Finite Group Databasebeta

← #46822529

Subgroups — #46822529

Order 34432.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22575483C211yesmaximal, characteristic, complemented
2C2×#22575482C211yesmaximal, characteristic, complemented
3#31771268C211yesmaximal, characteristic, complemented
4#31771254C211yesmaximal, characteristic
5#31770983C211yesmaximal, characteristic, complemented
6#31770862C211yesmaximal, characteristic, complemented
7#31770861C211yesmaximal, characteristic, complemented
8#31770814C211yesmaximal, characteristic, complemented
9#31770808C211yesmaximal, characteristic, complemented
10#31770768C211yesmaximal, characteristic, complemented
11#31770763C211yesmaximal, characteristic, complemented
12#31770759C211yesmaximal, characteristic, complemented
13#31770696C211yesmaximal, characteristic, complemented
14#31770680C211yesmaximal, characteristic, complemented
15#452×C269C211yesmaximal, characteristic, complemented
16C2×#16864401C2211yescharacteristic
17C22×Dic538C2211yescharacteristic
18C2×Q8×D269C2211yescharacteristic
19#22575483—24no—
20#22575482—24no—
21C2×Dic1076C2211yescharacteristic
22C2×Dic1076C2211yescharacteristic
23C2×#16864392C2211yescharacteristic
24C2×#16864392C2211yescharacteristic
25C2×#16864392C2211yescharacteristic
26C2×#16864390C2211yescharacteristic
27C2×#16864390C2211yescharacteristic
28C2×#16864386C2211yescharacteristic
29C2×#16864385C2211yescharacteristic
30#22575452—21no—
31#22575451C2211yescharacteristic
32#22575451C2211yescharacteristic
33#22575436—21no—
34#22575434—21no—
35#22575433C2211yescharacteristic
36#22575432C2211yescharacteristic
37#22575431—21no—
38#22575430—21no—
39#22575429C2211yescharacteristic
40#22575425—21no—
41#22575418C2211yescharacteristic
42D4×Dic269C2211yescharacteristic
43C4×C22⋊D269C2211yescharacteristic
44#22575413—21no—
45C2×#16864381—21no—
46#22575384C2211yescharacteristic
47#22575382C2211yescharacteristic
48#22575378C2211yescharacteristic
49#22575378C2211yescharacteristic
50#22575377C2211yescharacteristic
51#22575376C2211yescharacteristic
52#22575365C2211yescharacteristic
53C8×Dic269C2211yescharacteristic
54C2×#41×C269C2211yescharacteristic
55C2×Q16×C269C2211yescharacteristic
56C2×SD16×C269C2211yescharacteristic
57C22⋊Q8×C269C2211yescharacteristic
58#103×C269C2211yescharacteristic
59#102×C269C2211yescharacteristic
60C22⋊C8×C269C2211yescharacteristic
61#16864401—22no—
62#16864401—41no—
63C2×Dic538—21no—
64C2×Dic538—22no—
65C2×Dic538—21no—
66C2×Dic538—41no—
67C2×Dic538C2311yescharacteristic
68C2×Dic538C2311yescharacteristic
69C2×Dic538C2311yescharacteristic
70Q8×D269—22no—
71Q8×D269—41no—
72C2×C22⋊D269—21no—
73C2×C22⋊D269D411yescharacteristic
74C22×Dic269—21no—
75C22×Dic269—21no—
76C22×Dic269—21no—
77C2×C4×D269—21no—
78C2×C4×D269C2311yescharacteristic
79Dic1076—22no—
80Dic1076—41no—
81#16864392—22no—
82#16864392—22no—
83#16864392—22no—
84#16864390—22no—
85#16864390D414yescomplemented
86#16864386—22no—
87#16864385—22no—
88C2×#14524130C2311yescharacteristic
89C2×#14524130C2311yescharacteristic
90C4⋊Dic269—21no—
91C4⋊Dic269C2311yescharacteristic
92#16864381—21no—
93#16864381—21no—
94#16864381—21no—
95#16864381—21no—
96#16864381—21no—
97#16864381—21no—
98#16864381—21no—
99#16864381D411yescharacteristic
100C22⋊Dic269—21no—
101C22⋊Dic269—21no—
102C22⋊Dic269D411yescharacteristic
