Finite Group Databasebeta

← #46822748

Subgroups — #46822748

Order 34432.

420 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771268C211yesmaximal, characteristic, complemented
2#31771256C211yesmaximal, characteristic
3#31770954C211yesmaximal, characteristic, complemented
4#31770937C211yesmaximal, characteristic, complemented
5#31770889C211yesmaximal, characteristic, complemented
6#31770836C211yesmaximal, characteristic, complemented
7#31770835C211yesmaximal, characteristic, complemented
8#31770809C211yesmaximal, characteristic, complemented
9#31770746C211yesmaximal, characteristic, complemented
10#31770745C211yesmaximal, characteristic, complemented
11#31770692C211yesmaximal, characteristic, complemented
12#31770668C211yesmaximal, characteristic, complemented
13C2×#22575376C211yesmaximal, characteristic, complemented
14#103×D269C211yesmaximal, characteristic, complemented
15#483×C269C211yesmaximal, characteristic, complemented
16C2×#16864402—21no—
17C22×Dic538C2211yescharacteristic
18C2×Q8×D269C2211yescharacteristic
19C2×#16864391C2211yescharacteristic
20C2×#16864391C2211yescharacteristic
21C2×#16864389C2211yescharacteristic
22C22×#14524130C2211yescharacteristic
23C2×C8×D269C2211yescharacteristic
24#22575452—21no—
25#22575452C2211yescharacteristic
26#22575451C2211yescharacteristic
27#22575438—21no—
28#22575436—21no—
29#22575433C2211yescharacteristic
30#22575431—21no—
31#22575431—21no—
32#22575430—21no—
33#22575428C2211yescharacteristic
34#22575427C2211yescharacteristic
35#22575418C2211yescharacteristic
36#22575415C2211yescharacteristic
37C4×C22⋊D269C2211yescharacteristic
38#22575413—21no—
39C4⋊C4×D269C2211yescharacteristic
40#22575384C2211yescharacteristic
41#22575382C2211yescharacteristic
42#22575378C2211yescharacteristic
43#22575378C2211yescharacteristic
44#22575377C2211yescharacteristic
45#22575376C2211yescharacteristic
46#22575376C2211yescharacteristic
47#22575376C2211yescharacteristic
48#22575376C2211yescharacteristic
49#22575373C2211yescharacteristic
50#22575365C2211yescharacteristic
51C8×Dic269C2211yescharacteristic
52C22⋊Q8×C269C2211yescharacteristic
53C2×C4⋊C4×C269C2211yescharacteristic
54#106×C269C2211yescharacteristic
55#106×C269C2211yescharacteristic
56#103×C269C2211yescharacteristic
57#103×C269C2211yescharacteristic
58C22⋊C8×C269C2211yescharacteristic
59#16864402—44no—
60C2×Dic538—21no—
61C2×Dic538—21no—
62C2×Dic538—22no—
63C2×Dic538—21no—
64C2×Dic538—41no—
65C2×Dic538C2311yescharacteristic
66C2×Dic538C2311yescharacteristic
67Q8×D269—22no—
68Q8×D269—41no—
69C2×D1076—21no—
70C2×D1076C2311yescharacteristic
71C2×C22⋊D269—21no—
72C2×C22⋊D269D411yescharacteristic
73C22×Dic269—21no—
74C2×C4×D269—21no—
75C2×C4×D269—21no—
76C2×C4×D269—21no—
77C2×C4×D269C2311yescharacteristic
78#16864391—22no—
79#16864391—41no—
80#16864389—41no—
81C2×#14524130—41no—
82C2×#14524130C2311yescharacteristic
83C2×#14524130C2311yescharacteristic
84C8×D269—41no—
85C4⋊Dic269—21no—
86C4⋊Dic269C2311yescharacteristic
87#16864381—21no—
88#16864381—21no—
89#16864381—21no—
90#16864381—21no—
91#16864381—21no—
92#16864381D411yescharacteristic
93C22⋊Dic269D411yescharacteristic
94#16864379—21no—
95#16864379—21no—
