Finite Group Databasebeta

← #46822395

Subgroups — #46822395

Order 34432.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22575482C211yesmaximal, characteristic, complemented
2Q8×C22⋊D269C211yesmaximal, characteristic, complemented
3#31771249C211yesmaximal, characteristic
4#31770996C211yesmaximal, characteristic, complemented
5#31770848C211yesmaximal, characteristic, complemented
6#31770846C211yesmaximal, characteristic, complemented
7#31770825C211yesmaximal, characteristic, complemented
8#31770808C211yesmaximal, characteristic, complemented
9#31770764C211yesmaximal, characteristic, complemented
10#31770761C211yesmaximal, characteristic, complemented
11#31770756C211yesmaximal, characteristic, complemented
12#31770742C211yesmaximal, characteristic, complemented
13#31770706C211yesmaximal, characteristic, complemented
14#31770676C211yesmaximal, characteristic, complemented
15C22×SD16×C269C211yesmaximal, characteristic, complemented
16C2×#16864401C2211yescharacteristic
17C2×Q8×D269C2211yescharacteristic, complemented
18#22575482—24no—
19C2×Dic1076C2211yescharacteristic, complemented
20C2×#16864392C2211yescharacteristic, complemented
21C2×#16864392C2211yescharacteristic, complemented
22C2×#16864390C2211yescharacteristic
23C2×#16864390C2211yescharacteristic
24C2×#16864385C2211yescharacteristic, complemented
25#22575451—21no—
26#22575451C2211yescharacteristic, complemented
27#22575440C2211yescharacteristic
28#22575434—21no—
29#22575432—21no—
30#22575432C2211yescharacteristic
31#22575432C2211yescharacteristic
32#22575429—21no—
33#22575429C2211yescharacteristic, complemented
34#22575425—21no—
35#22575424C2211yescharacteristic
36#22575422—21no—
37Q8×Dic269C2211yescharacteristic, complemented
38D4×Dic269C2211yescharacteristic
39C4×C22⋊D269—21no—
40C4×C22⋊D269C2211yescharacteristic
41C2×C22⋊Dic269—21no—
42#22575387C2211yescharacteristic, complemented
43#22575382C2211yescharacteristic, complemented
44#22575378C2211yescharacteristic, complemented
45#22575378C2211yescharacteristic, complemented
46#22575377C2211yescharacteristic, complemented
47#22575377C2211yescharacteristic, complemented
48#22575374C2211yescharacteristic
49#22575374C2211yescharacteristic
50C22⋊#14524130C2211yescharacteristic
51#22575365C2211yescharacteristic, complemented
52#22575363C2211yescharacteristic, complemented
53C22×Q8×C269C2211yescharacteristic
54C22×D4×C269C2211yescharacteristic
55C2×SD16×C269—24no—
56C2×SD16×C269C2211yescharacteristic
57C2×SD16×C269C2211yescharacteristic
58C2×SD16×C269C2211yescharacteristic
59C2×SD16×C269C2211yescharacteristic
60C22×C8×C269C2211yescharacteristic
61#16864401—22no—
62#16864401—41no—
63C2×Dic538—21no—
64C2×Dic538C2311yescharacteristic
65C2×Dic538C2311yescharacteristic
66Q8×D269—22no—
67Q8×D269—41no—
68C2×C22⋊D269—21no—
69C2×C22⋊D269—21no—
70C2×C22⋊D269D411yescharacteristic
71C22×Dic269—21no—
72C22×Dic269—21no—
73C2×C4×D269—21no—
74C2×C4×D269C2311yescharacteristic
75Dic1076—22no—
76#16864392—22no—
77#16864392—41no—
78#16864390—22no—
79#16864390—41no—
80#16864385—22no—
81C2×#14524130C2311yescharacteristic
82C2×#14524130C2311yescharacteristic
83C4⋊Dic269—21no—
84C4⋊Dic269C2311yescharacteristic
85C4⋊Dic269C2311yescharacteristic
86#16864381—21no—
87#16864381—21no—
88#16864381—21no—
89#16864381—21no—
90#16864381—21no—
91#16864381D411yescharacteristic
92C22⋊Dic269—21no—
93C22⋊Dic269—21no—
94C22⋊Dic269—21no—
95C22⋊Dic269—21no—
96C22⋊Dic269—21no—
97C22⋊Dic269—21no—
98C22⋊Dic269D411yescharacteristic
99#16864379—21no—
100#16864379D411yescharacteristic
101C4×Dic269—21no—
102C4×Dic269C2311yescharacteristic
103C24×C269—21no—
104C2×Q8×C269—21no—
105C2×Q8×C269—22no—
106C2×Q8×C269—41no—
