Finite Group Databasebeta

← #36363175

Subgroups — #36363175

Order 22144.

548 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149706C211yesmaximal, characteristic, complemented
2C22×#14583809C211yesmaximal, characteristic, complemented
3#26012004C211yesmaximal, characteristic, complemented
4#121×D173C211yesmaximal, characteristic, complemented
5#26011745C211yesmaximal, characteristic, complemented
6#26011743C211yesmaximal, characteristic, complemented
7#26011740C211yesmaximal, characteristic, complemented
8#26011734C211yesmaximal, characteristic, complemented
9#26011727C211yesmaximal, characteristic, complemented
10#26011725C211yesmaximal, characteristic, complemented
11#26011722C211yesmaximal, characteristic, complemented
12#26011530C211yesmaximal, characteristic, complemented
13#26011494C211yesmaximal, characteristic, complemented
14#26011491C211yesmaximal, characteristic, complemented
15#465×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×D4×D173—21no—
18C2×D4×D173C2211yescharacteristic
19C22×C4×D173C2211yescharacteristic
20#19149706—24no—
21C2×#14583813C2211yescharacteristic, complemented
22C2×D1384C2211yescharacteristic, complemented
23C2×#14583811C2211yescharacteristic, complemented
24C2×#14583809—24no—
25C2×#14583809C2211yescharacteristic
26C2×#14583809C2211yescharacteristic
27C2×#14583809C2211yescharacteristic, complemented
28C2×#14583809C2211yescharacteristic, complemented
29C22×#13063436C2211yescharacteristic, complemented
30#19149664—21no—
31#19149655C2211yescharacteristic, complemented
32#19149654C2211yescharacteristic
33#19149654C2211yescharacteristic, complemented
34#19149653C2211yescharacteristic
35#19149653C2211yescharacteristic, complemented
36#19149652—21no—
37#19149651—21no—
38#19149650C2211yescharacteristic
39#19149649—21no—
40#19149641—21no—
41#19149639—21no—
42#19149637C2211yescharacteristic
43C4⋊C4×D173C2211yescharacteristic
44C22⋊C4×D173—21no—
45#19149613C2211yescharacteristic, complemented
46#19149612C2211yescharacteristic, complemented
47#19149610C2211yescharacteristic, complemented
48#19149603C2211yescharacteristic, complemented
49#19149601C2211yescharacteristic, complemented
50#19149601C2211yescharacteristic, complemented
51#19149600C2211yescharacteristic, complemented
52#19149599C2211yescharacteristic, complemented
53#19149599C2211yescharacteristic, complemented
54C2×SD16×C173C2211yescharacteristic, complemented
55C22×C8×C173C2211yescharacteristic, complemented
56C22⋊Q8×C173C2211yescharacteristic, complemented
57#121×C173C2211yescharacteristic
58#106×C173C2211yescharacteristic, complemented
59#103×C173C2211yescharacteristic, complemented
60#102×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62C23×D173D411yescharacteristic
63#14583827—22no—
64#14583827—22no—
65C2×D692—21no—
66C2×D692—21no—
67C2×D692—21no—
68C2×D692C2311yescharacteristic
69C2×D692C2311yescharacteristic
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C2×C22⋊D173—21no—
73C2×C22⋊D173—21no—
74D4×D173—22no—
