—6922no—
358C2×C4—6922no—
359C2×C4—6922no—
360C2×C4D4×D17311yescharacteristic
361C2×C4C23×D17311yescharacteristic
362C8—21no—
363C8—3461no—
364C8—1732no—
365C8D4×D17312yes—
366C22—21no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—3461no—
372C22—3461no—
373C22—3461no—
374C22—3461no—
375C22—3461no—
376C22—1732no—
377C22—6924no—
378C22#1914970812yescomplemented
379C22C2×D4×D17311yescharacteristic
380C4—21no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—3461no—
385C4—1732no—
386C4—6921no—
387C4—6921no—
388C4—6921no—
389C4—6921no—
390C4C2×D4×D17311yescharacteristic
391C4#41×D17311yescharacteristic
392C2—22no—
393C2—81no—
394C2—3462no—
395C2—6921no—
396C2#121×D17311yescharacteristic
397C2C2×#1914970811yescharacteristic
398C2#2601228811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#465×C173C2
socleC22×C173C22×D4
prime radical#26012001C2
hypercenter#465D173
hypocenterC173#3056
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#19149708C2semidirect
2. C22×#14583809C2semidirect
4. C22⋊Q8×D173C2semidirect
5. #26011750C2semidirect
6. #26011748C2semidirect
7. #26011736C2semidirect
8. #26011734C2semidirect
9. #26011732C2semidirect
10. #26011730C2semidirect
11. #26011718C2semidirect
12. #26011530C2semidirect
13. #26011498C2semidirect
14. #26011489C2semidirect
15. #465×C173C2semidirect
20. C2×Dic692C22semidirect
21. C2×#14583816C22semidirect
22. C2×#14583814C22semidirect
45. #19149613C22semidirect
46. #19149612C22semidirect
47. #19149610C22semidirect
48. #19149604C22semidirect
49. #19149604C22semidirect
50. #19149603C22semidirect
51. #19149602C22semidirect
52. #19149602C22semidirect
53. #19149600C22semidirect
54. C2×SD16×C173C22semidirect
58. #106×C173C22semidirect
59. #103×C173C22semidirect
60. #102×C173C22semidirect
87. #14583809D4semidirect
171. C22×C173#137semidirect
199. C173#3056semidirect
212. #465D173semidirect
223. C2×SD16C2×D173semidirect
252. #106C2×D173semidirect
258. #103C2×D173semidirect
260. #102C2×D173semidirect
378. C22#19149708semidirect

Maximal subgroups

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

subgroupindex
C2×#191497082
C22×#145838092
#260120012
C22⋊Q8×D1732
#260117502
#260117482
#260117362
#260117342
#260117322
#260117302
#260117182
#260115302
#260114982
#260114892
#465×C1732
#3056173