Finite Group Databasebeta

← #46820108

Subgroups — #46820108

Order 34432.

764 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#16864403C211yesmaximal, characteristic, complemented
2C2×#22575476C212yesmaximal, complemented
3#31771118C211yesmaximal, characteristic, complemented
4#31770766C212yesmaximal, complemented
5#31770755C211yesmaximal, characteristic, complemented
6#31770753C212yesmaximal, complemented
7#31770744C211yesmaximal, characteristic, complemented
8#31770741C211yesmaximal, characteristic, complemented
9#31770600C212yesmaximal, complemented
10C22⋊C8×D269C211yesmaximal, characteristic, complemented
11#451×C269C211yesmaximal, characteristic, complemented
12C2×#16864403—28no—
13C2×#16864403—21no—
14C2×#16864403—21no—
15C2×#16864403C2212yes—
16C2×#16864403C2212yes—
17C2×#16864403C2211yescharacteristic, complemented
18C22×D1076—21no—
19C22×D1076C2211yescharacteristic
20C22×C4×D269—21no—
21C22×C4×D269C2211yescharacteristic
22#22575476—28no—
23C2×#16864389C2212yescomplemented
24C2×#16864389C2212yescomplemented
25C2×D2152C2212yescomplemented
26C2×C8×D269C2212yescomplemented
27#22575438—21no—
28#22575429C2211yescharacteristic, complemented
29#22575428C2211yescharacteristic, complemented
30#22575427C2211yescharacteristic
31#22575423—21no—
32#22575415—21no—
33#22575413—21no—
34#22575411C2211yescharacteristic, complemented
35#22575377C2212yescomplemented
36#22575375C2212yescomplemented
37#22575373C2212yescomplemented
38C22⋊#14524130C2211yescharacteristic
39#22575365C2212yescomplemented
40C22×Q8×C269C2211yescharacteristic
41C2×Q16×C269C2212yescomplemented
42C22⋊Q8×C269C2211yescharacteristic
43#103×C269C2212yescomplemented
44C22⋊C8×C269C2211yescharacteristic
45C23×D269—21no—
46C23×D269D411yescharacteristic
47#16864403—28no—
48#16864403—28no—
49#16864403—24no—
50#16864403—24no—
51#16864403—24no—
52#16864403—24no—
53#16864403D418yescomplemented
54C2×D1076—28no—
55C2×D1076—22no—
56C2×D1076—22no—
57C2×D1076—24no—
58C2×D1076—21no—
59C2×D1076—21no—
60C2×D1076—21no—
61C2×D1076C2311yescharacteristic
62C2×D1076C2312yes—
63C2×C22⋊D269—21no—
64C22×Dic269D411yescharacteristic
65C2×C4×D269—28no—
66C2×C4×D269—22no—
67C2×C4×D269—22no—
68C2×C4×D269—24no—
69C2×C4×D269—21no—
70C2×C4×D269D411yescharacteristic
71C2×C4×D269D411yescharacteristic
72C2×C4×D269C2311yescharacteristic
73C2×C4×D269C2311yescharacteristic
74#16864389—24no—
75#16864389—24no—
76D2152—24no—
77C2×#14524130C2312yes—
78C8×D269—24no—
79C4⋊Dic269C2311yescharacteristic
80#16864381—21no—
81#16864379—21no—
82#16864379—21no—
83#16864379—21no—
84#16864379—21no—
85C4×Dic269—21no—
86C2×Q8×C269—24no—
87C2×Q8×C269—21no—
88C2×Q8×C269C2311yescharacteristic
89C2×Q8×C269C2312yes—
90C22×C4×C269—21no—
91C22×C4×C269C2311yescharacteristic
92Q16×C269—24no—
93C2×C8×C269C2312yes—
94C4⋊C4×C269—21no—
95C4⋊C4×C269C2311yescharacteristic
96C22⋊C4×C269—21no—
97C22×D269—24no—
98C22×D269—22no—
99C22×D269—22no—
100C22×D269—22no—
101C22×D269—21no—
102C22×D269—41no—
103C22×D269—42no—