75D4×D173—22no—
76D4×D173—44no—
77C22×Dic173D411yescharacteristic
78C2×C4×D173—22no—
79C2×C4×D173—22no—
80C2×C4×D173—21no—
81C2×C4×D173—21no—
82C2×C4×D173D411yescharacteristic
83C2×C4×D173D411yescharacteristic
84C2×C4×D173C2311yescharacteristic
85C2×C4×D173C2311yescharacteristic
86#14583813—22no—
87D1384—22no—
88#14583811—22no—
89#14583809—22no—
90#14583809—22no—
91#14583809—22no—
92#14583809D414yescomplemented
93C2×#13063436—22no—
94C2×#13063436C2311yescharacteristic
95C2×#13063436C2311yescharacteristic
96C4⋊Dic173C2311yescharacteristic
97C4⋊Dic173C2311yescharacteristic
98#14583805—21no—
99#14583805—21no—
100C22⋊Dic173—21no—
101#14583803—21no—
102#14583803—21no—
103#14583803—21no—
104#14583803—21no—
105#14583803—21no—
106#14583803—21no—
107C4×Dic173—21no—
108C2×Q8×C173C2311yescharacteristic
109C2×D4×C173—21no—
110C2×D4×C173C2311yescharacteristic
111C22×C4×C173C2311yescharacteristic
112SD16×C173—22no—
113C2×C8×C173—22no—
114C2×C8×C173C2311yescharacteristic
115C2×C8×C173C2311yescharacteristic
116C4⋊C4×C173—21no—
117C4⋊C4×C173C2311yescharacteristic
118C4⋊C4×C173C2311yescharacteristic
119C22⋊C4×C173—21no—
120C22⋊C4×C173—21no—
121C22×D173—22no—
122C22×D173—22no—
123C22×D173—21no—
124C22×D173—21no—
125C22×D173—41no—
126C22×D173—42no—
127C22×D173—42no—
128C22×D173—42no—
129C22×D173C2×D411yescharacteristic
130C22×D173C2×D411yescharacteristic
131D692—21no—
132D692—21no—
133D692—21no—
134D692—21no—
135D692—42no—
136D692—42no—
137D692—42no—
138C22⋊D173—42no—
139C22⋊D173—42no—
140C22⋊D173—42no—
141C22⋊D173—42no—
142C2×Dic173—22no—
143C2×Dic173—21no—
144C2×Dic173—21no—
145C2×Dic173C2×D411yescharacteristic
146C2×Dic173C2×D411yescharacteristic
147C4×D173—22no—
148C4×D173—21no—
149C4×D173—21no—
150C4×D173—22no—
151C4×D173—42no—
152C4×D173—42no—
153C4×D173C2×D412yes—
154C4×D173#4112yes—
155#13063436—21no—
156#13063436C2×D412yescomplemented
157C23×C173—21no—
158C23×C173C2×D411yescharacteristic
159Q8×C173—21no—
160Q8×C173—21no—
161D4×C173—21no—
162D4×C173—21no—
163D4×C173—42no—
164C2×C4×C173—21no—
165C2×C4×C173—21no—
166C2×C4×C173—21no—
167C2×C4×C173—21no—
168C2×C4×C173—21no—
169C2×C4×C173C2×D411yescharacteristic
170C2×C4×C173C2411yescharacteristic
171C8×C173—21no—
172C8×C173C2×D412yescomplemented
173C22×C173—21no—
174C22×C173—21no—
175C22×C173—41no—
176C22×C173—41no—
177C22×C173—41no—
178C22×C173AGL(1,ℤ/8)12yescomplemented
179C22×C173C22×D411yescharacteristic
180C2×D173—21no—
181C2×D173—21no—
182C2×D173—21no—
183C2×D173—21no—
184C2×D173—21no—
185C2×D173—44no—
186C2×D173—41no—
187C2×D173—41no—
188C2×D173—41no—
189C2×D173—41no—
190C2×D173—41no—
191C2×D173—41no—
192C2×D173—84no—
193C2×D173#12112yes—
194C4×C173—21no—
195C4×C173—41no—
196C4×C173—41no—
197C4×C173—41no—
198C4×C173C22×D411yescharacteristic
199C4×C173C2×#4111yescharacteristic
200Dic173—21no—
201Dic173—41no—
202Dic173—41no—
203Dic173#12112yes—
204C2×C173—22no—
205C2×C173—81no—
206C2×C173C2×#12111yescharacteristic
207C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
208C2×C173#57511yescharacteristic