103#16864379—21no—
104#16864379—21no—
105#16864379D411yescharacteristic
106C4×Dic269—21no—
107C4×Dic269C2311yescharacteristic
108#41×C269—22no—
109#41×C269—41no—
110C2×Q8×C269C2311yescharacteristic
111C2×Q8×C269C2311yescharacteristic
112C2×D4×C269—21no—
113C2×D4×C269C2311yescharacteristic
114C22×C4×C269—21no—
115C22×C4×C269C2311yescharacteristic
116Q16×C269—41no—
117SD16×C269—22no—
118C2×C8×C269C2311yescharacteristic
119C2×C8×C269C2311yescharacteristic
120C4⋊C4×C269—21no—
121C4⋊C4×C269C2311yescharacteristic
122C22⋊C4×C269—21no—
123C22×D269C2×D411yescharacteristic
124Dic538—21no—
125Dic538—21no—
126Dic538—21no—
127Dic538—21no—
128Dic538—21no—
129Dic538—42no—
130Dic538—42no—
131Dic538C2×D412yes—
132C22⋊D269—22no—
133C22⋊D269—42no—
134C2×Dic269—21no—
135C2×Dic269—21no—
136C2×Dic269—21no—
137C2×Dic269—21no—
138C2×Dic269—42no—
139C2×Dic269—42no—
140C2×Dic269—42no—
141C2×Dic269C2×D411yescharacteristic
142C2×Dic269C2×D411yescharacteristic
143C2×Dic269C2×D411yescharacteristic
144C4×D269—21no—
145C4×D269—21no—
146C4×D269—42no—
147#14524130—21no—
148#14524130C2×D412yes—
149C23×C269—21no—
150C23×C269C2×D411yescharacteristic
151Q8×C269—21no—
152Q8×C269—21no—
153Q8×C269—21no—
154Q8×C269—21no—
155D4×C269—21no—
156D4×C269—42no—
157D4×C269C2×D412yes—
158C2×C4×C269—21no—
159C2×C4×C269—21no—
160C2×C4×C269—21no—
161C2×C4×C269—21no—
162C2×C4×C269—21no—
163C2×C4×C269—42no—
164C2×C4×C269C2×D411yescharacteristic
165C2×C4×C269C2411yescharacteristic
166C8×C269—21no—
167C8×C269—21no—
168C22×C269—21no—
169C22×C269—22no—
170C22×C269—21no—
171C22×C269—21no—
172C22×C269—41no—
173C22×C269—41no—
174C22×C269C22×D411yescharacteristic
175C2×D269—21no—
176C2×D269—21no—
177C2×D269—21no—
178C4×C269—21no—
179C4×C269—41no—
180C4×C269—41no—
181C4×C269—41no—
182C4×C269C22×D411yescharacteristic
183C4×C269H(5,𝔽2)11yescharacteristic
184Dic269—21no—
185Dic269—22no—
186Dic269—21no—
187Dic269—41no—
188Dic269—41no—
189Dic269—41no—
190Dic269#13712yescomplemented
191C2×C269—41no—
192C2×C269—42no—
193C2×C269D4211yescharacteristic
194C2×C269C2×#13711yescharacteristic
195C2×C269#57811yescharacteristic
196D269—41no—
197C269#327611yescharacteristic, complemented, Sylow
198#3276—2691nomaximal, Sylow
199C2×#137—2691no—
200C2×#137—2691no—
201D4×Q8—2691no—
202#547—2691no—
203#494—2691no—
204#470—2691no—
205#470—2691no—
206#462—2691no—
207#461—2691no—
208#452—2691no—
209#452D26911yescharacteristic, complemented
210#451—2691no—
211#451—2691no—
212C4×SD16—2691no—
213#436—2691no—
214C2×#41—2691no—
215C2×#41C2×D26911yescharacteristic
216C22×Q8—2691no—
217C22×Q8—2691no—
218#137—5384no—
219#137—5384no—
220C2×Q16—2691no—
221C2×Q16—2691no—
222C2×Q16—2691no—
223C2×Q16—2691no—
224C2×Q16—2691no—
225C2×Q16C2×D26911yescharacteristic
226C2×SD16—2691no—
227C2×SD16—2691no—
228C2×SD16C2×D26911yescharacteristic
229C2×M16—2691no—
230C2×M16—2691no—
231C4⋊Q8—2691no—
232C4⋊Q8—2691no—
233C4⋊Q8—5381no—
234#123—5381no—
235#123—5381no—
236C22⋊Q8—2691no—
237C22⋊Q8—2691no—
238C22⋊Q8—2691no—
239C22⋊Q8—5381no—
240C22⋊Q8—5381no—
241C22⋊Q8C2×D26911yescharacteristic
242#121—5381no—
243C4×Q8—2691no—
244C4×D4—2691no—
245C4×D4—2691no—
246C4×D4—5381no—
247C2×C4⋊C4—5381no—
248#106—2691no—
249C4⋊C8—2691no—
250#103—2691no—
251#103—2691no—
252#103—2691no—
253#103—2691no—
254#103C2×D26911yescharacteristic
255#102C2×D26911yescharacteristic
