Finite Group Databasebeta

← #46820507

Subgroups — #46820507

Order 34432.

548 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C2×#22575478C211yesmaximal, characteristic, complemented
2C2×M16×D269C211yesmaximal, characteristic, complemented
3#31771112C211yesmaximal, characteristic, complemented
4C2×#22575428C211yesmaximal, characteristic, complemented
5#31770857C212yesmaximal, complemented
6#31770854C212yesmaximal, complemented
7#31770840C212yesmaximal, complemented
8#31770837C212yesmaximal, complemented
9#31770645C211yesmaximal, characteristic, complemented
10#31770611C211yesmaximal, characteristic, complemented
11#417×C269C211yesmaximal, characteristic, complemented
12C22×D1076—21no—
13C22×D1076C2211yescharacteristic
14C22×C4×D269C2211yescharacteristic
15#22575478—24no—
16C2×#16864389C2212yescomplemented
17C2×#16864387C2212yescomplemented
18C2×#16864386C2211yescharacteristic
19C2×#16864385C2212yescomplemented
20M16×D269—24no—
21C2×C8×D269C2212yescomplemented
22#22575434—21no—
23#22575429C2211yescharacteristic, complemented
24#22575429C2211yescharacteristic, complemented
25#22575428—21no—
26#22575428—21no—
27#22575428C2212yes—
28#22575428C2212yes—
29#22575424C2211yescharacteristic, complemented
30#22575424C2211yescharacteristic, complemented
31#22575421—21no—
32C4×C22⋊D269—21no—
33C4×C22⋊D269—21no—
34#22575412C2211yescharacteristic
35C2×#16864379—21no—
36#22575386C2212yescomplemented
37#22575384C2212yescomplemented
38#22575375C2212yescomplemented
39#22575375C2212yescomplemented
40C2×#41×C269C2211yescharacteristic
41C2×M16×C269C2211yescharacteristic
42C2×C4⋊C4×C269C2211yescharacteristic
43#103×C269C2212yescomplemented
44#102×C269C2212yescomplemented
45C23×D269—21no—
46C23×D269D411yescharacteristic
47C2×D1076—22no—
48C2×D1076—22no—
49C2×D1076—22no—
50C2×D1076—21no—
51C2×D1076—22no—
52C2×D1076—44no—
53C2×D1076C2312yes—
54C2×C22⋊D269—21no—
55C2×C22⋊D269—21no—
56C22×Dic269D411yescharacteristic
57C2×C4×D269—22no—
58C2×C4×D269—22no—
59C2×C4×D269D411yescharacteristic
60C2×C4×D269D411yescharacteristic
61C2×C4×D269C2311yescharacteristic
62C2×C4×D269C2311yescharacteristic
63#16864389—24no—
64#16864387—24no—
65#16864386D414yescomplemented
66#16864385—24no—
67C2×#14524130C2312yes—
68C8×D269—24no—
69C4⋊Dic269C2311yescharacteristic
70C4⋊Dic269C2311yescharacteristic
71#16864381—21no—
72#16864381—21no—
73C22⋊Dic269—21no—
74C22⋊Dic269—21no—
75C22⋊Dic269—21no—
76#16864379—21no—
77#16864379—21no—
78#16864379—22no—
79#16864379—22no—
80C4×Dic269—21no—
81#41×C269—22no—
82#41×C269—22no—
83C2×Q8×C269C2311yescharacteristic
84C2×D4×C269—21no—
85C2×D4×C269C2311yescharacteristic
86C22×C4×C269—21no—
87C22×C4×C269—21no—
88C22×C4×C269C2311yescharacteristic
89M16×C269—22no—
90C2×C8×C269C2312yes—
91C4⋊C4×C269—21no—
92C4⋊C4×C269C2312yes—
93C22×D269—22no—
94C22×D269—22no—
95C22×D269—22no—
96C22×D269—41no—
97C22×D269—42no—
98C22×D269—42no—
99C22×D269—42no—
100C22×D269C2×D411yescharacteristic
101C22×D269C2×D411yescharacteristic
