Finite Group Databasebeta

← #46824564

Subgroups — #46824564

Order 34432.

1522 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22575501C211yesmaximal, characteristic, complemented
2C2×#41×D269C211yesmaximal, characteristic, complemented
3Q8×D1076C211yesmaximal, characteristic, complemented
4Q8×C22⋊D269C211yesmaximal, characteristic, complemented
5#31771262C211yesmaximal, characteristic, complemented
6#31771260C213yesmaximal, complemented
7#31771257C213yesmaximal, complemented
8#31771252C213yesmaximal, complemented
9#31771248C213yesmaximal, complemented
10#31771184C213yesmaximal, complemented
11#31771175C213yesmaximal, complemented
12#31771114C213yesmaximal, complemented
13C22⋊Q8×D269C213yesmaximal, complemented
14#31771087C211yesmaximal, characteristic, complemented
15#548×C269C211yesmaximal, characteristic, complemented
16#22575501—28no—
17C2×#16864403C2211yescharacteristic
18C2×#16864403C2213yescomplemented
19C2×#16864402C2213yes—
20C2×#16864402C2213yescomplemented
21C2×#16864402C2213yescomplemented
22C2×#16864401C2213yes—
23#41×D269—28no—
24C2×Q8×D269C2211yescharacteristic
25C2×Q8×D269C2213yescomplemented
26C2×Q8×D269C2211yescharacteristic, complemented
27C2×D4×D269C2213yes—
28C22×C4×D269C2213yes—
29#22575453C2213yescomplemented
30#22575451C2213yescomplemented
31#22575440C2213yescomplemented
32#22575438C2213yescomplemented
33#22575435C2216yescomplemented
34#22575435C2216yescomplemented
35#22575435C2213yescomplemented
36#22575433C2213yes—
37#22575433C2213yescomplemented
38#22575432C2213yescomplemented
39#22575432C2213yescomplemented
40#22575431C2213yes—
41#22575430C2213yescomplemented
42#22575430C2216yescomplemented
43#22575429C2213yes—
44#22575426C2211yescharacteristic
45#22575426C2216yescomplemented
46#22575425C2213yescomplemented
47#22575424C2213yescomplemented
48#22575424C2216yescomplemented
49#22575423C2213yes—
50#22575421C2213yes—
51Q8×Dic269C2211yescharacteristic, complemented
52C4×D1076C2213yes—
53C4×D1076C2213yescomplemented
54#22575415C2213yescomplemented
55C4×C22⋊D269C2213yes—
56C4×C22⋊D269C2213yescomplemented
57#22575410C2213yescomplemented
58C4⋊C4×D269C2213yescomplemented
59C4⋊C4×D269C2216yescomplemented
60C22⋊C4×D269C2216yes—
61C2×#41×C269C2211yescharacteristic
62C22×Q8×C269C2211yescharacteristic, complemented
63#124×C269C2213yescomplemented
64C22⋊Q8×C269C2213yescomplemented
65#121×C269C2213yescomplemented
66C4×Q8×C269C2211yescharacteristic, complemented
67C4×D4×C269C2213yescomplemented
68C23×D269C2313yes—
69#16864403—212no—
70#16864403D418yescomplemented
71#16864402—212no—
72#16864402—212no—
73#16864402—212no—
74#16864401—212no—
75C2×Dic538C2313yes—
76C2×Dic538C2313yes—
77C2×Dic538C2313yes—
78Q8×D269—212no—
79Q8×D269—24no—
80Q8×D269—24no—
81C2×D1076C2311yescharacteristic
82C2×D1076C2313yes—
83C2×D1076C2313yes—
84C2×C22⋊D269C2311yescharacteristic
85C2×C22⋊D269C2313yes—
86C2×C22⋊D269C2316yes—
