Finite Group Databasebeta

← #46821029

Subgroups — #46821029

Order 34432.

430 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#31771150C211yesmaximal, characteristic, complemented
2#41×Dic269C211yesmaximal, characteristic, complemented
3#31770930C211yesmaximal, characteristic, complemented
4#31770926C211yesmaximal, characteristic, complemented
5#31770888C211yesmaximal, characteristic, complemented
6#31770883C211yesmaximal, characteristic, complemented
7#31770803C211yesmaximal, characteristic, complemented
8#31770798C211yesmaximal, characteristic, complemented
9#31770750C211yesmaximal, characteristic, complemented
10#31770717C211yesmaximal, characteristic, complemented
11#31770714C211yesmaximal, characteristic, complemented
12#31770708C211yesmaximal, characteristic, complemented
13#31770706C211yesmaximal, characteristic, complemented
14#31770659C211yesmaximal, characteristic, complemented
15#449×C269C211yesmaximal, characteristic, complemented
16C2×#16864401—21no—
17C2×#16864389C2211yescharacteristic
18C2×#16864387C2211yescharacteristic
19#22575452C2211yescharacteristic
20#22575449C2211yescharacteristic
21#22575439—21no—
22#22575427C2211yescharacteristic
23#22575425—21no—
24Q8×Dic269C2211yescharacteristic
25Q8×Dic269C2211yescharacteristic
26D4×Dic269—21no—
27D4×Dic269—21no—
28D4×Dic269C2211yescharacteristic
29D4×Dic269C2211yescharacteristic
30D4×Dic269C2211yescharacteristic
31#22575415C2211yescharacteristic
32#22575413—21no—
33#22575412—21no—
34#22575412C2211yescharacteristic
35C2×C4×Dic269—21no—
36C2×C4×Dic269C2211yescharacteristic
37#22575387C2211yescharacteristic
38#22575386C2211yescharacteristic
39#22575385C2211yescharacteristic
40#22575384C2211yescharacteristic
41#22575382C2211yescharacteristic
42#22575382C2211yescharacteristic
43#22575377C2211yescharacteristic
44#22575375C2211yescharacteristic
45#22575375C2211yescharacteristic
46#22575374C2211yescharacteristic
47#22575374C2211yescharacteristic
48#22575374C2211yescharacteristic
49C22⋊#14524130C2211yescharacteristic
50#22575363C2211yescharacteristic
51#22575363C2211yescharacteristic
52C2×#41×C269C2211yescharacteristic
53C2×SD16×C269C2211yescharacteristic
54C2×D8×C269C2211yescharacteristic
55#121×C269C2211yescharacteristic
56#103×C269C2211yescharacteristic
57#102×C269C2211yescharacteristic
58C22⋊C8×C269C2211yescharacteristic
59#16864401—44no—
60C2×Dic538—21no—
61C2×D1076C2311yescharacteristic
62C2×C22⋊D269—21no—
63C2×C22⋊D269—21no—
64C2×C22⋊D269—21no—
65C22×Dic269—21no—
66C22×Dic269—21no—
67C22×Dic269—21no—
68C22×Dic269D411yescharacteristic
69C22×Dic269D411yescharacteristic
70C2×C4×D269—21no—
71#16864389—22no—
72#16864387—22no—
73C2×#14524130C2311yescharacteristic
74C2×#14524130C2311yescharacteristic
75C4⋊Dic269—21no—
76C4⋊Dic269—21no—
77C4⋊Dic269C2311yescharacteristic
78C4⋊Dic269C2311yescharacteristic
79C4⋊Dic269C2311yescharacteristic
80#16864381—21no—
81C22⋊Dic269—21no—
82C22⋊Dic269—21no—
83C22⋊Dic269—21no—
84C22⋊Dic269—21no—
85#16864379—21no—
86#16864379—21no—
87C4×Dic269—21no—
88C4×Dic269—21no—
89C4×Dic269—21no—
90C4×Dic269D411yescharacteristic
91C4×Dic269D411yescharacteristic
92C4×Dic269C2311yescharacteristic
93C4×Dic269C2311yescharacteristic
94#41×C269—22no—
