Finite Group Databasebeta

← #46822735

Subgroups — #46822735

Order 34432.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771301C211yesmaximal, characteristic, complemented
2#31770959C211yesmaximal, characteristic, complemented
3#31770938C211yesmaximal, characteristic, complemented
4#31770926C211yesmaximal, characteristic, complemented
5#31770905C211yesmaximal, characteristic, complemented
6#31770839C211yesmaximal, characteristic, complemented
7#31770836C211yesmaximal, characteristic, complemented
8#31770817C211yesmaximal, characteristic, complemented
9#31770750C211yesmaximal, characteristic, complemented
10#31770739C211yesmaximal, characteristic, complemented
11C4×#16864387C211yesmaximal, characteristic, complemented
12D4×#14524130C211yesmaximal, characteristic, complemented
13#31770601C211yesmaximal, characteristic, complemented
14C2×#22575374C211yesmaximal, characteristic, complemented
15#546×C269C211yesmaximal, characteristic, complemented
16C2×#16864389C2211yescharacteristic, complemented
17C2×#16864387C2211yescharacteristic
18C2×#16864387C2211yescharacteristic
19C22×#14524130C2211yescharacteristic
20C22×#14524130C2211yescharacteristic, complemented
21#22575448—21no—
22#22575445C2211yescharacteristic, complemented
23#22575435—21no—
24#22575427C2211yescharacteristic
25#22575427C2211yescharacteristic, complemented
26#22575425—21no—
27C4×D1076C2211yescharacteristic, complemented
28#22575413—21no—
29#22575412C2211yescharacteristic, complemented
30C2×C4⋊Dic269C2211yescharacteristic
31#22575386C2211yescharacteristic, complemented
32#22575386C2211yescharacteristic, complemented
33#22575384C2211yescharacteristic, complemented
34C4⋊#14524130C2211yescharacteristic, complemented
35#22575377C2211yescharacteristic, complemented
36#22575377C2211yescharacteristic, complemented
37#22575377C2211yescharacteristic, complemented
38#22575374C2211yescharacteristic
39#22575374C2211yescharacteristic
40#22575374C2211yescharacteristic
41#22575374C2211yescharacteristic
42#22575374C2211yescharacteristic
43#22575374C2211yescharacteristic
44#22575373C2211yescharacteristic, complemented
45C22⋊#14524130C2211yescharacteristic
46C22⋊#14524130C2211yescharacteristic, complemented
47C4×#14524130C2211yescharacteristic, complemented
48C2×#41×C269C2211yescharacteristic
49C22×D4×C269C2211yescharacteristic
50#124×C269C2211yescharacteristic
51#123×C269—21no—
52C22⋊Q8×C269C2211yescharacteristic
53#121×C269—21no—
54#121×C269C2211yescharacteristic
55#120×C269—21no—
56C4×D4×C269C2211yescharacteristic
57C4×D4×C269C2211yescharacteristic
58C2×C22⋊C4×C269—21no—
59C2×D1076C2311yescharacteristic
60C2×C22⋊D269—21no—
61C2×C22⋊D269—21no—
62C22×Dic269—21no—
63C2×C4×D269—21no—
64#16864389—41no—
65#16864387—22no—
66#16864387—41no—
67C2×#14524130—41no—
68C2×#14524130—41no—
69C2×#14524130C2311yescharacteristic
70C2×#14524130C2311yescharacteristic
71C2×#14524130C2311yescharacteristic
72C2×#14524130C2311yescharacteristic
73C4⋊Dic269—21no—
74C4⋊Dic269C2311yescharacteristic
75C4⋊Dic269C2311yescharacteristic
76C4⋊Dic269C2311yescharacteristic
77#16864381—21no—
78#16864381—21no—
79C22⋊Dic269—21no—
80#16864379—21no—
81#16864379—21no—
82#16864379—21no—
83C4×Dic269—21no—
84C24×C269—21no—
85#41×C269—22no—
86#41×C269—41no—
87C2×Q8×C269C2311yescharacteristic
88C2×D4×C269—22no—
89C2×D4×C269—21no—
90C2×D4×C269—21no—
91C2×D4×C269—21no—
92C2×D4×C269—41no—
93C2×D4×C269D411yescharacteristic
94C2×D4×C269C2311yescharacteristic
95C2×D4×C269C2311yescharacteristic
