Finite Group Databasebeta

← #46823465

Subgroups — #46823465

Order 34432.

374 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771323C211yesmaximal, characteristic, complemented
2#31771256C211yesmaximal, characteristic
3#31771009C211yesmaximal, characteristic, complemented
4#31770942C211yesmaximal, characteristic, complemented
5#31770940C211yesmaximal, characteristic, complemented
6#31770931C211yesmaximal, characteristic, complemented
7#31770929C211yesmaximal, characteristic, complemented
8#31770889C211yesmaximal, characteristic, complemented
9#31770884C211yesmaximal, characteristic, complemented
10#31770852C211yesmaximal, characteristic, complemented
11#31770851C211yesmaximal, characteristic, complemented
12#31770692C211yesmaximal, characteristic, complemented
13#31770681C211yesmaximal, characteristic, complemented
14#31770644C211yesmaximal, characteristic, complemented
15#428×C269C211yesmaximal, characteristic, complemented
16C2×#16864402—21no—
17C2×#16864391C2211yescharacteristic
18C2×#16864385C2211yescharacteristic
19#22575452C2211yescharacteristic
20#22575452C2211yescharacteristic
21#22575447—21no—
22#22575446—21no—
23#22575439—21no—
24#22575438—21no—
25#22575436—21no—
26#22575433C2211yescharacteristic
27#22575432C2211yescharacteristic
28#22575431—21no—
29#22575428C2211yescharacteristic
30#22575427C2211yescharacteristic
31C4×Dic538—21no—
32#22575418C2211yescharacteristic
33#22575415C2211yescharacteristic
34C4×C22⋊D269C2211yescharacteristic
35#22575411C2211yescharacteristic
36C4⋊C4×D269C2211yescharacteristic
37#22575387C2211yescharacteristic
38#22575386C2211yescharacteristic
39#22575386C2211yescharacteristic
40#22575386C2211yescharacteristic
41#22575384C2211yescharacteristic
42#22575382C2211yescharacteristic
43#22575378C2211yescharacteristic
44#22575376C2211yescharacteristic
45#22575376C2211yescharacteristic
46#22575375C2211yescharacteristic
47#22575373C2211yescharacteristic
48#22575373C2211yescharacteristic
49C22⋊#14524130C2211yescharacteristic
50#22575365C2211yescharacteristic
51C8×Dic269C2211yescharacteristic
52C2×M16×C269C2211yescharacteristic
53#117×C269C2211yescharacteristic
54C2×C4⋊C4×C269C2211yescharacteristic
55#107×C269C2211yescharacteristic
56#107×C269C2211yescharacteristic
57#106×C269C2211yescharacteristic
58#106×C269C2211yescharacteristic
59#16864402—44no—
60C2×Dic538—21no—
61C2×Dic538—21no—
62C2×Dic538C2311yescharacteristic
63C2×D1076—21no—
64C2×D1076C2311yescharacteristic
65C2×C22⋊D269—21no—
66C2×C22⋊D269D411yescharacteristic
67C2×C4×D269—21no—
68C2×C4×D269—21no—
69C2×C4×D269C2311yescharacteristic
70#16864391—22no—
71#16864385—22no—
72C2×#14524130C2311yescharacteristic
73C2×#14524130C2311yescharacteristic
74C4⋊Dic269—21no—
75C4⋊Dic269C2311yescharacteristic
76C4⋊Dic269C2311yescharacteristic
77#16864381—21no—
78#16864381—21no—
79#16864381—21no—
80#16864381—21no—
81#16864381—21no—
82#16864381D411yescharacteristic
83C22⋊Dic269—21no—
84C22⋊Dic269D411yescharacteristic
85#16864379—21no—
86#16864379—21no—
87#16864379—21no—
