Finite Group Databasebeta

← #46822388

Subgroups — #46822388

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×#22575485C211yesmaximal, characteristic, complemented
2C22×#16864389C211yesmaximal, characteristic, complemented
3Q8×C22⋊D269C211yesmaximal, characteristic, complemented
4#31771246C211yesmaximal, characteristic
5#31770998C211yesmaximal, characteristic, complemented
6#31770854C211yesmaximal, characteristic, complemented
7#31770844C211yesmaximal, characteristic, complemented
8#31770829C211yesmaximal, characteristic, complemented
9#31770809C211yesmaximal, characteristic, complemented
10#31770766C211yesmaximal, characteristic, complemented
11#31770758C211yesmaximal, characteristic, complemented
12#31770745C211yesmaximal, characteristic, complemented
13#31770708C211yesmaximal, characteristic, complemented
14#31770673C211yesmaximal, characteristic, complemented
15#451×C269C211yesmaximal, characteristic, complemented
16C2×#16864403C2211yescharacteristic
17C2×Q8×D269C2211yescharacteristic, complemented
18C22×D1076C2211yescharacteristic
19#22575485—24no—
20C2×#16864392C2211yescharacteristic, complemented
21C2×#16864391C2211yescharacteristic
22C2×#16864391C2211yescharacteristic
23C2×#16864389—24no—
24C2×#16864389C2211yescharacteristic
25C2×#16864389C2211yescharacteristic
26C2×#16864389C2211yescharacteristic
27C2×#16864389C2211yescharacteristic
28C2×#16864385C2211yescharacteristic, complemented
29C22×#14524130C2211yescharacteristic
30#22575451—21no—
31#22575451C2211yescharacteristic, complemented
32#22575442C2211yescharacteristic
33#22575438—21no—
34#22575432—21no—
35#22575432C2211yescharacteristic
36#22575429—21no—
37#22575429C2211yescharacteristic, complemented
38#22575428—21no—
39#22575428C2211yescharacteristic
40#22575423—21no—
41Q8×Dic269C2211yescharacteristic, complemented
42#22575415C2211yescharacteristic
43C4×C22⋊D269—21no—
44C4×C22⋊D269C2211yescharacteristic
45C22⋊C4×D269—21no—
46#22575386C2211yescharacteristic, complemented
47#22575382C2211yescharacteristic, complemented
48#22575377C2211yescharacteristic, complemented
49#22575376C2211yescharacteristic, complemented
50#22575375C2211yescharacteristic
51#22575375C2211yescharacteristic
52#22575365C2211yescharacteristic, complemented
53#22575363C2211yescharacteristic, complemented
54C22×Q8×C269C2211yescharacteristic
55C2×Q16×C269C2211yescharacteristic, complemented
56C2×Q16×C269C2211yescharacteristic, complemented
57C22⋊Q8×C269C2211yescharacteristic
58#103×C269C2211yescharacteristic, complemented
59#103×C269C2211yescharacteristic, complemented
60C22⋊C8×C269C2211yescharacteristic
61C23×D269—21no—
62#16864403—22no—
63#16864403—41no—
64C2×Dic538—21no—
65C2×Dic538C2311yescharacteristic
66Q8×D269—22no—
67Q8×D269—41no—
68C2×D1076—22no—
69C2×D1076—21no—
70C2×D1076—21no—
71C2×D1076—21no—
72C2×D1076—41no—
73C2×D1076C2311yescharacteristic
74C2×D1076C2311yescharacteristic
75C2×C22⋊D269D411yescharacteristic
76C2×C4×D269—21no—
77C2×C4×D269—21no—
78C2×C4×D269—21no—
79C2×C4×D269C2311yescharacteristic
80#16864392—22no—
81#16864391—22no—
82#16864391—41no—
83#16864389—22no—
84#16864389—22no—
85#16864389—22no—
86#16864389D414yescomplemented
87#16864385—22no—
88C2×#14524130—22no—
89C2×#14524130C2311yescharacteristic
90C2×#14524130C2311yescharacteristic
91C4⋊Dic269—21no—
92C4⋊Dic269C2311yescharacteristic
93#16864381—21no—
94#16864381—21no—
95#16864381—21no—
96#16864381—21no—
97#16864381D411yescharacteristic
98C22⋊Dic269—21no—
99C22⋊Dic269D411yescharacteristic
100#16864379—21no—
101#16864379—21no—
