Finite Group Databasebeta

← #36363167

Subgroups — #36363167

Order 22144.

500 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149700C211yesmaximal, characteristic, complemented
2C2×M16×D173C211yesmaximal, characteristic, complemented
3#26012004C211yesmaximal, characteristic, complemented
4#26012001C211yesmaximal, characteristic, complemented
5#26011748C211yesmaximal, characteristic, complemented
6#26011747C211yesmaximal, characteristic, complemented
7#26011743C211yesmaximal, characteristic, complemented
8#26011742C211yesmaximal, characteristic, complemented
9#26011738C211yesmaximal, characteristic, complemented
10#26011737C211yesmaximal, characteristic, complemented
11#26011734C211yesmaximal, characteristic, complemented
12#26011528C211yesmaximal, characteristic, complemented
13#26011486C211yesmaximal, characteristic, complemented
14#26011484C211yesmaximal, characteristic, complemented
15#468×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583825C2211yescharacteristic
18C22×C4×D173C2211yescharacteristic
19#19149700—24no—
20C2×#14583816C2211yescharacteristic, complemented
21C2×#14583815C2211yescharacteristic, complemented
22C2×#14583811C2211yescharacteristic, complemented
23C2×#14583810C2211yescharacteristic
24C2×#14583809C2211yescharacteristic
25C2×#14583809C2211yescharacteristic
26M16×D173—24no—
27C2×C8×D173C2211yescharacteristic
28C2×C8×D173C2211yescharacteristic
29#19149664—21no—
30#19149661—21no—
31#19149659C2211yescharacteristic
32#19149658C2211yescharacteristic
33#19149655C2211yescharacteristic
34#19149654C2211yescharacteristic
35#19149653C2211yescharacteristic
36#19149650C2211yescharacteristic
37#19149649—21no—
38#19149647—21no—
39D4×Dic173—21no—
40#19149641—21no—
41#19149639—21no—
42#19149639—21no—
43#19149637C2211yescharacteristic
44#19149637C2211yescharacteristic
45#19149613C2211yescharacteristic, complemented
46#19149610C2211yescharacteristic, complemented
47#19149610C2211yescharacteristic, complemented
48#19149604C2211yescharacteristic, complemented
49#19149603C2211yescharacteristic, complemented
50#19149602C2211yescharacteristic, complemented
51#19149601C2211yescharacteristic, complemented
52#19149600C2211yescharacteristic, complemented
53#19149599C2211yescharacteristic, complemented
54C2×SD16×C173C2211yescharacteristic, complemented
55C2×M16×C173C2211yescharacteristic
56C22⋊Q8×C173C2211yescharacteristic
57#121×C173C2211yescharacteristic
58#107×C173C2211yescharacteristic, complemented
59#103×C173C2211yescharacteristic, complemented
60#102×C173C2211yescharacteristic, complemented
61C23×D173D411yescharacteristic
62#14583827—22no—
63#14583827—22no—
64#14583825—22no—
65#14583825—22no—
66C2×Dic346C2311yescharacteristic
67C2×D692—21no—
68C2×D692—21no—
69C2×D692C2311yescharacteristic
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C2×C22⋊D173—21no—
73C22×Dic173—21no—
74C22×Dic173D411yescharacteristic
75C2×C4×D173—22no—
76C2×C4×D173—22no—
77C2×C4×D173—21no—
78C2×C4×D173D411yescharacteristic
79C2×C4×D173D411yescharacteristic
80C2×C4×D173C2311yescharacteristic
81C2×C4×D173C2311yescharacteristic
82#14583816—22no—
83#14583815—22no—
84#14583811—22no—
85#14583810—22no—
86#14583809—22no—
87#14583809—22no—
88C2×#13063436C2311yescharacteristic
89C2×#13063436C2311yescharacteristic
90C8×D173—22no—
91C8×D173D414yescomplemented
92C4⋊Dic173—21no—
93C4⋊Dic173C2311yescharacteristic
94C4⋊Dic173C2311yescharacteristic
95#14583805—21no—
96#14583805—21no—
97C22⋊Dic173—21no—
98C22⋊Dic173—21no—
99#14583803—21no—
100#14583803—21no—
101#14583803—21no—
102#14583803—21no—