96#16864379—21no—
97#16864379—21no—
98#16864379D411yescharacteristic
99C4×Dic269—21no—
100C4×Dic269C2311yescharacteristic
101C2×Q8×C269C2311yescharacteristic
102C22×C4×C269—21no—
103C22×C4×C269C2311yescharacteristic
104C2×C8×C269C2311yescharacteristic
105C2×C8×C269C2311yescharacteristic
106C4⋊C4×C269—21no—
107C4⋊C4×C269—21no—
108C4⋊C4×C269C2311yescharacteristic
109C4⋊C4×C269C2311yescharacteristic
110C4⋊C4×C269C2311yescharacteristic
111C22⋊C4×C269—21no—
112C22×D269—21no—
113C22×D269C2×D411yescharacteristic
114Dic538—21no—
115Dic538—21no—
116Dic538—21no—
117Dic538—42no—
118Dic538—42no—
119Dic538—42no—
120Dic538#4112yescomplemented
121D1076—21no—
122D1076—21no—
123D1076—42no—
124C22⋊D269—42no—
125C22⋊D269SD1614yescomplemented
126C2×Dic269—21no—
127C2×Dic269—21no—
128C2×Dic269—21no—
129C2×Dic269—42no—
130C2×Dic269C2×D411yescharacteristic
131C2×Dic269C2×D411yescharacteristic
132C2×Dic269C2×D411yescharacteristic
133C4×D269—21no—
134C4×D269—21no—
135C4×D269—42no—
136C4×D269—42no—
137C4×D269—42no—
138#14524130—21no—
139#14524130—21no—
140C23×C269C2×D411yescharacteristic
141Q8×C269—21no—
142Q8×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—21no—
149C2×C4×C269—42no—
150C2×C4×C269C2×D411yescharacteristic
151C2×C4×C269C2411yescharacteristic
152C8×C269—21no—
153C8×C269—21no—
154C22×C269—21no—
155C22×C269—21no—
156C22×C269C2×SD1612yes—
157C22×C269C22×D411yescharacteristic
158C2×D269—21no—
159C2×D269—21no—
160C2×D269—41no—
161C2×D269—41no—
162C2×D269—41no—
163C2×D269C2×SD1612yes—
164C4×C269—21no—
165C4×C269—41no—
166C4×C269—41no—
167C4×C269—41no—
168C4×C269—41no—
169C4×C269C2×#4111yescharacteristic
170C4×C269H(5,𝔽2)11yescharacteristic
171Dic269—21no—
172Dic269—22no—
173Dic269—21no—
174Dic269—41no—
175Dic269—41no—
176Dic269C2×SD1612yes—
177C2×C269—22no—
178C2×C269#54611yescharacteristic
179C2×C269C22×SD1611yescharacteristic
180C2×C269#57811yescharacteristic
181D269—22no—
182D269—81no—
183C269#329811yescharacteristic, complemented, Sylow
184#3298—2691nomaximal, Sylow
185D4×Q8—2691no—
186#547—2691no—
187#487—2691no—
188#483—2691no—
189#483D26911yescharacteristic, complemented
190#475—2691no—
191#465—2691no—
192#465—2691no—
193#461—2691no—
194#448—2691no—
195#448—2691no—
196C4×SD16—2691no—
197C8×D4—2691no—
198C2×#103—2691no—
199C2×#103—2691no—
200C2×#41—5381no—
201C22×Q8—2691no—
202C22×Q8—2691no—
203C2×SD16—2691no—
204C2×SD16—2691no—
205C2×SD16—2691no—
206C22×C8—2691no—
207C22×C8—2691no—
208C4⋊Q8—2691no—
209C4⋊Q8—2691no—
210C4⋊Q8—5381no—
211#123—5381no—
212#123—5381no—
213C22⋊Q8—2691no—
214C22⋊Q8—5381no—
215C22⋊Q8—5381no—
216C22⋊Q8—5381no—
217C22⋊Q8C2×D26911yescharacteristic
218#121—2691no—
219#121—2691no—
220C4×Q8—2691no—
221C4×D4—2691no—
222C4×D4—2691no—
223C4×D4—5381no—
224C2×C4⋊C4—2691no—
225C2×C4⋊C4C2×D26911yescharacteristic
226#106—2691no—
227#106C2×D26911yescharacteristic
228#106C2×D26911yescharacteristic
229C4⋊C8—2691no—
230#103—2691no—
231#103—2691no—
232#103—2691no—
233#103—2691no—
234#103—2691no—
235#103—2691no—
236#103—2691no—
237#103C2×D26911yescharacteristic
238#103C2×D26911yescharacteristic
239#102—2691no—
240C22⋊C8—2691no—
241C22⋊C8C2×D26911yescharacteristic
242C4×C8—2691no—
