Finite Group Databasebeta

← #46822787

Subgroups — #46822787

Order 34432.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 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#31770943C211yesmaximal, characteristic, complemented
4#31770921C211yesmaximal, characteristic, complemented
5#31770912C211yesmaximal, characteristic, complemented
6#31770839C211yesmaximal, characteristic, complemented
7#31770836C211yesmaximal, characteristic, complemented
8#31770817C211yesmaximal, characteristic, complemented
9#31770750C211yesmaximal, characteristic, complemented
10#31770744C211yesmaximal, characteristic, complemented
11C4×#16864389C211yesmaximal, characteristic, complemented
12D4×#14524130C211yesmaximal, characteristic, complemented
13#31770601C211yesmaximal, characteristic, complemented
14C2×#22575377C211yesmaximal, characteristic, complemented
15#548×C269C211yesmaximal, characteristic, complemented
16C2×#16864389C2211yescharacteristic
17C2×#16864389C2211yescharacteristic
18C2×#16864387C2211yescharacteristic
19C22×#14524130C2211yescharacteristic
20C22×#14524130C2211yescharacteristic
21#22575448—21no—
22#22575445C2211yescharacteristic
23#22575435—21no—
24#22575427C2211yescharacteristic
25#22575427C2211yescharacteristic
26#22575425—21no—
27C4×D1076C2211yescharacteristic
28#22575413—21no—
29#22575412C2211yescharacteristic
30C2×C4⋊Dic269C2211yescharacteristic
31#22575386C2211yescharacteristic
32#22575384C2211yescharacteristic
33#22575384C2211yescharacteristic
34C4⋊#14524130C2211yescharacteristic
35#22575377C2211yescharacteristic
36#22575377C2211yescharacteristic
37#22575377C2211yescharacteristic
38#22575377C2211yescharacteristic
39#22575377C2211yescharacteristic
40#22575377C2211yescharacteristic
41#22575374C2211yescharacteristic
42#22575374C2211yescharacteristic
43#22575374C2211yescharacteristic
44#22575373C2211yescharacteristic
45C22⋊#14524130C2211yescharacteristic
46C22⋊#14524130C2211yescharacteristic
47C4×#14524130C2211yescharacteristic
48C2×#41×C269C2211yescharacteristic
49C22×Q8×C269C2211yescharacteristic
50#124×C269—21no—
51#124×C269C2211yescharacteristic
52C22⋊Q8×C269—21no—
53C22⋊Q8×C269C2211yescharacteristic
54#121×C269—21no—
55#121×C269C2211yescharacteristic
56C4×Q8×C269C2211yescharacteristic
57C4×D4×C269—21no—
58C4×D4×C269C2211yescharacteristic
59C2×D1076C2311yescharacteristic
60C2×C22⋊D269—21no—
61C2×C22⋊D269—21no—
62C22×Dic269—21no—
63C2×C4×D269—21no—
64#16864389—22no—
65#16864389—41no—
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—
84#41×C269—22no—
85#41×C269—41no—
86C2×Q8×C269—21no—
87C2×Q8×C269—22no—
88C2×Q8×C269—41no—
89C2×Q8×C269C2311yescharacteristic
90C2×Q8×C269C2311yescharacteristic
91C2×D4×C269—21no—
92C2×D4×C269—21no—
93C2×D4×C269D411yescharacteristic
94C2×D4×C269C2311yescharacteristic
95C22×C4×C269—21no—
96C22×C4×C269—21no—
97C22×C4×C269C2311yescharacteristic
98C22×C4×C269C2311yescharacteristic
99C4⋊C4×C269—21no—
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×C269D411yescharacteristic
108C22⋊C4×C269D411yescharacteristic
109C42×C269—21no—
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×C269C2×D411yescharacteristic
127C23×C269C2×D411yescharacteristic
128Q8×C269—21no—
129Q8×C269—21no—
130Q8×C269—21no—
131Q8×C269—42no—
132Q8×C269#4112yes—
133D4×C269—22no—
134D4×C269—21no—
135D4×C269—21no—
136D4×C269—42no—
137D4×C269—42no—
138C2×C4×C269—21no—
139C2×C4×C269—21no—
140C2×C4×C269—21no—
141C2×C4×C269—21no—
142C2×C4×C269—21no—
143C2×C4×C269—21no—
144C2×C4×C269—21no—
145C2×C4×C269—42no—
146C2×C4×C269—42no—
147C2×C4×C269C2×D411yescharacteristic
148C2×C4×C269C2×D411yescharacteristic
149C2×C4×C269C2×D411yescharacteristic
150C2×C4×C269C2×D411yescharacteristic
151C2×C4×C269C2411yescharacteristic
152C22×C269—21no—
153C22×C269—21no—
154C22×C269—21no—
155C22×C269—21no—
156C22×C269—21no—
157C22×C269—41no—
158C22×C269—41no—
159C22×C269—41no—
160C22×C269#13512yes—
161C22×C269C22×D411yescharacteristic
162C2×D269—41no—
163C2×D269—41no—
164C2×D269—41no—
165C4×C269—21no—
166C4×C269—21no—
167C4×C269—22no—
168C4×C269—21no—
169C4×C269—21no—
170C4×C269—41no—
171C4×C269—41no—
172C4×C269C2×#4111yescharacteristic
173C4×C269H(5,𝔽2)11yescharacteristic
174Dic269—41no—
175Dic269—41no—
176Dic269—41no—
177C2×C269—22no—
178C2×C269—41no—
179C2×C269—81no—
180C2×C269#54611yescharacteristic
181C2×C269C2×#13511yescharacteristic
182C2×C269#57711yescharacteristic