209D173—22no—
210D173—41no—
211D173—81no—
212D173—81no—
213C173#305511yescharacteristic, complemented, Sylow
214#3055—1731nomaximal, Sylow
215C2×AGL(1,ℤ/8)—1731no—
216C22×M16—1731no—
217#525—1731no—
218C2×#121—1731no—
219#469—1731no—
220#469—1731no—
221#468—1731no—
222#468—1731no—
223#466—1731no—
224#466—1731no—
225#465—1731no—
226#465D17311yescharacteristic, complemented
227#428—1731no—
228#418—1731no—
229#417—1731no—
230C2×#41—1731no—
231C22×D4—3461no—
232C22×D4—1731no—
233C23×C4—1731no—
234AGL(1,ℤ/8)—3464no—
235C2×SD16—1731no—
236C2×SD16C2×D17311yescharacteristic, complemented
237C2×D8—1731no—
238C2×D8—1731no—
239C2×M16—3464no—
240C2×M16—1731no—
241C2×M16—1731no—
242C2×M16—1731no—
243C2×M16—1731no—
244C22×C8—1731no—
245C22×C8C2×D17311yescharacteristic, complemented
246#123—3461no—
247C22⋊Q8—1731no—
248C22⋊Q8C2×D17311yescharacteristic, complemented
249#121—3461no—
250#121—3461no—
251#121—1731no—
252#121—1731no—
253#121—1731no—
254#121—1731no—
255#121—1731no—
256#121C2×D17311yescharacteristic
257#120—3461no—
258C4×D4—3461no—
259C4×D4—3461no—
260#117—1731no—
261C2×C4⋊C4—1731no—
262C2×C22⋊C4—3461no—
263#107—1731no—
264#107—1731no—
265#106—1731no—
266#106C2×D17311yescharacteristic, complemented
267#103—1731no—
268#103C2×D17311yescharacteristic, complemented
269#102—1731no—
270#102—1731no—
271#102—1731no—
272#102—1731no—
273#102—1731no—
274#102C2×D17311yescharacteristic, complemented
275C24—3461no—
276C24—1731no—
277#41—3462no—
278#41—3462no—
279C2×Q8C22×D17311yescharacteristic
280C2×D4—21no—
281C2×D4—3461no—
282C2×D4—3461no—
283C2×D4—3461no—
284C2×D4—3461no—
285C2×D4—3461no—
286C2×D4—3461no—
287C2×D4—3462no—
288C2×D4—3462no—
289C2×D4—3461no—
290C2×D4—1731no—
291C2×D4—1731no—
292C2×D4—6924no—
293C2×D4C22×D17311yescharacteristic
294C22×C4—3462no—
295C22×C4—3462no—
296C22×C4—3461no—
297C22×C4—3461no—
298C22×C4—1731no—
299C22×C4—1731no—
300C22×C4—1731no—
301C22×C4—1731no—
302C22×C4—1731no—
303C22×C4C22×D17311yescharacteristic
304SD16—22no—
305SD16—3462no—
306D8—3462no—
307D8—3462no—
308M16—3462no—
309M16—3462no—
310M16—3462no—
311M16—1734no—
312C2×C8—22no—
313C2×C8—3462no—
314C2×C8—1731no—
315C2×C8—1731no—
316C2×C8C22×D17311yescharacteristic
317C2×C8C22×D17311yescharacteristic
318C4⋊C4—21no—
319C4⋊C4—3461no—
320C4⋊C4—3461no—
321C4⋊C4—1731no—
322C4⋊C4—1731no—
323C4⋊C4C22×D17311yescharacteristic
324C4⋊C4C22×D17311yescharacteristic
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—3461no—
328C22⋊C4—3461no—
329C22⋊C4—3461no—
330C22⋊C4—3461no—
331C22⋊C4—3461no—
332C22⋊C4—3461no—
333C22⋊C4—3461no—
334C42—3461no—
335C23—21no—
336C23—3462no—
337C23—3462no—
338C23—3461no—
339C23—3461no—
340C23—1731no—
341C23—1731no—
342C23—6921no—
343C23—6922no—
344C23—6922no—
345C23—6922no—
346C23D4×D17311yescharacteristic
347Q8—21no—
348Q8—21no—
349D4—21no—
350D4—21no—
351D4—42no—
352D4—3461no—
353D4—3461no—
354D4—3461no—
355D4—3461no—
356D4—6922no—
357D4—6922no—
358D4—6922no—
359D4—6922no—
360D4—6922no—
361D4—6922no—
