Finite Group Databasebeta

← #46820261

Subgroups — #46820261

Order 34432.

704 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771441C211yesmaximal, characteristic
2C2×#22575484C211yesmaximal, characteristic, complemented
3#31771179C211yesmaximal, characteristic, complemented
4#31770786C214yesmaximal, complemented
5#31770781C214yesmaximal, complemented
6#31770622C211yesmaximal, characteristic, complemented
7#31770621C211yesmaximal, characteristic, complemented
8#31770587C211yesmaximal, characteristic, complemented
9C2×#137×C269C211yesmaximal, characteristic, complemented
10#22575501—21no—
11#22575500—21no—
12#22575500C2212yes—
13#22575499—21no—
14#22575499C2212yes—
15#22575498—22no—
16#22575498—21no—
17C2×#16864402—22no—
18C2×#16864402C2211yescharacteristic
19C2×#16864402C2211yescharacteristic
20#41×D269—22no—
21#41×D269—22no—
22#22575484—22no—
23#22575484C2214yescomplemented
24C2×#16864392—21no—
25C2×#16864389—21no—
26C2×#16864386C2211yescharacteristic, complemented
27#22575451C2212yescomplemented
28#22575442C2212yescomplemented
29#22575436—22no—
30#22575429—22no—
31#22575412C2211yescharacteristic, complemented
32#22575380C2214yescomplemented
33#22575379C2214yescomplemented
34#22575371C2212yescomplemented
35#22575369C2212yescomplemented
36C2×#41×C269C2211yescharacteristic
37C22×Q8×C269C2211yescharacteristic, complemented
38#137×C269—22no—
39#137×C269C2214yescomplemented
40C2×Q16×C269—21no—
41C2×SD16×C269—21no—
42C2×M16×C269C2211yescharacteristic, complemented
43#16864403—22no—
44#16864403—22no—
45#16864402—24no—
46#16864402—24no—
47#16864402—22no—
48#16864402—22no—
49#16864402—21no—
50#16864402—21no—
51#16864402—21no—
52#16864402—21no—
53#16864402—21no—
54#16864402—21no—
55#16864402D412yes—
56#16864402C2312yes—
57#16864401—24no—
58#16864401—22no—
59#16864401—24no—
60#16864401—22no—
61C2×Dic538—22no—
62C2×Dic538D411yescharacteristic
63C2×Dic538C2311yescharacteristic
64Q8×D269—22no—
65Q8×D269—22no—
66C2×D1076—22no—
67C2×D1076D411yescharacteristic
68C2×D1076C2311yescharacteristic
69C2×C22⋊D269—24no—
70C2×C22⋊D269—21no—
71C2×C22⋊D269D412yes—
72D4×D269—24no—
73D4×D269—22no—
74D4×D269—24no—
75D4×D269—22no—
76C2×C4×D269—24no—
77C2×C4×D269—21no—
78C2×C4×D269D412yes—
79#16864392—24no—
80#16864389—24no—
81#16864386—21no—
82#16864386C2312yes—
83C2×#14524130—21no—
84C4⋊Dic269—21no—
85#16864381—24no—
86C22⋊Dic269—21no—
87#16864379—24no—
88C4×Dic269C2312yes—
89#41×C269—21no—
90#41×C269—21no—
91#41×C269D412yes—
92#41×C269C2312yes—
93C2×Q8×C269—21no—
94C2×Q8×C269—21no—
95C2×Q8×C269—21no—
96C2×Q8×C269—21no—
97C2×Q8×C269—21no—
98C2×Q8×C269D411yescharacteristic
99C2×Q8×C269C2312yes—
100C2×D4×C269—21no—
101C2×D4×C269D411yescharacteristic
102C22×C4×C269—21no—
103C22×C4×C269—21no—
104C22×C4×C269C2311yescharacteristic
105Q16×C269—24no—
106SD16×C269—24no—
107M16×C269—21no—
108M16×C269C2312yes—
109C2×C8×C269—21no—
110C22×D269—24no—
111C22×D269—21no—
112C22×D269C2×D412yes—
113Dic538—24no—
114Dic538—21no—
115Dic538—21no—
116Dic538—21no—
117Dic538C2×D412yes—
118D1076—24no—
119D1076—21no—
120D1076—21no—
121D1076—21no—