87C2×C22⋊D269C2311yescharacteristic
88C2×C22⋊D269C2313yes—
89D4×D269—212no—
90C22×Dic269C2313yes—
91C2×C4×D269—212no—
92C2×C4×D269C2316yes—
93C2×C4×D269C2313yes—
94C2×C4×D269C2313yes—
95C2×C4×D269C2313yes—
96C2×C4×D269C2311yescharacteristic
97C2×C4×D269C2316yes—
98C2×C4×D269C2313yes—
99C4⋊Dic269C2313yes—
100C4⋊Dic269C2316yes—
101C4⋊Dic269C2313yes—
102#16864381C2313yes—
103#16864381C2313yes—
104#16864381C2316yes—
105#16864381C2311yescharacteristic
106#16864381C2313yes—
107C22⋊Dic269C2316yes—
108C22⋊Dic269C2313yes—
109C22⋊Dic269C2313yes—
110#16864379C2313yes—
111#16864379C2313yes—
112#16864379C2313yes—
113#16864379C2313yes—
114#16864379C2311yescharacteristic
115#16864379C2313yes—
116C4×Dic269C2313yes—
117#41×C269—24no—
118C2×Q8×C269—24no—
119C2×Q8×C269C2313yes—
120C2×Q8×C269C2311yescharacteristic
121C2×D4×C269C2313yes—
122C2×D4×C269C2313yes—
123C22×C4×C269C2313yes—
124C22×C4×C269C2313yes—
125C4⋊C4×C269C2313yes—
126C4⋊C4×C269C2313yes—
127C22⋊C4×C269C2316yes—
128C22⋊C4×C269C2313yes—
129C22⋊C4×C269C2311yescharacteristic
130C42×C269C2313yes—
131C22×D269—26no—
132C22×D269—26no—
133C22×D269—43no—
134C22×D269C2411yescharacteristic
135C22×D269C2413yes—
136C22×D269C2411yescharacteristic
137Dic538—26no—
138Dic538—26no—
139Dic538—26no—
140D1076—26no—
141D1076—22no—
142D1076C2×D4112yes—
143C22⋊D269—212no—
144C22⋊D269—26no—
145C22⋊D269—22no—
146C22⋊D269—22no—
147C22⋊D269—43no—
148C2×Dic269—26no—
149C2×Dic269C2411yescharacteristic
150C2×Dic269C2413yes—
151C2×Dic269C2413yes—
152C2×Dic269C2413yes—
153C2×Dic269C2411yescharacteristic
154C4×D269—26no—
155C4×D269—212no—
156C4×D269—26no—
157C4×D269—26no—
158C4×D269—22no—
159C4×D269—46no—
160C4×D269C2×D4112yes—
161C23×C269C2413yes—
162C23×C269C2411yescharacteristic
163Q8×C269—26no—
164Q8×C269C2×D414yes—
165D4×C269—26no—
166D4×C269—43no—
167C2×C4×C269—26no—
168C2×C4×C269—26no—
169C2×C4×C269C2411yescharacteristic
170C2×C4×C269C2413yes—
171C2×C4×C269C2413yes—
172C2×C4×C269C2411yescharacteristic
173C2×C4×C269C2413yes—
174C22×C269—21no—
175C22×C269—23no—
176C22×C269—21no—
177C22×C269—23no—
178C22×C269—21no—
179C22×C269—23no—
180C22×C269C2511yescharacteristic
181C2×D269—21no—
182C2×D269—23no—
183C2×D269—23no—
184C2×D269—21no—
185C2×D269—21no—
186C2×D269—21no—
187C2×D269—21no—
188C2×D269—412no—
189C2×D269C22×D416yes—
190C2×D2692−1+412yescomplemented
191C4×C269—21no—
192C4×C269—23no—
193C4×C269—21no—
194C4×C269—23no—
195C4×C269C22×D416yes—
196Dic269—23no—
197Dic269—23no—
198Dic269—21no—
199Dic269—23no—
200Dic269C22×D412yes—
201C2×C269—43no—
202C2×C269—41no—
203C2×C269C23×D411yescharacteristic
204C2×C269C2×2−1+411yescharacteristic
205C2×C269#58511yescharacteristic
206D269—22no—
207D269—26no—
208D269—41no—
209C269#348911yescharacteristic, complemented, Sylow
210#3489—2691nomaximal, Sylow
211C2×2−1+4—2691no—
212C22×#41—2691no—
213D4×Q8—2691no—
214D4×Q8—2691no—
215#548—2691no—