107C2×Q8×C269C2311yescharacteristic
108C2×Q8×C269C2311yescharacteristic
109C2×D4×C269—22no—
110C2×D4×C269—21no—
111C2×D4×C269—41no—
112C2×D4×C269C2311yescharacteristic
113C2×D4×C269C2311yescharacteristic
114C22×C4×C269—21no—
115C22×C4×C269C2311yescharacteristic
116SD16×C269—22no—
117SD16×C269—22no—
118SD16×C269—22no—
119SD16×C269D414yescomplemented
120C2×C8×C269—22no—
121C2×C8×C269C2311yescharacteristic
122C2×C8×C269C2311yescharacteristic
123C22×D269C2×D411yescharacteristic
124Dic538—21no—
125Dic538—21no—
126Dic538—21no—
127Dic538—21no—
128Dic538—42no—
129C22⋊D269—22no—
130C22⋊D269—42no—
131C22⋊D269—81no—
132C2×Dic269—21no—
133C2×Dic269—21no—
134C2×Dic269—21no—
135C2×Dic269—21no—
136C2×Dic269—42no—
137C2×Dic269—81no—
138C2×Dic269C2×D411yescharacteristic
139C2×Dic269C2×D411yescharacteristic
140C2×Dic269C2×D411yescharacteristic
141C4×D269—21no—
142C4×D269—21no—
143C4×D269—42no—
144#14524130—21no—
145#14524130—21no—
146C23×C269—21no—
147C23×C269—21no—
148C23×C269—42no—
149C23×C269—42no—
150C23×C269—81no—
151C23×C269C2×D411yescharacteristic
152Q8×C269—21no—
153Q8×C269—21no—
154Q8×C269—21no—
155Q8×C269—42no—
156Q8×C269C2×D412yes—
157D4×C269—21no—
158D4×C269—21no—
159D4×C269—21no—
160D4×C269—42no—
161D4×C269C2×D412yes—
162C2×C4×C269—21no—
163C2×C4×C269—21no—
164C2×C4×C269—21no—
165C2×C4×C269—21no—
166C2×C4×C269—42no—
167C2×C4×C269C2×D411yescharacteristic
168C2×C4×C269C2411yescharacteristic
169C8×C269—21no—
170C8×C269C2×D412yes—
171C22×C269—21no—
172C22×C269—21no—
173C22×C269—22no—
174C22×C269—41no—
175C22×C269—41no—
176C22×C269—41no—
177C22×C269—41no—
178C22×C269—41no—
179C22×C269—84no—
180C22×C269#13712yescomplemented
181C22×C269C22×D411yescharacteristic
182C2×D269—21no—
183C2×D269—21no—
184C2×D269—21no—
185C4×C269—21no—
186C4×C269—22no—
187C4×C269—41no—
188C4×C269C22×D411yescharacteristic
189C4×C269H(5,𝔽2)11yescharacteristic
190Dic269—21no—
191Dic269—21no—
192Dic269—21no—
193Dic269—41no—
194Dic269—41no—
195Dic269—41no—
196Dic269—41no—
197C2×C269—22no—
198C2×C269—42no—
199C2×C269—81no—
200C2×C269D4211yescharacteristic
201C2×C269C2×#13711yescharacteristic
202C2×C269#57711yescharacteristic
203D269—41no—
204C269#326611yescharacteristic, complemented, Sylow
205#3266—2691nomaximal, Sylow
206C2×#137—2691no—
207C22×SD16D26911yescharacteristic, complemented
208D4×Q8—2691no—
209#546—2691no—
210#497—2691no—
211#468—2691no—
212#467—2691no—
213#464—2691no—
214#461—2691no—
215#452—2691no—
216#451—2691no—
217#450—2691no—
218#448—2691no—
219#440—2691no—
220#435—2691no—
221C2×#41—2691no—
222C22×Q8—2691no—
223C22×Q8C2×D26911yescharacteristic
224C22×D4C2×D26911yescharacteristic
225#137—5384no—
226C2×Q16—2691no—
227C2×Q16—2691no—
228C2×Q16—2691no—
229C2×SD16—24no—
230C2×SD16—2691no—
231C2×SD16—2691no—
232C2×SD16C2×D26911yescharacteristic
233C2×SD16C2×D26911yescharacteristic
234C2×SD16C2×D26911yescharacteristic
235C2×SD16C2×D26911yescharacteristic
236C2×M16—2691no—
237C22×C8C2×D26911yescharacteristic
238C4⋊Q8—2691no—
239C4⋊Q8—5381no—
240#124—2691no—
241#123—5381no—
242C22⋊Q8—2691no—
243C22⋊Q8—2691no—
244C22⋊Q8—2691no—
245C22⋊Q8—5381no—
246C22⋊Q8—5381no—
247#121—2691no—
248#121—5381no—
249#120—5381no—
250C4×Q8—2691no—
251C4×D4—2691no—
252C4×D4—2691no—
253C4×D4—5381no—
254C2×C22⋊C4—5381no—
255#107—2691no—
256C4⋊C8—2691no—
257#103—2691no—
258#103—2691no—
259#103—2691no—
260#103—2691no—
261#102—2691no—
262#102—2691no—
263C22⋊C8—2691no—
264C22⋊C8—2691no—
265#97—2691no—
266C24—21no—
267#41—5382no—
268#41—10761no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—2691no—
273C2×Q8—2691no—
274C2×Q8—5381no—
275C2×Q8—5382no—
276C2×Q8—10761no—
277C2×Q8C22×D26911yescharacteristic