122D1076C2×D412yes—
123C22⋊D269—24no—
124C22⋊D269—24no—
125C22⋊D269—22no—
126C22⋊D269—22no—
127C22⋊D269—22no—
128C22⋊D269—22no—
129C2×Dic269—24no—
130C2×Dic269—21no—
131C2×Dic269—22no—
132C2×Dic269C2×D412yes—
133C4×D269—24no—
134C4×D269—24no—
135C4×D269—22no—
136C4×D269—22no—
137C4×D269—22no—
138C4×D269—22no—
139#14524130—22no—
140C23×C269—21no—
141C23×C269C2×D411yescharacteristic
142Q8×C269—21no—
143Q8×C269—22no—
144Q8×C269—22no—
145Q8×C269—41no—
146Q8×C269—41no—
147Q8×C269C2×D412yes—
148D4×C269—22no—
149D4×C269—22no—
150D4×C269—21no—
151D4×C269C2×D412yes—
152C2×C4×C269—21no—
153C2×C4×C269—22no—
154C2×C4×C269—22no—
155C2×C4×C269—22no—
156C2×C4×C269—41no—
157C2×C4×C269—41no—
158C2×C4×C269C2×D411yescharacteristic
159C2×C4×C269C2×D411yescharacteristic
160C2×C4×C269C2×D411yescharacteristic
161C2×C4×C269C2×D411yescharacteristic
162C2×C4×C269C2×D411yescharacteristic
163C2×C4×C269C2411yescharacteristic
164C8×C269—22no—
165C22×C269—22no—
166C22×C269—41no—
167C22×C269—81no—
168C22×C269#12011yescharacteristic
169C22×C269#12011yescharacteristic
170C22×C269C22×D411yescharacteristic
171C2×D269—22no—
172C2×D269—22no—
173C2×D269—42no—
174C2×D269—81no—
175C2×D269—84no—
176C4×C269—22no—
177C4×C269—42no—
178C4×C269#12011yescharacteristic
179C4×C269#12011yescharacteristic
180C4×C269C22×D411yescharacteristic
181C4×C269C22×D411yescharacteristic
182Dic269—22no—
183Dic269—22no—
184Dic269—42no—
185C2×C269—21no—
186C2×C269—21no—
187C2×C269—21no—
188C2×C269—42no—
189C2×C269C2×#12011yescharacteristic
190D269—42no—
191D269—42no—
192C269#302211yescharacteristic, complemented, Sylow
193#3022—2691nomaximal, Sylow
194#585—2691no—
195C2×#137—2691no—
196C2×#137D26911yescharacteristic, complemented
197#536—2691no—
198#456—2694no—
199#455—2694no—
200#421—2691no—
201#421—2691no—
202#413—2691no—
2032−1+4—2692no—
2042−1+4—5381no—
2052−1+4—5381no—
206H(5,𝔽2)—2692no—
207H(5,𝔽2)—5382no—
208H(5,𝔽2)—5381no—
209H(5,𝔽2)—5381no—
210C2×#41—2691no—
211C2×#41—2691no—
212C2×#41—5382no—
213C2×#41—5382no—
214C2×#41—5382no—
215C2×#41C2×D26911yescharacteristic
216C22×Q8C2×D26911yescharacteristic, complemented
217#137—22no—
218#137—2694no—
219#137—5382no—
220#137C2×D26914yescomplemented
221C2×Q16—21no—
222C2×Q16—5381no—
223C2×SD16—21no—
224C2×SD16—5381no—
225C2×M16—2691no—
226C2×M16C2×D26911yescharacteristic, complemented
227C4⋊Q8—2692no—
228#124—2692no—
229#123—5382no—
230C22⋊Q8—5382no—
231#117—2691no—
232#104—2694no—
233#104—2694no—
234#101—2692no—
235#100—2692no—
236#41—21no—
237#41—21no—
238#41—2692no—
239#41—2692no—
240#41—5382no—
241#41—5382no—
242#41—5384no—
243#41—5384no—
244#41—5384no—
245#41—5384no—
246#41—5382no—
247#41—5382no—
248#41—5382no—
249#41—5382no—
250#41—5381no—
251#41—5381no—
252#41—5381no—
253#41—5381no—
254#41—5381no—
255#41—5381no—
256#41C22⋊D26912yes—
257#41C22×D26912yes—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—2691no—
264C2×Q8—2691no—
265C2×Q8—5382no—
266C2×Q8—5382no—
267C2×Q8—5382no—
268C2×Q8C22⋊D26911yescharacteristic
269C2×Q8C22×D26912yes—
270C2×D4—21no—
271C2×D4—2692no—
272C2×D4—2691no—
273C2×D4—2691no—