216#548D26911yescharacteristic, complemented
217#547—2693no—
218#547—2693no—
219#546—2693no—
220#546—2693no—
221#537—2693no—
222#536—2693no—
223#525—2693no—
224C2×C22⋊Q8—2693no—
225#520—2691no—
2262−1+4—5388no—
227C2×#41—2691no—
228C2×#41—2693no—
229C2×#41—2693no—
230C2×#41—2693no—
231C2×#41—2693no—
232C2×#41—2693no—
233C2×#41—5388no—
234C2×#41C2×D26911yescharacteristic
235C22×Q8—2693no—
236C22×Q8—2691no—
237C22×Q8—2691no—
238C22×Q8C2×D26911yescharacteristic, complemented
239C22×D4—2693no—
240C23×C4—2693no—
241C4⋊Q8—2693no—
242C4⋊Q8—2693no—
243#124—2693no—
244#124C2×D26913yescomplemented
245#123—2693no—
246#123—2693no—
247#123—2696no—
248#123—2696no—
249C22⋊Q8—2693no—
250C22⋊Q8—2693no—
251C22⋊Q8—2693no—
252C22⋊Q8—2693no—
253C22⋊Q8—2693no—
254C22⋊Q8—2693no—
255C22⋊Q8—2696no—
256C22⋊Q8—2693no—
257C22⋊Q8C2×D26913yescomplemented
258#121—2693no—
259#121—2696no—
260#121—2693no—
261#121—2691no—
262#121—2696no—
263#121C2×D26913yescomplemented
264#120—2693no—
265#120—2693no—
266C4×Q8—2691no—
267C4×Q8C2×D26911yescharacteristic, complemented
268C4×D4—2693no—
269C4×D4—2693no—
270C4×D4—2693no—
271C4×D4—2693no—
272C4×D4—2693no—
273C4×D4C2×D26913yescomplemented
274#117—2693no—
275C2×C4⋊C4—2693no—
276C2×C4⋊C4—2696no—
277C2×C22⋊C4—2696no—
278C24—2693no—
279#41—24no—
280#41—2698no—
281#41—53812no—
282#41—53812no—
283#41—53812no—
284#41—53812no—
285#41—53812no—
286C2×Q8—24no—
287C2×Q8—2693no—
288C2×Q8—2693no—
289C2×Q8—2693no—
290C2×Q8—53812no—
291C2×Q8—5384no—
292C2×Q8—5384no—
293C2×Q8C22×D26913yes—
294C2×Q8C22×D26911yescharacteristic
295C2×D4—2691no—
296C2×D4—2693no—
297C2×D4—2691no—
298C2×D4—2696no—
299C2×D4—2693no—
300C2×D4—2693no—
301C2×D4—2691no—
302C2×D4—2693no—
303C2×D4—53812no—
304C2×D4C22×D26913yes—
305C2×D4C22×D26913yes—
306C22×C4—2693no—
307C22×C4—2693no—
308C22×C4—2696no—
309C22×C4—2693no—
310C22×C4—2693no—
311C22×C4—2693no—
312C22×C4—2691no—
313C22×C4—2696no—
314C22×C4—53812no—
315C22×C4C22×D26913yes—
316C22×C4C22×D26913yes—
317C4⋊C4—2693no—
318C4⋊C4—2693no—
319C4⋊C4—2693no—
320C4⋊C4—2696no—
321C4⋊C4—2696no—
322C4⋊C4—2691no—
323C4⋊C4—2693no—
324C4⋊C4—2693no—
325C4⋊C4C22×D26913yes—
326C4⋊C4C22×D26913yes—
327C22⋊C4—2696no—
328C22⋊C4—2693no—
329C22⋊C4—2693no—
330C22⋊C4—2693no—
331C22⋊C4—2693no—
332C22⋊C4—2693no—
333C22⋊C4—2693no—
334C22⋊C4—2691no—
335C22⋊C4—2693no—
336C22⋊C4C22×D26916yes—
337C22⋊C4C22×D26911yescharacteristic
338C22⋊C4C22×D26913yes—
339C42—2693no—
340C42C22×D26913yes—
341C23—2691no—
342C23—2693no—
343C23—2691no—
344C23—5386no—
345C23—5386no—
346C23—10763no—
347C23C23×D26913yes—
348C23C23×D26911yescharacteristic
349Q8—26no—
350Q8—5386no—
351Q8—5386no—
352Q8—5386no—
353Q8D4×D26914yes—
354D4—26no—
355D4—43no—
356D4—26912no—
357D4—5386no—
358D4—5382no—
359D4—5382no—
360D4—53812no—
361D4—5386no—
362D4—5382no—
363D4—10763no—
364C2×C4—26no—
365C2×C4—26no—
366C2×C4—2691no—
367C2×C4—26912no—
368C2×C4—2691no—
369C2×C4—2693no—