96C22×C4×C269—21no—
97C22×C4×C269—21no—
98C22×C4×C269C2311yescharacteristic
99C22×C4×C269C2311yescharacteristic
100C4⋊C4×C269—21no—
101C4⋊C4×C269D411yescharacteristic
102C4⋊C4×C269C2311yescharacteristic
103C22⋊C4×C269—21no—
104C22⋊C4×C269—21no—
105C22⋊C4×C269—21no—
106C22⋊C4×C269—21no—
107C22⋊C4×C269—21no—
108C22⋊C4×C269D411yescharacteristic
109C22⋊C4×C269D411yescharacteristic
110C42×C269C2311yescharacteristic
111C22×D269—21no—
112D1076—21no—
113D1076—21no—
114C22⋊D269—42no—
115C22⋊D269—81no—
116C2×Dic269—21no—
117C2×Dic269—21no—
118C2×Dic269—21no—
119C2×Dic269—81no—
120C4×D269—81no—
121#14524130—21no—
122#14524130—21no—
123#14524130—21no—
124#14524130—21no—
125C23×C269—21no—
126C23×C269—21no—
127C23×C269—42no—
128C23×C269—42no—
129C23×C269—81no—
130C23×C269C2×D411yescharacteristic
131C23×C269C2×D411yescharacteristic
132Q8×C269—21no—
133Q8×C269—21no—
134D4×C269—22no—
135D4×C269—21no—
136D4×C269—21no—
137D4×C269—21no—
138D4×C269—42no—
139D4×C269—42no—
140D4×C269—81no—
141D4×C269#4112yes—
142C2×C4×C269—21no—
143C2×C4×C269—21no—
144C2×C4×C269—21no—
145C2×C4×C269—21no—
146C2×C4×C269—21no—
147C2×C4×C269—21no—
148C2×C4×C269—42no—
149C2×C4×C269—81no—
150C2×C4×C269C2×D411yescharacteristic
151C2×C4×C269C2×D411yescharacteristic
152C2×C4×C269C2×D411yescharacteristic
153C2×C4×C269C2×D411yescharacteristic
154C2×C4×C269C2411yescharacteristic
155C22×C269—21no—
156C22×C269—21no—
157C22×C269—21no—
158C22×C269—22no—
159C22×C269—21no—
160C22×C269—21no—
161C22×C269—41no—
162C22×C269—41no—
163C22×C269—41no—
164C22×C269—41no—
165C22×C269—41no—
166C22×C269—84no—
167C22×C269#13512yes—
168C22×C269C22×D411yescharacteristic
169C2×D269—41no—
170C2×D269—41no—
171C2×D269—41no—
172C4×C269—21no—
173C4×C269—21no—
174C4×C269—21no—
175C4×C269—21no—
176C4×C269—41no—
177C4×C269—41no—
178C4×C269C2×#4111yescharacteristic
179C4×C269H(5,𝔽2)11yescharacteristic
180Dic269—41no—
181Dic269—41no—
182Dic269—41no—
183C2×C269—22no—
184C2×C269—41no—
185C2×C269—42no—
186C2×C269—81no—
187C2×C269#54611yescharacteristic
188C2×C269C2×#13511yescharacteristic
189C2×C269#57611yescharacteristic
190D269—81no—
191C269#329711yescharacteristic, complemented, Sylow
192#3297—2691nomaximal, Sylow
193#553—2691no—
194#546D26911yescharacteristic, complemented
195#488—2691no—
196#484—2691no—
197#482—2691no—
198#478—2691no—
199#466—2691no—
200#465—2691no—
201#463—2691no—
202#449—2691no—
203#447—2691no—
204C4×D8—2691no—
205C8×D4—2691no—
206#416—2691no—
207C2×#102—2691no—
208C2×#41C2×D26911yescharacteristic
209C22×D4C2×D26911yescharacteristic
210C2×SD16—2691no—
211C2×D8—2691no—
212C2×D8—2691no—
213C22×C8—2691no—
214C22×C8—2691no—
215#126—5381no—
216#125—2691no—
217#124C2×D26911yescharacteristic
218#123—21no—
219#123—5381no—
220C22⋊Q8C2×D26911yescharacteristic
221#121—21no—
222#121—2691no—
223#121—2691no—
224#121—5381no—
225#121C2×D26911yescharacteristic
226#120—21no—
227C4×D4—2691no—
228C4×D4—5381no—
229C4×D4C2×D26911yescharacteristic
230C4×D4C2×D26911yescharacteristic
231#117—2691no—
232C2×C4⋊C4—2691no—
233C2×C22⋊C4—21no—
234#107—2691no—
235#107—2691no—
236#106—2691no—
237C4⋊C8—2691no—
238#103—2691no—
239#103—2691no—
240#103—2691no—
241#102—2691no—
242#102—2691no—
243#102—2691no—
244#102—2691no—
245#102—2691no—
246#102—2691no—
247#102—2691no—
248C22⋊C8—2691no—
249C22⋊C8—2691no—
250C4×C8—2691no—
251C24—21no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D26911yescharacteristic
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—41no—