274C2×D4—5382no—
275C2×D4—5384no—
276C2×D4—5384no—
277C2×D4—5382no—
278C2×D4—5384no—
279C2×D4—5381no—
280C2×D4—5382no—
281C2×D4C22⋊D26911yescharacteristic
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—2692no—
285C22×C4—5384no—
286C22×C4—5381no—
287C22×C4C22×D26911yescharacteristic
288Q16—24no—
289Q16—5384no—
290SD16—24no—
291SD16—5384no—
292M16—21no—
293M16—2692no—
294M16—5381no—
295M16C22×D26912yes—
296C2×C8—21no—
297C2×C8—5381no—
298C4⋊C4—5384no—
299C4⋊C4—5381no—
300C22⋊C4—5384no—
301C22⋊C4—5381no—
302C42—2692no—
303C23—21no—
304C23—2692no—
305C23—5384no—
306C23—5381no—
307C23D4×D26911yescharacteristic
308Q8—21no—
309Q8—22no—
310Q8—22no—
311Q8—41no—
312Q8—41no—
313Q8—2692no—
314Q8—5384no—
315Q8—5381no—
316Q8—5381no—
317Q8—5381no—
318Q8C2×C22⋊D26912yes—
319D4—22no—
320D4—22no—
321D4—21no—
322D4—2692no—
323D4—5384no—
324D4—5384no—
325D4—5384no—
326D4—5382no—
327D4—5382no—
328D4—5382no—
329D4—5382no—
330D4—5381no—
331D4—5381no—
332D4—5381no—
333D4C2×C22⋊D26912yes—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—2692no—
341C2×C4—5384no—
342C2×C4—5384no—
343C2×C4—5384no—
344C2×C4—5381no—
345C2×C4—5382no—
346C2×C4—5382no—
347C2×C4—5382no—
348C2×C4—5382no—
349C2×C4—5382no—
350C2×C4D4×D26911yescharacteristic
351C2×C4D4×D26911yescharacteristic
352C2×C4D4×D26911yescharacteristic
353C2×C4C2×C22⋊D26911yescharacteristic
354C2×C4C2×C22⋊D26911yescharacteristic
355C2×C4C23×D26911yescharacteristic
356C8—22no—
357C8—5382no—
358C22—22no—
359C22—41no—
360C22—81no—
361C22—5382no—
362C22—5382no—
363C22—10762no—
364C22—21521no—
365C22—21524no—
366C22#2257542111yescharacteristic
367C22#2257542111yescharacteristic
368C22C2×D4×D26911yescharacteristic
369C4—22no—
370C4—42no—
371C4—5382no—
372C4—5382no—
373C4—10762no—
374C4#2257542111yescharacteristic
375C4#2257542111yescharacteristic
376C4C2×D4×D26911yescharacteristic
377C4C22×C22⋊D26911yescharacteristic
378C2—21no—
379C2—21no—
380C2—21no—
381C2—42no—
382C2—10762no—
383C2—10762no—
384C2C2×#2257542111yescharacteristic
subgroupG/H
centerC2C2×#22575421
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC2×#137×C269C2
socleC2×C269C2×#120
prime radical#31771441C2
hypercenterC2×#137D269
hypocenterC269#3022
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#22575484C2semidirect
3. #31771179C2semidirect
4. #31770786C2semidirect
5. #31770781C2semidirect
6. #31770622C2semidirect
7. #31770621C2semidirect
8. #31770587C2semidirect
9. C2×#137×C269C2semidirect
23. #22575484C22semidirect
26. C2×#16864386C22semidirect
27. #22575451C22semidirect
28. #22575442C22semidirect
31. #22575412C22semidirect
32. #22575380C22semidirect
33. #22575379C22semidirect
34. #22575371C22semidirect
35. #22575369C22semidirect
37. C22×Q8×C269C22semidirect
39. #137×C269C22semidirect
42. C2×M16×C269C22semidirect
192. C269#3022semidirect
196. C2×#137D269semidirect
216. C22×Q8C2×D269semidirect
220. #137C2×D269semidirect
226. C2×M16C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317714412
C2×#225754842
#317711792
#317707862
#317707812
#317706222
#317706212
#317705872
C2×#137×C2692
#3022269