102D1076—22no—
103D1076—22no—
104D1076—44no—
105D1076—44no—
106D1076—42no—
107C22⋊D269—42no—
108C22⋊D269—42no—
109C2×Dic269—22no—
110C2×Dic269—21no—
111C2×Dic269—21no—
112C2×Dic269C2×D411yescharacteristic
113C2×Dic269C2×D411yescharacteristic
114C4×D269—22no—
115C4×D269—22no—
116C4×D269—21no—
117C4×D269—21no—
118C4×D269—21no—
119C4×D269—21no—
120#14524130C2×D414yescomplemented
121C23×C269—21no—
122C23×C269C2×D411yescharacteristic
123Q8×C269—21no—
124Q8×C269—21no—
125D4×C269—21no—
126D4×C269—21no—
127D4×C269—42no—
128C2×C4×C269—22no—
129C2×C4×C269—21no—
130C2×C4×C269—42no—
131C2×C4×C269—42no—
132C2×C4×C269C2×D411yescharacteristic
133C2×C4×C269C2×D412yes—
134C2×C4×C269#4112yes—
135C2×C4×C269C2411yescharacteristic
136C8×C269—22no—
137C22×C269—21no—
138C22×C269—21no—
139C22×C269—41no—
140C22×C269—41no—
141C22×C269—41no—
142C22×C269#12112yes—
143C22×C269C22×D411yescharacteristic
144C2×D269—21no—
145C2×D269—21no—
146C2×D269—21no—
147C2×D269—21no—
148C2×D269—21no—
149C2×D269—44no—
150C2×D269—42no—
151C2×D269—42no—
152C2×D269—42no—
153C2×D269—84no—
154C2×D269AGL(1,ℤ/8)12yescomplemented
155C4×C269—41no—
156C4×C269—42no—
157C4×C269#12112yes—
158C4×C269C22×D411yescharacteristic
159C4×C269C2×#4111yescharacteristic
160Dic269—21no—
161Dic269—21no—
162Dic269—41no—
163Dic269—41no—
164C2×C269—22no—
165C2×C269—81no—
166C2×C269C2×#12111yescharacteristic
167C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
168C2×C269#57511yescharacteristic
169D269—22no—
170D269—41no—
171D269—82no—
172C269#305511yescharacteristic, complemented, Sylow
173#3055—2691nomaximal, Sylow
174C2×AGL(1,ℤ/8)—2691no—
175C22×M16—2691no—
176#525—2691no—
177C2×#121—2691no—
178#469—2692no—
179#468—2692no—
180#466—2692no—
181#465—2692no—
182#428—2691no—
183#418—2691no—
184#417D26911yescharacteristic, complemented
185C2×#41C2×D26911yescharacteristic
186C22×D4—2691no—
187C22×D4—5381no—
188C23×C4—2691no—
189AGL(1,ℤ/8)—5384no—
190C2×SD16—2692no—
191C2×D8—2692no—
192C2×M16—2691no—
193C2×M16—2692no—
194C2×M16—5384no—
195C2×M16C2×D26911yescharacteristic
196C22×C8—2692no—
197#123—5381no—
198C22⋊Q8—2691no—
199C22⋊Q8—2691no—
200#121—2691no—
201#121—2691no—
202#121—2692no—
203#121—2692no—
204#121—5381no—
205#121—5381no—
206#120—5381no—
207C4×D4—5381no—
208C4×D4—5381no—
209#117—2691no—
210C2×C4⋊C4C2×D26911yescharacteristic
211C2×C22⋊C4—5381no—
212#107—2692no—
213#106—2692no—
214#103C2×D26912yescomplemented
215#102—2692no—
216#102—2692no—
217#102C2×D26912yescomplemented
218C24—2691no—
219C24—5381no—
220#41—22no—
221#41—22no—
222C2×Q8C22×D26911yescharacteristic
223C2×D4—21no—
224C2×D4—2692no—
225C2×D4—5382no—
226C2×D4—5382no—
227C2×D4—5381no—
228C2×D4—5381no—
229C2×D4—5382no—
230C2×D4—5381no—
231C2×D4—5382no—
232C2×D4—10764no—
233C2×D4C22×D26911yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—2691no—
237C22×C4—2691no—
238C22×C4—2691no—
239C22×C4—2691no—
240C22×C4—2691no—
241C22×C4—5382no—
242C22×C4—5382no—
243C22×C4C22×D26911yescharacteristic