104C22×D269—42no—
105C22×D269C2×D411yescharacteristic
106C22×D269C2×D411yescharacteristic
107D1076—28no—
108D1076—24no—
109D1076—28no—
110D1076—22no—
111D1076—22no—
112D1076—42no—
113D1076—42no—
114D1076C2×D414yes—
115C22⋊D269—42no—
116C2×Dic269—22no—
117C2×Dic269—21no—
118C2×Dic269C2×D411yescharacteristic
119C2×Dic269C2×D411yescharacteristic
120C4×D269—28no—
121C4×D269—22no—
122C4×D269—22no—
123C4×D269—28no—
124C4×D269—22no—
125C4×D269—42no—
126C4×D269C2×D414yes—
127#14524130—22no—
128C23×C269C2×D411yescharacteristic
129Q8×C269—24no—
130Q8×C269—22no—
131Q8×C269—22no—
132Q8×C269C2×D414yes—
133C2×C4×C269—24no—
134C2×C4×C269—22no—
135C2×C4×C269—22no—
136C2×C4×C269—21no—
137C2×C4×C269—21no—
138C2×C4×C269C2×D411yescharacteristic
139C2×C4×C269C2411yescharacteristic
140C8×C269—22no—
141C22×C269—21no—
142C22×C269—21no—
143C22×C269#13512yescomplemented
144C22×C269C22×D411yescharacteristic
145C2×D269—21no—
146C2×D269—21no—
147C2×D269—21no—
148C2×D269—24no—
149C2×D269—21no—
150C2×D269—21no—
151C2×D269—44no—
152C2×D269—48no—
153C2×D269—42no—
154C2×D269—42no—
155C2×D269—41no—
156C2×D269—41no—
157C2×D269—41no—
158C2×D269#12012yes—
159C4×C269—24no—
160C4×C269—21no—
161C4×C269—41no—
162C4×C269—41no—
163C4×C269C22×D412yes—
164Dic269—21no—
165Dic269—41no—
166Dic269#12012yes—
167C2×C269—22no—
168C2×C269C2×#12011yescharacteristic
169C2×C269C2×#13511yescharacteristic
170C2×C269#57511yescharacteristic
171D269—22no—
172D269—41no—
173D269—44no—
174D269—81no—
175C269#300211yescharacteristic, complemented, Sylow
176#3002—2691nomaximal, Sylow
177C22×#41—2691no—
178C2×#135—2692no—
179#525—2691no—
180#452—2692no—
181#451D26911yescharacteristic, complemented
182#450—2691no—
183#449—2692no—
184#448—2691no—
185#447—2691no—
186#416—2692no—
187C2×C22⋊C8—2691no—
188C2×#41—2692no—
189C2×#41—2691no—
190C2×#41—2692no—
191C2×#41—5381no—
192C2×#41—5381no—
193C2×#41—5388no—
194C22×Q8C2×D26911yescharacteristic
195C22×D4—2691no—
196C22×D4—5381no—
197C23×C4—2691no—
198C23×C4—5381no—
199#135—5388no—
200C2×Q16C2×D26912yescomplemented
201C2×SD16—2692no—
202C2×SD16—2692no—
203C2×D8—2692no—
204C22×C8—2692no—
205#123—5381no—
206C22⋊Q8—2691no—
207C22⋊Q8C2×D26911yescharacteristic
208#121—2691no—
209#121—2691no—
210#120—5381no—
211C4×D4—5381no—
212C4×D4—5381no—
213#117—2691no—
214#103—2692no—
215#103C2×D26912yescomplemented
216#102—2692no—
217#102—2692no—
218C22⋊C8—2691no—
219C22⋊C8—2692no—
220C22⋊C8C2×D26911yescharacteristic
221C24—2691no—
222C24—5381no—
223#41—2698no—
224#41—5384no—
225#41—5384no—
226#41—5384no—
227#41—5388no—
228#41—5388no—
229#41—5384no—
230C2×Q8—24no—
231C2×Q8—21no—
232C2×Q8C22×D26911yescharacteristic
233C2×Q8C22×D26912yes—
234C2×D4—2692no—
235C2×D4—2691no—
236C2×D4—5384no—
237C2×D4—5382no—
238C2×D4—5382no—
239C2×D4—5388no—
240C2×D4—5381no—
241C2×D4—5381no—
242C2×D4—5381no—
243C2×D4—5381no—
244C22×C4—21no—
245C22×C4—2691no—
246C22×C4—2691no—
247C22×C4—2691no—
248C22×C4—2691no—
249C22×C4—2691no—
250C22×C4—5384no—
251C22×C4—5382no—