88#16864379—21no—
89#16864379—21no—
90#16864379D411yescharacteristic
91C4×Dic269—21no—
92C4×Dic269—21no—
93C4×Dic269C2311yescharacteristic
94C22×C4×C269—21no—
95C22×C4×C269C2311yescharacteristic
96M16×C269—22no—
97C2×C8×C269C2311yescharacteristic
98C2×C8×C269C2311yescharacteristic
99C4⋊C4×C269—21no—
100C4⋊C4×C269C2311yescharacteristic
101C4⋊C4×C269C2311yescharacteristic
102C4⋊C4×C269C2311yescharacteristic
103C4⋊C4×C269C2311yescharacteristic
104C22⋊C4×C269—21no—
105C42×C269—21no—
106C22×D269—21no—
107C22×D269C2×D411yescharacteristic
108Dic538—21no—
109Dic538—21no—
110Dic538—42no—
111Dic538—42no—
112D1076—21no—
113D1076—21no—
114D1076—42no—
115C22⋊D269—22no—
116C22⋊D269—42no—
117C2×Dic269—21no—
118C2×Dic269—21no—
119C2×Dic269—21no—
120C2×Dic269C2×D411yescharacteristic
121C2×Dic269C2×D411yescharacteristic
122C2×Dic269C2×D411yescharacteristic
123C4×D269—21no—
124C4×D269—21no—
125C4×D269—42no—
126C4×D269—42no—
127#14524130—21no—
128#14524130—21no—
129C23×C269C2×D411yescharacteristic
130C2×C4×C269—21no—
131C2×C4×C269—21no—
132C2×C4×C269—21no—
133C2×C4×C269—21no—
134C2×C4×C269—21no—
135C2×C4×C269—21no—
136C2×C4×C269—42no—
137C2×C4×C269C2×D411yescharacteristic
138C2×C4×C269C2411yescharacteristic
139C8×C269—21no—
140C8×C269#4112yescomplemented
141C22×C269—21no—
142C22×C269—21no—
143C22×C269—21no—
144C22×C269C22×D411yescharacteristic
145C2×D269—21no—
146C2×D269—21no—
147C2×D269—21no—
148C2×D269—41no—
149C2×D269—41no—
150C2×D269—41no—
151C4×C269—21no—
152C4×C269—41no—
153C4×C269—41no—
154C4×C269—41no—
155C4×C269—41no—
156C4×C269C2×#4111yescharacteristic
157C4×C2692−1+411yescharacteristic
158Dic269—21no—
159Dic269—21no—
160Dic269—41no—
161Dic269—41no—
162Dic269—41no—
163Dic269AGL(1,ℤ/8)12yes—
164C2×C269—41no—
165C2×C269#54711yescharacteristic
166C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
167C2×C269#57811yescharacteristic
168D269—41no—
169D269—81no—
170C269#335311yescharacteristic, complemented, Sylow
171#3353—2691nomaximal, Sylow
172#556—2691no—
173#547—2691no—
174#500—2691no—
175#484—2691no—
176#484—2691no—
177#482—2691no—
178#482—2691no—
179#475—2691no—
180#474—2691no—
181#468—2691no—
182#468—2691no—
183C4×SD16—2691no—
184#436—2691no—
185#428—2691no—
186#428D26911yescharacteristic, complemented
187C2×#41—5381no—
188C2×SD16—2691no—
189C2×M16—2691no—
190C2×M16C2×D26911yescharacteristic
191C4⋊Q8—2691no—
192C4⋊Q8—2691no—
193#126—5381no—
194#126—5381no—
195#124—5381no—
196#123—5381no—
197#123—5381no—
198C22⋊Q8—2691no—
199C22⋊Q8—2691no—
200C22⋊Q8—5381no—
201#121—2691no—
202#121—2691no—
203C4×Q8—2691no—
204C4×Q8—5381no—
205C4×D4—2691no—
206C4×D4—2691no—
207#117—2691no—
208#117C2×D26911yescharacteristic
209C2×C4⋊C4—2691no—
210C2×C4⋊C4C2×D26911yescharacteristic
211#107—2691no—
212#107—2691no—
213#107—2691no—
214#107—2691no—
215#107C2×D26911yescharacteristic
216#107C2×D26911yescharacteristic
217#106—2691no—
218#106C2×D26911yescharacteristic
219#106C2×D26911yescharacteristic
220C4⋊C8—2691no—
221#103—2691no—
222#103—2691no—
223#103—2691no—
224#102—2691no—
225#102—2691no—