362D4—6922no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—3462no—
369C2×C4—3462no—
370C2×C4—3462no—
371C2×C4—3461no—
372C2×C4—3461no—
373C2×C4—3461no—
374C2×C4—3461no—
375C2×C4—1731no—
376C2×C4—1731no—
377C2×C4—1732no—
378C2×C4—1732no—
379C2×C4—6922no—
380C2×C4—6922no—
381C2×C4D4×D17311yescharacteristic
382C2×C4C23×D17311yescharacteristic
383C8—21no—
384C8—3461no—
385C8—1732no—
386C8D4×D17312yescomplemented
387C22—21no—
388C22—21no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—3461no—
393C22—3461no—
394C22—3461no—
395C22—3461no—
396C22—3461no—
397C22—13844no—
398C22—1732no—
399C22—6924no—
400C22—6921no—
401C22—6921no—
402C22—6921no—
403C22—6921no—
404C22—6921no—
405C22—6921no—
406C22#1914970612yescomplemented
407C22C2×D4×D17311yescharacteristic
408C4—21no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—3461no—
413C4—1732no—
414C4—6921no—
415C4—6921no—
416C4C2×D4×D17311yescharacteristic
417C4#41×D17311yescharacteristic
418C2—22no—
419C2—81no—
420C2—3462no—
421C2—13841no—
422C2—13841no—
423C2—6921no—
424C2#121×D17311yescharacteristic
425C2C2×#1914970611yescharacteristic
426C2#2601228811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#465×C173C2
socleC22×C173C22×D4
hypercenter#465D173
hypocenterC173#3055
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×#19149706C2semidirect
2. C22×#14583809C2semidirect
3. #26012004C2semidirect
4. #121×D173C2semidirect
5. #26011745C2semidirect
6. #26011743C2semidirect
7. #26011740C2semidirect
8. #26011734C2semidirect
9. #26011727C2semidirect
10. #26011725C2semidirect
11. #26011722C2semidirect
12. #26011530C2semidirect
13. #26011494C2semidirect
14. #26011491C2semidirect
15. #465×C173C2semidirect
21. C2×#14583813C22semidirect
22. C2×D1384C22semidirect
23. C2×#14583811C22semidirect
27. C2×#14583809C22semidirect
28. C2×#14583809C22semidirect
29. C22×#13063436C22semidirect
31. #19149655C22semidirect
33. #19149654C22semidirect
35. #19149653C22semidirect
45. #19149613C22semidirect
46. #19149612C22semidirect
47. #19149610C22semidirect
48. #19149603C22semidirect
49. #19149601C22semidirect
50. #19149601C22semidirect
51. #19149600C22semidirect
52. #19149599C22semidirect
53. #19149599C22semidirect
54. C2×SD16×C173C22semidirect
55. C22×C8×C173C22semidirect
56. C22⋊Q8×C173C22semidirect
58. #106×C173C22semidirect
59. #103×C173C22semidirect
60. #102×C173C22semidirect
92. #14583809D4semidirect
156. #13063436C2×D4semidirect
172. C8×C173C2×D4semidirect
178. C22×C173AGL(1,ℤ/8)semidirect
213. C173#3055semidirect
226. #465D173semidirect
236. C2×SD16C2×D173semidirect
245. C22×C8C2×D173semidirect
248. C22⋊Q8C2×D173semidirect
266. #106C2×D173semidirect
268. #103C2×D173semidirect
274. #102C2×D173semidirect
386. C8D4×D173semidirect
406. C22#19149706semidirect

Maximal subgroups

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

subgroupindex
C2×#191497062
C22×#145838092
#260120042
#121×D1732
#260117452
#260117432
#260117402
#260117342
#260117272
#260117252
#260117222
#260115302
#260114942
#260114912
#465×C1732
#3055173