278C2×Q8C22×D26911yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—2691no—
283C2×D4—5381no—
284C2×D4—5381no—
285C2×D4C22×D26911yescharacteristic
286C2×D4C22×D26911yescharacteristic
287C22×C4—21no—
288C22×C4—2691no—
289C22×C4—5381no—
290C22×C4—5381no—
291C22×C4—5381no—
292C22×C4C22×D26911yescharacteristic
293Q16—5382no—
294Q16—5382no—
295Q16—10761no—
296SD16—22no—
297SD16—22no—
298SD16—22no—
299SD16—5382no—
300SD16—10761no—
301SD16C22⋊D26914yescomplemented
302M16—5382no—
303C2×C8—22no—
304C2×C8—2691no—
305C2×C8—2691no—
306C2×C8C22×D26911yescharacteristic
307C2×C8C22×D26911yescharacteristic
308C4⋊C4—2691no—
309C4⋊C4—2691no—
310C4⋊C4—2691no—
311C4⋊C4—5381no—
312C4⋊C4—5381no—
313C4⋊C4—5381no—
314C4⋊C4—5381no—
315C4⋊C4—5381no—
316C4⋊C4—5381no—
317C22⋊C4—2691no—
318C22⋊C4—2691no—
319C22⋊C4—5381no—
320C22⋊C4—5381no—
321C22⋊C4—5381no—
322C22⋊C4—5381no—
323C22⋊C4—5381no—
324C22⋊C4—5381no—
325C22⋊C4—5381no—
326C42—2691no—
327C42—5381no—
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23—2691no—
334C23D4×D26911yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—42no—
339Q8—5381no—
340Q8—5381no—
341Q8—5381no—
342Q8—5381no—
343Q8—10762no—
344Q8C2×C22⋊D26912yes—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—5382no—
350D4—10762no—
351D4—21521no—
352D4C2×C22⋊D26912yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—42no—
358C2×C4—2691no—
359C2×C4—2691no—
360C2×C4—2691no—
361C2×C4—5381no—
362C2×C4—5381no—
363C2×C4—5381no—
364C2×C4—5381no—
365C2×C4—5381no—
366C2×C4—5381no—
367C2×C4—10762no—
368C2×C4—10762no—
369C2×C4—21521no—
370C2×C4D4×D26911yescharacteristic
371C2×C4C23×D26911yescharacteristic
372C8—21no—
373C8—5381no—
374C8—5381no—
375C8C2×C22⋊D26912yes—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—84no—
385C22—5381no—
386C22—5381no—
387C22—5381no—
388C22#2257548212yescomplemented
389C22C2×D4×D26911yescharacteristic
390C4—21no—
391C4—22no—
392C4—41no—
393C4—5381no—
394C4—5381no—
395C4—5381no—
396C4—10761no—
397C4—10761no—
398C4—10761no—
399C4—10761no—
400C4C22×C22⋊D26911yescharacteristic
401C4#2257549811yescharacteristic
402C2—22no—
403C2—42no—
404C2—81no—
405C2—10761no—
406C2D4×C22⋊D26911yescharacteristic
407C2C2×#2257548211yescharacteristic
408C2#3177141111yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC22×SD16×C269C2
socleC22×C269C22×D4
prime radical#31771249C2
hypercenterC22×SD16D269
hypocenterC269#3266
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#22575482C2semidirect
2. Q8×C22⋊D269C2semidirect
4. #31770996C2semidirect
5. #31770848C2semidirect
6. #31770846C2semidirect
7. #31770825C2semidirect
8. #31770808C2semidirect
9. #31770764C2semidirect
10. #31770761C2semidirect
11. #31770756C2semidirect
12. #31770742C2semidirect
13. #31770706C2semidirect
14. #31770676C2semidirect
15. C22×SD16×C269C2semidirect
17. C2×Q8×D269C22semidirect
19. C2×Dic1076C22semidirect
20. C2×#16864392C22semidirect
21. C2×#16864392C22semidirect
24. C2×#16864385C22semidirect
26. #22575451C22semidirect
33. #22575429C22semidirect
37. Q8×Dic269C22semidirect
42. #22575387C22semidirect
43. #22575382C22semidirect
44. #22575378C22semidirect
45. #22575378C22semidirect
46. #22575377C22semidirect
47. #22575377C22semidirect
51. #22575365C22semidirect
52. #22575363C22semidirect
119. SD16×C269D4semidirect
180. C22×C269#137semidirect
204. C269#3266semidirect
207. C22×SD16D269semidirect
301. SD16C22⋊D269semidirect
388. C22#22575482semidirect

Maximal subgroups

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

subgroupindex
C2×#225754822
Q8×C22⋊D2692
#317712492
#317709962
#317708482
#317708462
#317708252
#317708082
#317707642
#317707612
#317707562
#317707422
#317707062
#317706762
C22×SD16×C2692
#3266269