102#16864379—21no—
103#16864379—21no—
104#16864379—21no—
105#16864379D411yescharacteristic
106C4×Dic269—21no—
107C4×Dic269C2311yescharacteristic
108C2×Q8×C269—21no—
109C2×Q8×C269—22no—
110C2×Q8×C269—41no—
111C2×Q8×C269C2311yescharacteristic
112C2×Q8×C269C2311yescharacteristic
113C2×Q8×C269C2311yescharacteristic
114C22×C4×C269—21no—
115C22×C4×C269C2311yescharacteristic
116Q16×C269—22no—
117Q16×C269—41no—
118C2×C8×C269C2311yescharacteristic
119C2×C8×C269C2311yescharacteristic
120C4⋊C4×C269—21no—
121C4⋊C4×C269C2311yescharacteristic
122C22⋊C4×C269—21no—
123C22×D269—21no—
124C22×D269—21no—
125C22×D269—42no—
126C22×D269—42no—
127C22×D269—81no—
128C22×D269C2×D411yescharacteristic
129Dic538—21no—
130Dic538—21no—
131Dic538—42no—
132D1076—21no—
133D1076—21no—
134D1076—21no—
135D1076—42no—
136D1076—42no—
137D1076—81no—
138D1076C2×D412yes—
139C22⋊D269—22no—
140C2×Dic269—21no—
141C2×Dic269—21no—
142C2×Dic269C2×D411yescharacteristic
143C2×Dic269C2×D411yescharacteristic
144C2×Dic269C2×D411yescharacteristic
145C4×D269—21no—
146C4×D269—21no—
147C4×D269—42no—
148C4×D269—42no—
149C4×D269—81no—
150#14524130—21no—
151#14524130C2×D412yes—
152C23×C269C2×D411yescharacteristic
153Q8×C269—21no—
154Q8×C269—21no—
155Q8×C269—21no—
156Q8×C269—21no—
157Q8×C269—21no—
158Q8×C269—42no—
159Q8×C269C2×D412yes—
160C2×C4×C269—21no—
161C2×C4×C269—21no—
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×C269—21no—
171C22×C269—21no—
172C22×C269—21no—
173C22×C269#13712yescomplemented
174C22×C269C22×D411yescharacteristic
175C2×D269—21no—
176C2×D269—22no—
177C2×D269—21no—
178C2×D269—21no—
179C2×D269—41no—
180C2×D269—41no—
181C2×D269—41no—
182C2×D269—41no—
183C2×D269—41no—
184C2×D269—84no—
185C4×C269—21no—
186C4×C269—22no—
187C4×C269—41no—
188C4×C269—41no—
189C4×C269—41no—
190C4×C269C22×D411yescharacteristic
191C4×C269H(5,𝔽2)11yescharacteristic
192Dic269—21no—
193Dic269—21no—
194Dic269—21no—
195Dic269—41no—
196Dic269—41no—
197C2×C269—22no—
198C2×C269D4211yescharacteristic
199C2×C269C2×#13711yescharacteristic
200C2×C269#57711yescharacteristic
201D269—41no—
202D269—42no—
203D269—81no—
204C269#326611yescharacteristic, complemented, Sylow
205#3266—2691nomaximal, Sylow
206C2×#137—2691no—
207C22×SD16—2691no—
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#451D26911yescharacteristic, complemented
217#450—2691no—
218#448—2691no—
219#440—2691no—
220#435—2691no—
221C2×#41—2691no—
222C22×Q8—2691no—
223C22×Q8C2×D26911yescharacteristic
224C22×D4—2691no—
225#137—5384no—
226C2×Q16—2691no—
227C2×Q16C2×D26911yescharacteristic, complemented
228C2×Q16C2×D26911yescharacteristic, complemented
229C2×SD16—2691no—
230C2×SD16—2691no—
231C2×SD16—2691no—
232C2×SD16—2691no—
233C2×SD16—2691no—
234C2×SD16—2691no—
235C2×SD16—5384no—
236C2×M16—2691no—
237C22×C8—2691no—
238C4⋊Q8—2691no—
239C4⋊Q8—5381no—
240#124—2691no—
241#123—5381no—
242C22⋊Q8—2691no—
243C22⋊Q8—2691no—
244C22⋊Q8—5381no—
245C22⋊Q8—5381no—
246C22⋊Q8C2×D26911yescharacteristic
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#103C2×D26911yescharacteristic, complemented
260#103C2×D26911yescharacteristic, complemented
261#102—2691no—
262#102—2691no—
263C22⋊C8—2691no—
264C22⋊C8C2×D26911yescharacteristic
265#97—2691no—
266C24—5381no—
267#41—5382no—
268#41—10761no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—2691no—
273C2×Q8—5381no—
274C2×Q8—5382no—
275C2×Q8—10761no—
276C2×Q8C22×D26911yescharacteristic
277C2×Q8C22×D26911yescharacteristic
278C2×Q8C22×D26911yescharacteristic
279C2×D4—2691no—
280C2×D4—2691no—
281C2×D4—2691no—
282C2×D4—5382no—
283C2×D4—5381no—