226#102—2691no—
227C22⋊C8—2691no—
228C22⋊C8—2691no—
229C4×C8—2691no—
230#41—10764no—
231C2×Q8—2691no—
232C2×Q8—5381no—
233C2×Q8—5381no—
234C2×D4—2691no—
235C2×D4—2691no—
236C2×D4—5381no—
237C2×D4—5381no—
238C22×C4—21no—
239C22×C4—2691no—
240C22×C4—5381no—
241C22×C4—5381no—
242C22×C4C22×D26911yescharacteristic
243SD16—5382no—
244M16—22no—
245M16—5382no—
246C2×C8—2691no—
247C2×C8—2691no—
248C2×C8C22×D26911yescharacteristic
249C2×C8C22×D26911yescharacteristic
250C4⋊C4—21no—
251C4⋊C4—2691no—
252C4⋊C4—2691no—
253C4⋊C4—2691no—
254C4⋊C4—5381no—
255C4⋊C4—5381no—
256C4⋊C4—5381no—
257C4⋊C4—5381no—
258C4⋊C4—5381no—
259C4⋊C4—5381no—
260C4⋊C4C22×D26911yescharacteristic
261C4⋊C4C22×D26911yescharacteristic
262C4⋊C4C22×D26911yescharacteristic
263C4⋊C4C22×D26911yescharacteristic
264C22⋊C4—21no—
265C22⋊C4—2691no—
266C22⋊C4—2691no—
267C22⋊C4—5381no—
268C22⋊C4—5381no—
269C22⋊C4—5381no—
270C22⋊C4—5381no—
271C22⋊C4—5381no—
272C22⋊C4—5381no—
273C42—21no—
274C42—2691no—
275C42—5381no—
276C42—5381no—
277C23—2691no—
278C23—5381no—
279C23D4×D26911yescharacteristic
280Q8—5381no—
281Q8—5381no—
282Q8—10762no—
283Q8—10762no—
284D4—5382no—
285D4—5381no—
286D4—5381no—
287D4—10762no—
288D4—10762no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—42no—
296C2×C4—2691no—
297C2×C4—2691no—
298C2×C4—2691no—
299C2×C4—5381no—
300C2×C4—5381no—
301C2×C4—5381no—
302C2×C4—5381no—
303C2×C4—5381no—
304C2×C4—10762no—
305C2×C4—10762no—
306C2×C4D4×D26911yescharacteristic
307C2×C4C23×D26911yescharacteristic
308C8—21no—
309C8—5381no—
310C8—5381no—
311C8#1686440212yescomplemented
312C22—21no—
313C22—21no—
314C22—21no—
315C22—5381no—
316C22—5381no—
317C22—5381no—
318C22—10761no—
319C22—10761no—
320C22—10761no—
321C22C2×D4×D26911yescharacteristic
322C4—21no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—2692no—
328C4—5381no—
329C4—5381no—
330C4—10761no—
331C4—10761no—
332C4—10761no—
333C4C2×#1686440211yescharacteristic
334C4#2257550111yescharacteristic
335C2—41no—
336C2—10761no—
337C2—21521no—
338C2#3177125611yescharacteristic
339C2AGL(1,ℤ/8)×D26911yescharacteristic
340C2#3177142111yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#428×C269C2
socleC22×C269C22×D4
prime radical#31771256C2
hypercenter#428D269
hypocenterC269#3353
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #31771323C2semidirect
3. #31771009C2semidirect
4. #31770942C2semidirect
5. #31770940C2semidirect
6. #31770931C2semidirect
7. #31770929C2semidirect
8. #31770889C2semidirect
9. #31770884C2semidirect
10. #31770852C2semidirect
11. #31770851C2semidirect
12. #31770692C2semidirect
13. #31770681C2semidirect
14. #31770644C2semidirect
15. #428×C269C2semidirect
140. C8×C269#41semidirect
170. C269#3353semidirect
186. #428D269semidirect
311. C8#16864402semidirect

Maximal subgroups

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

subgroupindex
#317713232
#317712562
#317710092
#317709422
#317709402
#317709312
#317709292
#317708892
#317708842
#317708522
#317708512
#317706922
#317706812
#317706442
#428×C2692
#3353269