260C2×D4—2691no—
261C2×D4—5381no—
262C2×D4—5381no—
263C2×D4C22⋊D26911yescharacteristic
264C2×D4C22×D26911yescharacteristic
265C2×D4C22×D26911yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—5381no—
269C22×C4—5381no—
270C22×C4C22×D26911yescharacteristic
271C22×C4C22×D26911yescharacteristic
272SD16—10761no—
273D8—5382no—
274D8—10761no—
275C2×C8—2691no—
276C2×C8—2691no—
277C2×C8—2691no—
278C2×C8—2691no—
279C2×C8—10761no—
280C2×C8—10761no—
281C4⋊C4—21no—
282C4⋊C4—2691no—
283C4⋊C4—2691no—
284C4⋊C4—2691no—
285C4⋊C4—5381no—
286C4⋊C4—5381no—
287C4⋊C4—5381no—
288C4⋊C4C22⋊D26911yescharacteristic
289C4⋊C4C22×D26911yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—5381no—
296C22⋊C4—5381no—
297C22⋊C4—5381no—
298C22⋊C4—5381no—
299C22⋊C4C22⋊D26911yescharacteristic
300C22⋊C4C22⋊D26911yescharacteristic
301C42—5381no—
302C42C22×D26911yescharacteristic
303C23—21no—
304C23—21no—
305C23—42no—
306C23—42no—
307C23—81no—
308C23—5381no—
309C23C2×C22⋊D26911yescharacteristic
310C23C2×C22⋊D26911yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—42no—
318D4—42no—
319D4—81no—
320D4—5381no—
321D4—5381no—
322D4—10762no—
323D4—21521no—
324D4#1686440112yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—42no—
332C2×C4—81no—
333C2×C4—5381no—
334C2×C4—5381no—
335C2×C4—5381no—
336C2×C4—21521no—
337C2×C4—21521no—
338C2×C4C2×C22⋊D26911yescharacteristic
339C2×C4C2×C22⋊D26911yescharacteristic
340C2×C4C2×C22⋊D26911yescharacteristic
341C2×C4C2×C22⋊D26911yescharacteristic
342C2×C4C23×D26911yescharacteristic
343C8—5381no—
344C8—5381no—
345C8—5381no—
346C8—5381no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—22no—
352C22—21no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—84no—
359C22—10761no—
360C22—10761no—
361C22—10761no—
362C22#2257547312yes—
363C22C22×C22⋊D26911yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—10761no—
371C4—10761no—
372C4—10761no—
373C4C2×#1686440111yescharacteristic
374C4#2257549811yescharacteristic
375C2—22no—
376C2—41no—
377C2—42no—
378C2—81no—
379C2—21521no—
380C2#3177124911yescharacteristic
381C2C2×#2257547311yescharacteristic
382C2#3177140811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#546×C269C2
socleC22×C269C22×D4
hypercenter#546D269
hypocenterC269#3297
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #31771301C2semidirect
2. #31770959C2semidirect
3. #31770938C2semidirect
4. #31770926C2semidirect
5. #31770905C2semidirect
6. #31770839C2semidirect
7. #31770836C2semidirect
8. #31770817C2semidirect
9. #31770750C2semidirect
10. #31770739C2semidirect
11. C4×#16864387C2semidirect
12. D4×#14524130C2semidirect
13. #31770601C2semidirect
14. C2×#22575374C2semidirect
15. #546×C269C2semidirect
16. C2×#16864389C22semidirect
20. C22×#14524130C22semidirect
22. #22575445C22semidirect
25. #22575427C22semidirect
27. C4×D1076C22semidirect
29. #22575412C22semidirect
31. #22575386C22semidirect
32. #22575386C22semidirect
33. #22575384C22semidirect
34. C4⋊#14524130C22semidirect
35. #22575377C22semidirect
36. #22575377C22semidirect
37. #22575377C22semidirect
44. #22575373C22semidirect
46. C22⋊#14524130C22semidirect
47. C4×#14524130C22semidirect
191. C269#3297semidirect
194. #546D269semidirect

Maximal subgroups

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

subgroupindex
#317713012
#317709592
#317709382
#317709262
#317709052
#317708392
#317708362
#317708172
#317707502
#317707392
C4×#168643872
D4×#145241302
#317706012
C2×#225753742
#546×C2692
#3297269