244SD16—5384no—
245D8—5384no—
246M16—22no—
247M16—2694no—
248M16—5384no—
249C2×C8—2692no—
250C2×C8—5384no—
251C2×C8C22×D26912yes—
252C4⋊C4—21no—
253C4⋊C4—2691no—
254C4⋊C4—2691no—
255C4⋊C4—5381no—
256C4⋊C4—5381no—
257C4⋊C4C22×D26912yes—
258C22⋊C4—5382no—
259C22⋊C4—5382no—
260C22⋊C4—5381no—
261C22⋊C4—5381no—
262C22⋊C4—5381no—
263C22⋊C4—5381no—
264C22⋊C4—5381no—
265C42—5381no—
266C23—21no—
267C23—2691no—
268C23—2691no—
269C23—5382no—
270C23—5382no—
271C23—5382no—
272C23—10761no—
273C23—10762no—
274C23—10762no—
275C23—10762no—
276C23D4×D26911yescharacteristic
277Q8—21no—
278Q8—21no—
279D4—21no—
280D4—21no—
281D4—42no—
282D4—5382no—
283D4—5382no—
284D4—10764no—
285D4—10764no—
286D4—10762no—
287D4—10762no—
288D4—10762no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—2691no—
294C2×C4—2691no—
295C2×C4—5382no—
296C2×C4—5382no—
297C2×C4—5382no—
298C2×C4—5381no—
299C2×C4—5381no—
300C2×C4—5381no—
301C2×C4—5381no—
302C2×C4—5381no—
303C2×C4—5381no—
304C2×C4D4×D26911yescharacteristic
305C2×C4D4×D26912yes—
306C2×C4#1686440212yes—
307C2×C4C23×D26911yescharacteristic
308C8—22no—
309C8—2694no—
310C22—21no—
311C22—21no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—2692no—
316C22—5381no—
317C22—5381no—
318C22—5381no—
319C22—5381no—
320C22—5381no—
321C22—10764no—
322C22—10762no—
323C22—10762no—
324C22—10762no—
325C22—21524no—
326C22#2257542612yes—
327C22C2×D4×D26911yescharacteristic
328C4—41no—
329C4—42no—
330C4—5381no—
331C4—5381no—
332C4—10761no—
333C4—10761no—
334C4#2257542612yes—
335C4C2×D4×D26911yescharacteristic
336C4C2×#1686440211yescharacteristic
337C2—22no—
338C2—81no—
339C2—5382no—
340C2—10761no—
341C2—21522no—
342C2C2×#2257542611yescharacteristic
343C2AGL(1,ℤ/8)×D26911yescharacteristic
344C2#3177140411yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#417×C269C2
socleC22×C269C22×D4
hypercenter#417D269
hypocenterC269#3055
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#22575478C2semidirect
2. C2×M16×D269C2semidirect
3. #31771112C2semidirect
4. C2×#22575428C2semidirect
5. #31770857C2semidirect
6. #31770854C2semidirect
7. #31770840C2semidirect
8. #31770837C2semidirect
9. #31770645C2semidirect
10. #31770611C2semidirect
11. #417×C269C2semidirect
16. C2×#16864389C22semidirect
17. C2×#16864387C22semidirect
19. C2×#16864385C22semidirect
21. C2×C8×D269C22semidirect
23. #22575429C22semidirect
24. #22575429C22semidirect
29. #22575424C22semidirect
30. #22575424C22semidirect
36. #22575386C22semidirect
37. #22575384C22semidirect
38. #22575375C22semidirect
39. #22575375C22semidirect
43. #103×C269C22semidirect
44. #102×C269C22semidirect
65. #16864386D4semidirect
120. #14524130C2×D4semidirect
154. C2×D269AGL(1,ℤ/8)semidirect
172. C269#3055semidirect
184. #417D269semidirect
214. #103C2×D269semidirect
217. #102C2×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754782
C2×M16×D2692
#317711122
C2×#225754282
#317708572
#317708542
#317708402
#317708372
#317706452
#317706112
#417×C2692
#3055269