370C2×C4—2693no—
371C2×C4—2693no—
372C2×C4—5382no—
373C2×C4—5386no—
374C2×C4—53812no—
375C2×C4—5386no—
376C2×C4—5386no—
377C2×C4—5386no—
378C2×C4—10766no—
379C2×C4C23×D26911yescharacteristic
380C2×C4C23×D26913yes—
381C2×C4C23×D26913yes—
382C2×C4C23×D26911yescharacteristic
383C2×C4C23×D26913yes—
384C22—21no—
385C22—23no—
386C22—21no—
387C22—23no—
388C22—21no—
389C22—23no—
390C22—2696no—
391C22—2692no—
392C22—5381no—
393C22—5383no—
394C22—5383no—
395C22—5381no—
396C22—5381no—
397C22—5381no—
398C22—5381no—
399C22—107612no—
400C22C24×D26911yescharacteristic
401C4—21no—
402C4—23no—
403C4—21no—
404C4—23no—
405C4—2692no—
406C4—5383no—
407C4—5383no—
408C4—5381no—
409C4—5383no—
410C4C2×D4×D26916yes—
411C2—43no—
412C2—41no—
413C2—5382no—
414C2—5386no—
415C2—10761no—
416C2C22×D4×D26911yescharacteristic
417C22−1+4×D26911yescharacteristic
418C2#3177144111yescharacteristic
subgroupG/H
center, FrattiniC22C24×D269
commutator, socle, cosocleC22×C269C25
Fitting#548×C269C2
hypercenter#548D269
hypocenterC269#3489
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#22575501C2semidirect
2. C2×#41×D269C2semidirect
3. Q8×D1076C2semidirect
4. Q8×C22⋊D269C2semidirect
5. #31771262C2semidirect
6. #31771260C2semidirect
7. #31771257C2semidirect
8. #31771252C2semidirect
9. #31771248C2semidirect
10. #31771184C2semidirect
11. #31771175C2semidirect
12. #31771114C2semidirect
13. C22⋊Q8×D269C2semidirect
14. #31771087C2semidirect
15. #548×C269C2semidirect
18. C2×#16864403C22semidirect
20. C2×#16864402C22semidirect
21. C2×#16864402C22semidirect
25. C2×Q8×D269C22semidirect
26. C2×Q8×D269C22semidirect
29. #22575453C22semidirect
30. #22575451C22semidirect
31. #22575440C22semidirect
32. #22575438C22semidirect
33. #22575435C22semidirect
34. #22575435C22semidirect
35. #22575435C22semidirect
37. #22575433C22semidirect
38. #22575432C22semidirect
39. #22575432C22semidirect
41. #22575430C22semidirect
42. #22575430C22semidirect
45. #22575426C22semidirect
46. #22575425C22semidirect
47. #22575424C22semidirect
48. #22575424C22semidirect
51. Q8×Dic269C22semidirect
53. C4×D1076C22semidirect
54. #22575415C22semidirect
56. C4×C22⋊D269C22semidirect
57. #22575410C22semidirect
58. C4⋊C4×D269C22semidirect
59. C4⋊C4×D269C22semidirect
62. C22×Q8×C269C22semidirect
63. #124×C269C22semidirect
64. C22⋊Q8×C269C22semidirect
65. #121×C269C22semidirect
66. C4×Q8×C269C22semidirect
67. C4×D4×C269C22semidirect
70. #16864403D4semidirect
190. C2×D2692−1+4semidirect
209. C269#3489semidirect
216. #548D269semidirect
238. C22×Q8C2×D269semidirect
244. #124C2×D269semidirect
257. C22⋊Q8C2×D269semidirect
263. #121C2×D269semidirect
267. C4×Q8C2×D269semidirect
273. C4×D4C2×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225755012
C2×#41×D2692
Q8×D10762
Q8×C22⋊D2692
#317712622
#317712602
#317712572
#317712522
#317712482
#317711842
#317711752
#317711142
C22⋊Q8×D2692
#317710872
#548×C2692
#3489269