95#41×C269—22no—
96C2×Q8×C269C2311yescharacteristic
97C2×D4×C269—21no—
98C2×D4×C269—21no—
99C2×D4×C269C2311yescharacteristic
100C2×D4×C269C2311yescharacteristic
101C22×C4×C269—21no—
102C22×C4×C269C2311yescharacteristic
103SD16×C269—22no—
104D8×C269—22no—
105C2×C8×C269C2311yescharacteristic
106C2×C8×C269C2311yescharacteristic
107C4⋊C4×C269C2311yescharacteristic
108C22⋊C4×C269—21no—
109C22×D269—21no—
110Dic538—42no—
111D1076—21no—
112D1076—21no—
113C22⋊D269—42no—
114C22⋊D269—42no—
115C22⋊D269—42no—
116C2×Dic269—22no—
117C2×Dic269—21no—
118C2×Dic269—21no—
119C2×Dic269—21no—
120C2×Dic269—41no—
121C2×Dic269—42no—
122C2×Dic269—42no—
123C2×Dic269—42no—
124C2×Dic269C2×D411yescharacteristic
125C2×Dic269C2×D411yescharacteristic
126C2×Dic269C2×D411yescharacteristic
127C2×Dic269C2×D411yescharacteristic
128C4×D269—42no—
129#14524130—21no—
130#14524130—21no—
131C23×C269—21no—
132C23×C269—21no—
133C23×C269C2×D411yescharacteristic
134Q8×C269—21no—
135Q8×C269#4112yescomplemented
136D4×C269—24no—
137D4×C269—21no—
138D4×C269—21no—
139D4×C269—21no—
140D4×C269—42no—
141D4×C269#4112yescomplemented
142C2×C4×C269—24no—
143C2×C4×C269—21no—
144C2×C4×C269—21no—
145C2×C4×C269—21no—
146C2×C4×C269—21no—
147C2×C4×C269C2×D411yescharacteristic
148C2×C4×C269C2411yescharacteristic
149C8×C269—21no—
150C8×C269—21no—
151C22×C269—21no—
152C22×C269—22no—
153C22×C269—21no—
154C22×C269—21no—
155C22×C269—41no—
156C22×C269—41no—
157C22×C269—41no—
158C22×C269—41no—
159C22×C269—41no—
160C22×C269C22×D411yescharacteristic
161C2×D269—41no—
162C2×D269—41no—
163C2×D269—41no—
164C4×C269—21no—
165C4×C269—22no—
166C4×C269—41no—
167C4×C269C2×#4111yescharacteristic
168C4×C269C2×#4111yescharacteristic
169Dic269—21no—
170Dic269—21no—
171Dic269—21no—
172Dic269—41no—
173Dic269—41no—
174Dic269—41no—
175Dic269AGL(1,ℤ/8)12yes—
176C2×C269—41no—
177C2×C269—42no—
178C2×C269—81no—
179C2×C269#52511yescharacteristic
180C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
181C2×C269#57511yescharacteristic
182D269—81no—
183C269#311511yescharacteristic, complemented, Sylow
184#3115—2691nomaximal, Sylow
185#532—2691no—
186C4×#41—2691no—
187#482—2691no—
188#482—2691no—
189#475—2691no—
190#474—2691no—
191#460—2691no—
192#459—2691no—
193#449—2691no—
194#449D26911yescharacteristic, complemented
195#442—2691no—
196#442—2691no—
197#440—2691no—
198#440—2691no—
199#432—2691no—
200C2×#41—5381no—
201C2×#41C2×D26911yescharacteristic
202C2×SD16—2691no—
203C2×SD16C2×D26911yescharacteristic
204C2×D8—2691no—
205C2×D8C2×D26911yescharacteristic
206C4⋊Q8—2691no—
207#127—2691no—
208#124—5381no—
209#121—2691no—
210#121—5381no—
211#121C2×D26911yescharacteristic
212C4×Q8—2691no—
213C4×Q8—2691no—
214C4×D4—2691no—
215C4×D4—2691no—
216C4×D4—2691no—
217C4×D4—2691no—
218C4×D4—5381no—
219C4×D4—5381no—
220C4×D4—5381no—
221#117—2691no—
222#117—5381no—
223C2×C42—2691no—
224C2×C42—5381no—
225#107—2691no—
226#107—2691no—
227#106—2691no—
228#106—2691no—
229C4⋊C8—2691no—
230C4⋊C8—2691no—
231#103—2691no—
232#103C2×D26911yescharacteristic
233#102—2691no—
234#102—2691no—
235#102—2691no—
236#102—2691no—
237#102—2691no—
238#102C2×D26911yescharacteristic
239C22⋊C8—2691no—
240C22⋊C8C2×D26911yescharacteristic
241#97—2691no—
242#97—2691no—
243#41—22no—