103#14583803—21no—
104#14583803—21no—
105C4×Dic173—21no—
106C4×Dic173—21no—
107C2×Q8×C173C2311yescharacteristic
108C2×D4×C173—21no—
109C2×D4×C173C2311yescharacteristic
110C22×C4×C173C2311yescharacteristic
111SD16×C173—22no—
112M16×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—21no—
122C22×D173—41no—
123C22×D173—41no—
124C22×D173C2×D411yescharacteristic
125C22×D173C2×D411yescharacteristic
126Dic346—21no—
127Dic346—21no—
128D692—21no—
129D692—21no—
130D692—42no—
131D692—42no—
132C22⋊D173—42no—
133C22⋊D173—42no—
134C22⋊D173—42no—
135C2×Dic173—22no—
136C2×Dic173—21no—
137C2×Dic173—21no—
138C2×Dic173—21no—
139C2×Dic173—42no—
140C2×Dic173C2×D411yescharacteristic
141C2×Dic173C2×D411yescharacteristic
142C4×D173—22no—
143C4×D173—21no—
144C4×D173—21no—
145C4×D173—22no—
146C4×D173—42no—
147C4×D173C2×D412yes—
148C4×D173#4112yes—
149#13063436—21no—
150#13063436C2×D412yes—
151C23×C173—21no—
152C23×C173C2×D411yescharacteristic
153Q8×C173—21no—
154Q8×C173—21no—
155D4×C173—21no—
156D4×C173—21no—
157D4×C173—42no—
158C2×C4×C173—21no—
159C2×C4×C173—21no—
160C2×C4×C173—21no—
161C2×C4×C173—21no—
162C2×C4×C173—21no—
163C2×C4×C173C2×D411yescharacteristic
164C2×C4×C173C2411yescharacteristic
165C8×C173—21no—
166C8×C173C2×D412yes—
167C22×C173—21no—
168C22×C173—21no—
169C22×C173—21no—
170C22×C173—41no—
171C22×C173—41no—
172C22×C173—41no—
173C22×C173C22×D411yescharacteristic
174C2×D173—21no—
175C2×D173—21no—
176C2×D173—21no—
177C2×D173—21no—
178C2×D173—21no—
179C2×D173—44no—
180C2×D173—41no—
181C2×D173—41no—
182C2×D173—41no—
183C2×D173#12112yes—
184C4×C173—21no—
185C4×C173—41no—
186C4×C173—41no—
187C4×C173—41no—
188C4×C173C22×D411yescharacteristic
189C4×C173C2×#4111yescharacteristic
190Dic173—21no—
191Dic173—41no—
192Dic173—41no—
193Dic173—41no—
194Dic173#12112yes—
195C2×C173—41no—
196C2×C173—81no—
197C2×C173C2×#12111yescharacteristic
198C2×C173#57511yescharacteristic
199C2×C173#57511yescharacteristic
200D173—22no—
201D173—41no—
202D173—81no—
203C173#305411yescharacteristic, complemented, Sylow
204#3054—1731nomaximal, Sylow
205C2×#135—1731no—
206C22×M16—1731no—
207#525—1731no—
208#525—1731no—
209#470—1731no—
210#470—1731no—
211#469—1731no—
212#469—1731no—
213#468—1731no—
214#468—1731no—
215#468—1731no—
216#468D17311yescharacteristic, complemented
217#427—1731no—
218#416—1731no—
219#416—1731no—
220C2×#41—1731no—
221C2×#41—1731no—
222C23×C4—1731no—
223#135—3464no—
224C2×Q16—1731no—
225C2×SD16—1731no—
226C2×SD16C2×D17311yescharacteristic, complemented
227C2×D8—1731no—
228C2×M16—3464no—
229C2×M16—1731no—
230C2×M16—1731no—
231C2×M16—1731no—
232C2×M16C2×D17311yescharacteristic
233C22×C8—1731no—
234C22×C8—1731no—
235#123—3461no—
236#123—3461no—
237C22⋊Q8—1731no—
238C22⋊Q8—1731no—
239C22⋊Q8—1731no—
240C22⋊Q8C2×D17311yescharacteristic
241#121—1731no—
242#121—1731no—
243#121—1731no—
244#121C2×D17311yescharacteristic
245#120—3461no—
246#120—3461no—
247C4×D4—3461no—
248C4×D4—3461no—
249C4×D4—3461no—
250C4×D4—3461no—
251#117—1731no—
252#117—1731no—
253#107—1731no—
254#107C2×D17311yescharacteristic, complemented
255#106—1731no—
256#106—1731no—
257#103—1731no—
258#103—1731no—
259#103—1731no—
260#103C2×D17311yescharacteristic, complemented
261#102—1731no—
262#102—1731no—
263#102—1731no—
264#102C2×D17311yescharacteristic, complemented
265C24—1731no—
266#41—3462no—
267#41—3462no—
268#41—3462no—
269#41—3462no—
270C2×Q8—1731no—
271C2×Q8C22×D17311yescharacteristic
272C2×D4—21no—
273C2×D4—3461no—
274C2×D4—3461no—
275C2×D4—3461no—
276C2×D4—3461no—
277C2×D4—3461no—
278C2×D4—1731no—
279C2×D4C22×D17311yescharacteristic