243#41—10764no—
244C2×Q8—2691no—
245C2×Q8—2691no—
246C2×Q8—5381no—
247C2×Q8—5381no—
248C2×Q8—5382no—
249C2×Q8—5381no—
250C2×Q8—5382no—
251C2×Q8—10761no—
252C2×Q8—10761no—
253C2×Q8C22×D26911yescharacteristic
254C2×D4—2691no—
255C2×D4—2691no—
256C2×D4—5381no—
257C2×D4—5381no—
258C22×C4—21no—
259C22×C4—2691no—
260C22×C4—5381no—
261C22×C4—5381no—
262C22×C4—5381no—
263C22×C4—5381no—
264C22×C4C22×D26911yescharacteristic
265SD16—5382no—
266SD16—10761no—
267SD16—10761no—
268C2×C8—2691no—
269C2×C8—2691no—
270C2×C8—10761no—
271C2×C8—10761no—
272C2×C8C22×D26911yescharacteristic
273C2×C8C22×D26911yescharacteristic
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—2691no—
277C4⋊C4—2691no—
278C4⋊C4—5381no—
279C4⋊C4—5381no—
280C4⋊C4—5381no—
281C4⋊C4—5381no—
282C4⋊C4—5381no—
283C4⋊C4—5381no—
284C4⋊C4C22×D26911yescharacteristic
285C4⋊C4C22×D26911yescharacteristic
286C4⋊C4C22×D26911yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—2691no—
289C22⋊C4—2691no—
290C22⋊C4—5381no—
291C22⋊C4—5381no—
292C22⋊C4—5381no—
293C22⋊C4—5381no—
294C42—2691no—
295C42—5381no—
296C23—2691no—
297C23—5381no—
298C23D4×D26911yescharacteristic
299Q8—21no—
300Q8—21no—
301Q8—2692no—
302Q8—5381no—
303Q8—5381no—
304Q8—5381no—
305Q8—10762no—
306Q8—10762no—
307Q8—10762no—
308D4—2694no—
309D4—5381no—
310D4—5381no—
311D4—10762no—
312D4—10762no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—2691no—
321C2×C4—2691no—
322C2×C4—2691no—
323C2×C4—5381no—
324C2×C4—5381no—
325C2×C4—5381no—
326C2×C4—5381no—
327C2×C4—5381no—
328C2×C4—10762no—
329C2×C4—10762no—
330C2×C4—10762no—
331C2×C4—10762no—
332C2×C4D4×D26911yescharacteristic
333C2×C4C23×D26911yescharacteristic
334C8—21no—
335C8—21no—
336C8—5381no—
337C8—5381no—
338C22—21no—
339C22—21no—
340C22—2692no—
341C22—5381no—
342C22—5381no—
343C22—10761no—
344C22—10761no—
345C22—10761no—
346C22SD16×D26912yes—
347C22C2×D4×D26911yescharacteristic
348C4—21no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—2692no—
354C4—5381no—
355C4—5381no—
356C4—5382no—
357C4—10761no—
358C4—10761no—
359C4#41×D26911yescharacteristic
360C4#2257549911yescharacteristic
361C2—22no—
362C2—5382no—
363C2—21521no—
364C2#3177124711yescharacteristic
365C2C2×SD16×D26911yescharacteristic
366C2#3177142111yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#483×C269C2
socleC22×C269C22×D4
prime radical#31771256C2
hypercenter#483D269
hypocenterC269#3298
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. #31771268C2semidirect
3. #31770954C2semidirect
4. #31770937C2semidirect
5. #31770889C2semidirect
6. #31770836C2semidirect
7. #31770835C2semidirect
8. #31770809C2semidirect
9. #31770746C2semidirect
10. #31770745C2semidirect
11. #31770692C2semidirect
12. #31770668C2semidirect
13. C2×#22575376C2semidirect
14. #103×D269C2semidirect
15. #483×C269C2semidirect
120. Dic538#41semidirect
125. C22⋊D269SD16semidirect
183. C269#3298semidirect
189. #483D269semidirect

Maximal subgroups

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

subgroupindex
#317712682
#317712562
#317709542
#317709372
#317708892
#317708362
#317708352
#317708092
#317707462
#317707452
#317706922
#317706682
C2×#225753762
#103×D2692
#483×C2692
#3298269