244#41—22no—
245#41—10764no—
246C2×Q8—5381no—
247C2×Q8C22×D26911yescharacteristic
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—2691no—
251C2×D4—5381no—
252C2×D4—5381no—
253C2×D4—5381no—
254C2×D4C22×D26911yescharacteristic
255C2×D4C22×D26911yescharacteristic
256C22×C4—21no—
257C22×C4—2691no—
258C22×C4—2691no—
259C22×C4—5381no—
260C22×C4—5381no—
261C22×C4—5381no—
262C22×C4—5381no—
263C22×C4C22×D26911yescharacteristic
264SD16—22no—
265SD16—5382no—
266D8—22no—
267D8—5382no—
268C2×C8—2691no—
269C2×C8—2691no—
270C2×C8C22×D26911yescharacteristic
271C2×C8C22×D26911yescharacteristic
272C4⋊C4—2691no—
273C4⋊C4—2691no—
274C4⋊C4—2691no—
275C4⋊C4—5381no—
276C4⋊C4—5381no—
277C4⋊C4—5381no—
278C4⋊C4C22×D26911yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—5381no—
281C22⋊C4—5381no—
282C22⋊C4—5381no—
283C22⋊C4—5381no—
284C22⋊C4—5381no—
285C22⋊C4—5381no—
286C42—2691no—
287C42—2691no—
288C42—2691no—
289C42—2691no—
290C42—5381no—
291C42—5381no—
292C42—5381no—
293C23—21no—
294C23—21no—
295C23—5381no—
296C23D4×D26911yescharacteristic
297Q8—21no—
298Q8—10762no—
299Q8#1686440112yescomplemented
300D4—24no—
301D4—21no—
302D4—21no—
303D4—21no—
304D4—42no—
305D4—5381no—
306D4—5381no—
307D4—10762no—
308D4—10762no—
309D4—10762no—
310D4#1686440112yescomplemented
311C2×C4—24no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—2691no—
317C2×C4—2691no—
318C2×C4—2691no—
319C2×C4—2691no—
320C2×C4—5382no—
321C2×C4—5381no—
322C2×C4—5381no—
323C2×C4—5381no—
324C2×C4—10762no—
325C2×C4—10762no—
326C2×C4—10761no—
327C2×C4—10762no—
328C2×C4—10762no—
329C2×C4D4×D26911yescharacteristic
330C2×C4C23×D26911yescharacteristic
331C8—21no—
332C8—21no—
333C8—5381no—
334C8—5381no—
335C22—21no—
336C22—21no—
337C22—22no—
338C22—21no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—10761no—
345C22—10761no—
346C22—10761no—
347C22C2×D4×D26911yescharacteristic
348C4—21no—
349C4—22no—
350C4—41no—
351C4—2692no—
352C4—5381no—
353C4—5381no—
354C4—5381no—
355C4—10761no—
356C4—10761no—
357C4—10761no—
358C4C2×#1686440111yescharacteristic
359C4C2×#1686440111yescharacteristic
360C2—41no—
361C2—42no—
362C2—81no—
363C2—21521no—
364C2#3177111111yescharacteristic
365C2AGL(1,ℤ/8)×D26911yescharacteristic
366C2#3177140211yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#449×C269C2
socleC22×C269C22×D4
hypercenter#449D269
hypocenterC269#3115
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #31771150C2semidirect
2. #41×Dic269C2semidirect
3. #31770930C2semidirect
4. #31770926C2semidirect
5. #31770888C2semidirect
6. #31770883C2semidirect
7. #31770803C2semidirect
8. #31770798C2semidirect
9. #31770750C2semidirect
10. #31770717C2semidirect
11. #31770714C2semidirect
12. #31770708C2semidirect
13. #31770706C2semidirect
14. #31770659C2semidirect
15. #449×C269C2semidirect
135. Q8×C269#41semidirect
141. D4×C269#41semidirect
183. C269#3115semidirect
194. #449D269semidirect
299. Q8#16864401semidirect
310. D4#16864401semidirect

Maximal subgroups

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

subgroupindex
#317711502
#41×Dic2692
#317709302
#317709262
#317708882
#317708832
#317708032
#317707982
#317707502
#317707172
#317707142
#317707082
#317707062
#317706592
#449×C2692
#3115269