183D269—81no—
184C269#330111yescharacteristic, complemented, Sylow
185#3301—2691nomaximal, Sylow
186#553—2691no—
187#548D26911yescharacteristic, complemented
188#488—2691no—
189#485—2691no—
190#481—2691no—
191#479—2691no—
192#466—2691no—
193#465—2691no—
194#463—2691no—
195#449—2691no—
196#448—2691no—
197C4×SD16—2691no—
198C8×D4—2691no—
199#416—2691no—
200C2×#103—2691no—
201C2×#41C2×D26911yescharacteristic
202C22×Q8C2×D26911yescharacteristic
203C2×SD16—2691no—
204C2×SD16—2691no—
205C2×D8—2691no—
206C22×C8—2691no—
207C22×C8—2691no—
208#126—5381no—
209#125—2691no—
210#124—21no—
211#124C2×D26911yescharacteristic
212#123—5381no—
213C22⋊Q8—21no—
214C22⋊Q8C2×D26911yescharacteristic
215#121—21no—
216#121—2691no—
217#121—2691no—
218#121—5381no—
219#121C2×D26911yescharacteristic
220C4×Q8C2×D26911yescharacteristic
221C4×D4—21no—
222C4×D4—2691no—
223C4×D4—5381no—
224C4×D4C2×D26911yescharacteristic
225#117—2691no—
226C2×C4⋊C4—2691no—
227#107—2691no—
228#106—2691no—
229#106—2691no—
230C4⋊C8—2691no—
231#103—2691no—
232#103—2691no—
233#103—2691no—
234#103—2691no—
235#103—2691no—
236#103—2691no—
237#102—2691no—
238#102—2691no—
239#102—2691no—
240#102—2691no—
241C22⋊C8—2691no—
242C22⋊C8—2691no—
243C4×C8—2691no—
244#41—22no—
245#41—41no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8C22×D26911yescharacteristic
250C2×Q8C22×D26911yescharacteristic
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—2691no—
254C2×D4—5381no—
255C2×D4—5381no—
256C2×D4C22⋊D26911yescharacteristic
257C2×D4C22×D26911yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—5381no—
261C22×C4—5381no—
262C22×C4C22×D26911yescharacteristic
263C22×C4C22×D26911yescharacteristic
264SD16—5382no—
265SD16—10761no—
266D8—10761no—
267C2×C8—2691no—
268C2×C8—2691no—
269C2×C8—2691no—
270C2×C8—2691no—
271C2×C8—10761no—
272C2×C8—10761no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—2691no—
276C4⋊C4—2691no—
277C4⋊C4—2691no—
278C4⋊C4—5381no—
279C4⋊C4—5381no—
280C4⋊C4—5381no—
281C4⋊C4C22⋊D26911yescharacteristic
282C4⋊C4C22×D26911yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—5381no—
288C22⋊C4—5381no—
289C22⋊C4—5381no—
290C22⋊C4—5381no—
291C22⋊C4C22⋊D26911yescharacteristic
292C22⋊C4C22⋊D26911yescharacteristic
293C42—21no—
294C42—5381no—
295C42C22×D26911yescharacteristic
296C23—21no—
297C23—5381no—
298C23C2×C22⋊D26911yescharacteristic
299C23C2×C22⋊D26911yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—42no—
304Q8#1686440112yes—
305D4—22no—
306D4—21no—
307D4—21no—
308D4—42no—
309D4—42no—
310D4—5381no—
311D4—5381no—
312D4—10762no—
313D4—21521no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—5381no—
324C2×C4—5381no—
325C2×C4—5381no—
326C2×C4—21521no—
327C2×C4—21521no—
328C2×C4C2×C22⋊D26911yescharacteristic
329C2×C4C2×C22⋊D26911yescharacteristic
330C2×C4C2×C22⋊D26911yescharacteristic
331C2×C4C2×C22⋊D26911yescharacteristic
332C2×C4C23×D26911yescharacteristic
333C8—5381no—
334C8—5381no—
335C8—5381no—
336C8—5381no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—10761no—
346C22—10761no—
347C22—10761no—
348C22#2257547312yes—
349C22C22×C22⋊D26911yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—22no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—10761no—
358C4—10761no—
359C4—10761no—
360C4C2×#1686440111yescharacteristic
361C4#2257549811yescharacteristic
362C2—22no—
363C2—41no—
364C2—81no—
365C2—21521no—
366C2#3177124911yescharacteristic
367C2C2×#2257547311yescharacteristic
368C2#3177141311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#548×C269C2
socleC22×C269C22×D4
hypercenter#548D269
hypocenterC269#3301
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #31771301C2semidirect
2. #31770959C2semidirect
3. #31770943C2semidirect
4. #31770921C2semidirect
5. #31770912C2semidirect
6. #31770839C2semidirect
7. #31770836C2semidirect
8. #31770817C2semidirect
9. #31770750C2semidirect
10. #31770744C2semidirect
11. C4×#16864389C2semidirect
12. D4×#14524130C2semidirect
13. #31770601C2semidirect
14. C2×#22575377C2semidirect
15. #548×C269C2semidirect
184. C269#3301semidirect
187. #548D269semidirect

Maximal subgroups

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

subgroupindex
#317713012
#317709592
#317709432
#317709212
#317709122
#317708392
#317708362
#317708172
#317707502
#317707442
C4×#168643892
D4×#145241302
#317706012
C2×#225753772
#548×C2692
#3301269