280C22×C4—3462no—
281C22×C4—3462no—
282C22×C4—3461no—
283C22×C4—3461no—
284C22×C4—1731no—
285C22×C4—1731no—
286C22×C4—1731no—
287C22×C4—1731no—
288C22×C4—1731no—
289C22×C4C22×D17311yescharacteristic
290Q16—3462no—
291SD16—22no—
292SD16—3462no—
293D8—3462no—
294M16—22no—
295M16—3462no—
296M16—3462no—
297M16—3462no—
298C2×C8—3462no—
299C2×C8—1731no—
300C2×C8—1731no—
301C2×C8—1734no—
302C2×C8C22×D17311yescharacteristic
303C2×C8C22×D17311yescharacteristic
304C4⋊C4—21no—
305C4⋊C4—3461no—
306C4⋊C4—3461no—
307C4⋊C4—3461no—
308C4⋊C4—1731no—
309C4⋊C4—1731no—
310C4⋊C4C22×D17311yescharacteristic
311C4⋊C4C22×D17311yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—3461no—
315C22⋊C4—3461no—
316C22⋊C4—3461no—
317C22⋊C4—3461no—
318C22⋊C4—3461no—
319C22⋊C4—3461no—
320C22⋊C4—3461no—
321C22⋊C4—3461no—
322C42—3461no—
323C42—3461no—
324C23—21no—
325C23—3462no—
326C23—3461no—
327C23—1731no—
328C23—1731no—
329C23—6921no—
330C23—6921no—
331C23D4×D17311yescharacteristic
332Q8—21no—
333Q8—21no—
334Q8—3461no—
335Q8—3461no—
336D4—21no—
337D4—21no—
338D4—42no—
339D4—3461no—
340D4—3461no—
341D4—6922no—
342D4—6922no—
343D4—6922no—
344D4—6922no—
345D4—6922no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—3462no—
352C2×C4—3462no—
353C2×C4—3462no—
354C2×C4—3461no—
355C2×C4—3461no—
356C2×C4—3461no—
357C2×C4—3461no—
358C2×C4—3461no—
359C2×C4—1731no—
360C2×C4—1731no—
361C2×C4—1732no—
362C2×C4—1732no—
363C2×C4—6922no—
364C2×C4—6922no—
365C2×C4D4×D17311yescharacteristic
366C2×C4C23×D17311yescharacteristic
367C8—21no—
368C8—3461no—
369C8—1732no—
370C8D4×D17312yes—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—3461no—
378C22—3461no—
379C22—3461no—
380C22—3461no—
381C22—3461no—
382C22—1732no—
383C22—6924no—
384C22—6921no—
385C22—6921no—
386C22—6921no—
387C22C2×D4×D17311yescharacteristic
388C4—21no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—3461no—
393C4—1732no—
394C4—6921no—
395C4—6921no—
396C4—6921no—
397C4C2×D4×D17311yescharacteristic
398C4#41×D17311yescharacteristic
399C2—41no—
400C2—81no—
401C2—3462no—
402C2—13841no—
403C2—6921no—
404C2#121×D17311yescharacteristic
405C2#2601228711yescharacteristic
406C2#2601229011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#468×C173C2
socleC22×C173C22×D4
hypercenter#468D173
hypocenterC173#3054
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#19149700C2semidirect
2. C2×M16×D173C2semidirect
3. #26012004C2semidirect
4. #26012001C2semidirect
5. #26011748C2semidirect
6. #26011747C2semidirect
7. #26011743C2semidirect
8. #26011742C2semidirect
9. #26011738C2semidirect
10. #26011737C2semidirect
11. #26011734C2semidirect
12. #26011528C2semidirect
13. #26011486C2semidirect
14. #26011484C2semidirect
15. #468×C173C2semidirect
20. C2×#14583816C22semidirect
21. C2×#14583815C22semidirect
22. C2×#14583811C22semidirect
45. #19149613C22semidirect
46. #19149610C22semidirect
47. #19149610C22semidirect
48. #19149604C22semidirect
49. #19149603C22semidirect
50. #19149602C22semidirect
51. #19149601C22semidirect
52. #19149600C22semidirect
53. #19149599C22semidirect
54. C2×SD16×C173C22semidirect
58. #107×C173C22semidirect
59. #103×C173C22semidirect
60. #102×C173C22semidirect
91. C8×D173D4semidirect
203. C173#3054semidirect
216. #468D173semidirect
226. C2×SD16C2×D173semidirect
254. #107C2×D173semidirect
260. #103C2×D173semidirect
264. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497002
C2×M16×D1732
#260120042
#260120012
#260117482
#260117472
#260117432
#260117422
#260117382
#260117372
#260117342
#260115282
#260114862
#260114842
#468×C1732
#3054173