256C22⋊C8—2691no—
257C22⋊C8C2×D26911yescharacteristic
258C4×C8—2691no—
259#41—22no—
260#41—41no—
261#41—5382no—
262#41—10761no—
263C2×Q8—2691no—
264C2×Q8—2691no—
265C2×Q8—2691no—
266C2×Q8—5381no—
267C2×Q8—5382no—
268C2×Q8—5381no—
269C2×Q8—5382no—
270C2×Q8—10761no—
271C2×Q8—10761no—
272C2×Q8C22×D26911yescharacteristic
273C2×Q8C22×D26911yescharacteristic
274C2×D4—21no—
275C2×D4—2691no—
276C2×D4—5381no—
277C2×D4C22×D26911yescharacteristic
278C22×C4—21no—
279C22×C4—2691no—
280C22×C4—5381no—
281C22×C4—5381no—
282C22×C4—5381no—
283C22×C4—5381no—
284C22×C4C22×D26911yescharacteristic
285Q16—41no—
286Q16—5382no—
287Q16—5382no—
288Q16—5382no—
289Q16—5382no—
290Q16—10761no—
291SD16—22no—
292SD16—2694no—
293SD16—5382no—
294M16—5382no—
295M16—5382no—
296C2×C8—2691no—
297C2×C8—2691no—
298C2×C8C22×D26911yescharacteristic
299C2×C8C22×D26911yescharacteristic
300C4⋊C4—21no—
301C4⋊C4—2691no—
302C4⋊C4—2691no—
303C4⋊C4—5381no—
304C4⋊C4—5381no—
305C4⋊C4—5381no—
306C4⋊C4—5381no—
307C4⋊C4—5381no—
308C4⋊C4—5381no—
309C4⋊C4—5381no—
310C4⋊C4—5381no—
311C4⋊C4C22×D26911yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—2691no—
314C22⋊C4—2691no—
315C22⋊C4—5381no—
316C22⋊C4—5381no—
317C22⋊C4—5381no—
318C22⋊C4—5381no—
319C42—2691no—
320C42—5381no—
321C23—21no—
322C23—2691no—
323C23D4×D26911yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—2692no—
329Q8—5381no—
330Q8—5381no—
331Q8—5381no—
332Q8—5381no—
333Q8—5381no—
334Q8—10762no—
335Q8—10762no—
336D4—21no—
337D4—42no—
338D4—5382no—
339D4—10762no—
340D4D4×D26912yes—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—42no—
347C2×C4—2691no—
348C2×C4—2691no—
349C2×C4—2691no—
350C2×C4—5381no—
351C2×C4—5381no—
352C2×C4—5381no—
353C2×C4—5381no—
354C2×C4—5381no—
355C2×C4—5381no—
356C2×C4—10762no—
357C2×C4—10762no—
358C2×C4—10762no—
359C2×C4—10762no—
360C2×C4D4×D26911yescharacteristic
361C2×C4C23×D26911yescharacteristic
362C8—21no—
363C8—21no—
364C8—2692no—
365C8—5381no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—22no—
370C22—41no—
371C22—41no—
372C22—5381no—
373C22—5381no—
374C22—5381no—
375C22C2×D4×D26911yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—2692no—
381C4—5381no—
382C4—5381no—
383C4—5382no—
384C4—10761no—
385C4—10761no—
386C4—10761no—
387C4C2×D4×D26911yescharacteristic
388C4#2257549811yescharacteristic
389C2—41no—
390C2—42no—
391C2—10761no—
392C2#3177123711yescharacteristic
393C2#137×D26911yescharacteristic
394C2#3177141911yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#452×C269C2
socleC22×C269C22×D4
prime radical#31771254C2
hypercenter#452D269
hypocenterC269#3276
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#22575483C2semidirect
2. C2×#22575482C2semidirect
3. #31771268C2semidirect
5. #31770983C2semidirect
6. #31770862C2semidirect
7. #31770861C2semidirect
8. #31770814C2semidirect
9. #31770808C2semidirect
10. #31770768C2semidirect
11. #31770763C2semidirect
12. #31770759C2semidirect
13. #31770696C2semidirect
14. #31770680C2semidirect
15. #452×C269C2semidirect
85. #16864390D4semidirect
190. Dic269#137semidirect
197. C269#3276semidirect
209. #452D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754832
C2×#225754822
#317712682
#317712542
#317709832
#317708622
#317708612
#317708142
#317708082
#317707682
#317707632
#317707592
#317706962
#317706802
#452×C2692
#3276269