252C22×C4—5382no—
253C22×C4—5381no—
254C22×C4—5388no—
255C22×C4C22×D26911yescharacteristic
256Q16—24no—
257SD16—5384no—
258SD16—5384no—
259D8—5384no—
260C2×C8—2692no—
261C2×C8—5384no—
262C2×C8C22×D26912yes—
263C4⋊C4—21no—
264C4⋊C4—2691no—
265C4⋊C4—5381no—
266C4⋊C4C22×D26911yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—5381no—
269C22⋊C4—5381no—
270C22⋊C4—5381no—
271C22⋊C4—5381no—
272C42—5381no—
273C23—2691no—
274C23—2691no—
275C23—5382no—
276C23—5384no—
277C23—5382no—
278C23—5382no—
279C23—5381no—
280C23—10761no—
281C23—10762no—
282C23—10762no—
283C23D4×D26911yescharacteristic
284Q8—24no—
285Q8—22no—
286Q8—22no—
287Q8D4×D26914yes—
288D4—2694no—
289D4—5388no—
290D4—5388no—
291D4—5384no—
292D4—5382no—
293D4—5382no—
294D4—10762no—
295D4—10762no—
296D4—10762no—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—2691no—
303C2×C4—2694no—
304C2×C4—2691no—
305C2×C4—5382no—
306C2×C4—5388no—
307C2×C4—5382no—
308C2×C4—5388no—
309C2×C4—5382no—
310C2×C4—5382no—
311C2×C4—5381no—
312C2×C4—10762no—
313C2×C4D4×D26911yescharacteristic
314C2×C4C23×D26911yescharacteristic
315C8—22no—
316C8—5382no—
317C22—21no—
318C22—21no—
319C22—2692no—
320C22—5381no—
321C22—5381no—
322C22—5381no—
323C22—5384no—
324C22—5381no—
325C22—5381no—
326C22—10764no—
327C22—10768no—
328C22—10762no—
329C22—10762no—
330C22—10761no—
331C22—10761no—
332C22—10761no—
333C22#2257547612yescomplemented
334C22C2×D4×D26911yescharacteristic
335C4—24no—
336C4—21no—
337C4—41no—
338C4—41no—
339C4—2692no—
340C4—5381no—
341C4—10761no—
342C4C2×D4×D26912yes—
343C2—22no—
344C2—5382no—
345C2—10761no—
346C2—10764no—
347C2—21521no—
348C2#120×D26911yescharacteristic
349C2C2×#2257547611yescharacteristic
350C2#3177140411yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#451×C269C2
socleC22×C269C22×D4
hypercenter#451D269
hypocenterC269#3002
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C22×#16864403C2semidirect
2. C2×#22575476C2semidirect
3. #31771118C2semidirect
4. #31770766C2semidirect
5. #31770755C2semidirect
6. #31770753C2semidirect
7. #31770744C2semidirect
8. #31770741C2semidirect
9. #31770600C2semidirect
10. C22⋊C8×D269C2semidirect
11. #451×C269C2semidirect
17. C2×#16864403C22semidirect
23. C2×#16864389C22semidirect
24. C2×#16864389C22semidirect
25. C2×D2152C22semidirect
26. C2×C8×D269C22semidirect
28. #22575429C22semidirect
29. #22575428C22semidirect
34. #22575411C22semidirect
35. #22575377C22semidirect
36. #22575375C22semidirect
37. #22575373C22semidirect
39. #22575365C22semidirect
41. C2×Q16×C269C22semidirect
43. #103×C269C22semidirect
53. #16864403D4semidirect
143. C22×C269#135semidirect
175. C269#3002semidirect
181. #451D269semidirect
200. C2×Q16C2×D269semidirect
215. #103C2×D269semidirect
333. C22#22575476semidirect

Maximal subgroups

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

subgroupindex
C22×#168644032
C2×#225754762
#317711182
#317707662
#317707552
#317707532
#317707442
#317707412
#317706002
C22⋊C8×D2692
#451×C2692
#3002269