284C2×D4—5381no—
285C2×D4—5381no—
286C2×D4—10761no—
287C22×C4—21no—
288C22×C4—2691no—
289C22×C4—5381no—
290C22×C4—5381no—
291C22×C4—5381no—
292C22×C4C22×D26911yescharacteristic
293Q16—22no—
294Q16—41no—
295Q16—5382no—
296SD16—2694no—
297SD16—5382no—
298SD16—5382no—
299SD16—5382no—
300SD16—5382no—
301SD16—10761no—
302M16—5382no—
303C2×C8—2691no—
304C2×C8—2691no—
305C2×C8—5382no—
306C2×C8C22×D26911yescharacteristic
307C2×C8C22×D26911yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—2691no—
310C4⋊C4—2691no—
311C4⋊C4—5381no—
312C4⋊C4—5381no—
313C4⋊C4—5381no—
314C4⋊C4—5381no—
315C4⋊C4—5381no—
316C4⋊C4C22×D26911yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—2691no—
319C22⋊C4—2691no—
320C22⋊C4—5381no—
321C22⋊C4—5381no—
322C22⋊C4—5381no—
323C22⋊C4—5381no—
324C22⋊C4—5381no—
325C22⋊C4—5381no—
326C42—2691no—
327C42—5381no—
328C23—2691no—
329C23—5381no—
330C23—5381no—
331C23—10762no—
332C23—10762no—
333C23—21521no—
334C23D4×D26911yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—5381no—
342Q8—5381no—
343Q8—10762no—
344Q8D4×D26912yes—
345D4—2692no—
346D4—5382no—
347D4—5381no—
348D4—5381no—
349D4—5381no—
350D4—10762no—
351D4—10762no—
352D4—21521no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—42no—
360C2×C4—2691no—
361C2×C4—2691no—
362C2×C4—2691no—
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—21no—
374C8—2692no—
375C8—5381no—
376C22—21no—
377C22—21no—
378C22—5381no—
379C22—5381no—
380C22—5381no—
381C22—5382no—
382C22—10761no—
383C22—10761no—
384C22—10761no—
385C22—10761no—
386C22—10761no—
387C22—21524no—
388C22#2257548512yescomplemented
389C22C2×D4×D26911yescharacteristic
390C4—21no—
391C4—22no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—5381no—
396C4—5381no—
397C4—5381no—
398C4—10761no—
399C4—10761no—
400C4C2×D4×D26911yescharacteristic
401C4#2257549811yescharacteristic
402C2—22no—
403C2—10761no—
404C2—10762no—
405C2—21521no—
406C2#3177123711yescharacteristic
407C2C2×#2257548511yescharacteristic
408C2#3177141411yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#451×C269C2
socleC22×C269C22×D4
prime radical#31771246C2
hypercenter#451D269
hypocenterC269#3266
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. C2×#22575485C2semidirect
2. C22×#16864389C2semidirect
3. Q8×C22⋊D269C2semidirect
5. #31770998C2semidirect
6. #31770854C2semidirect
7. #31770844C2semidirect
8. #31770829C2semidirect
9. #31770809C2semidirect
10. #31770766C2semidirect
11. #31770758C2semidirect
12. #31770745C2semidirect
13. #31770708C2semidirect
14. #31770673C2semidirect
15. #451×C269C2semidirect
17. C2×Q8×D269C22semidirect
20. C2×#16864392C22semidirect
28. C2×#16864385C22semidirect
31. #22575451C22semidirect
37. #22575429C22semidirect
41. Q8×Dic269C22semidirect
46. #22575386C22semidirect
47. #22575382C22semidirect
48. #22575377C22semidirect
49. #22575376C22semidirect
52. #22575365C22semidirect
53. #22575363C22semidirect
55. C2×Q16×C269C22semidirect
56. C2×Q16×C269C22semidirect
58. #103×C269C22semidirect
59. #103×C269C22semidirect
86. #16864389D4semidirect
173. C22×C269#137semidirect
204. C269#3266semidirect
216. #451D269semidirect
227. C2×Q16C2×D269semidirect
228. C2×Q16C2×D269semidirect
259. #103C2×D269semidirect
260. #103C2×D269semidirect
388. C22#22575485semidirect

Maximal subgroups

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

subgroupindex
C2×#225754852
C22×#168643892
Q8×C22⋊D2692
#317712462
#317709982
#317708542
#317708442
#317708292
#317708092
#317707662
#317707582
#317707452
#317707082
#317706732
#451×C2692
#3266269