Finite Group Databasebeta

← #49132772

Subgroups — #49132772

Order 36224.

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×#22583641C211yesmaximal, characteristic, complemented
2C22×#17985379C211yesmaximal, characteristic, complemented
3#31827111C211yesmaximal, characteristic
4#31826857C211yesmaximal, characteristic, complemented
5#31826713C211yesmaximal, characteristic, complemented
6#31826704C211yesmaximal, characteristic, complemented
7#31826687C211yesmaximal, characteristic, complemented
8#31826667C211yesmaximal, characteristic, complemented
9#31826625C211yesmaximal, characteristic, complemented
10#31826621C211yesmaximal, characteristic, complemented
11#31826615C211yesmaximal, characteristic, complemented
12#31826604C211yesmaximal, characteristic, complemented
13#31826563C211yesmaximal, characteristic, complemented
14#31826537C211yesmaximal, characteristic, complemented
15D4×Q8×C283C211yesmaximal, characteristic, complemented
16C2×#17985392C2211yescharacteristic
17C22×D1132C2211yescharacteristic
18#22583641—24no—
19C2×#17985382C2211yescharacteristic, complemented
20C2×#17985382C2211yescharacteristic, complemented
21C2×#17985382C2211yescharacteristic, complemented
22C2×#17985379—24no—
23C2×#17985379C2211yescharacteristic
24C2×#17985379C2211yescharacteristic
25C2×#17985379C2211yescharacteristic
26C2×#17985379C2211yescharacteristic
27C2×#17985379C2211yescharacteristic
28C2×#17985379C2211yescharacteristic
29C2×#17985376C2211yescharacteristic, complemented
30C22×C283⋊C8C2211yescharacteristic
31#22583598C2211yescharacteristic
32#22583595—21no—
33#22583589C2211yescharacteristic
34#22583584C2211yescharacteristic
35#22583583—21no—
36#22583578—21no—
37C4×D1132C2211yescharacteristic
38C22⋊C4×D283—21no—
39#22583543C2211yescharacteristic, complemented
40C4⋊(C283⋊C8)C2211yescharacteristic, complemented
41#22583534C2211yescharacteristic, complemented
42#22583534C2211yescharacteristic, complemented
43#22583534C2211yescharacteristic, complemented
44#22583533C2211yescharacteristic, complemented
45#22583530C2211yescharacteristic
46#22583530C2211yescharacteristic
47C22⋊(C283⋊C8)C2211yescharacteristic
48C22⋊(C283⋊C8)C2211yescharacteristic, complemented
49#22583521C2211yescharacteristic, complemented
50C22×Q8×C283C2211yescharacteristic
51C22×Q8×C283C2211yescharacteristic, complemented
52C4⋊Q8×C283—21no—
53C4⋊Q8×C283C2211yescharacteristic, complemented
54C22⋊Q8×C283—21no—
55C22⋊Q8×C283—21no—
56C22⋊Q8×C283C2211yescharacteristic
57C22⋊Q8×C283C2211yescharacteristic, complemented
58C4×Q8×C283C2211yescharacteristic, complemented
59C4×D4×C283—21no—
60C4×D4×C283C2211yescharacteristic
61C23×D283—21no—
62#17985392—22no—
63#17985392—41no—
64C2×Dic566C2311yescharacteristic
65C2×D1132—22no—
66C2×D1132—21no—
67C2×D1132—41no—
68C2×D1132C2311yescharacteristic
69C2×D1132C2311yescharacteristic
70C2×C22⋊D283—21no—
71C2×C22⋊D283—21no—
72C2×C4×D283—21no—
73C2×C4×D283—21no—
74#17985382—22no—
75#17985382—22no—
76#17985382—41no—
77#17985379—22no—
78#17985379—22no—
79#17985379—22no—
80#17985379—22no—
81#17985379—41no—
82#17985379D414yescomplemented
83#17985376—22no—
84C2×C283⋊C8—22no—
85C2×C283⋊C8C2311yescharacteristic
86C2×C283⋊C8C2311yescharacteristic
87C2×C283⋊C8C2311yescharacteristic
88C2×C283⋊C8C2311yescharacteristic
89C4⋊Dic283C2311yescharacteristic
90Dic283⋊C4—21no—
91C22⋊Dic283—21no—
92(C2×D283)⋊C4—21no—
93(C2×D283)⋊C4—21no—
94(C2×D283)⋊C4—21no—
95(C2×D283)⋊C4—21no—
96C2×Q8×C283—21no—
97C2×Q8×C283—21no—
98C2×Q8×C283—22no—
99C2×Q8×C283—22no—
100C2×Q8×C283—41no—
101C2×Q8×C283—41no—
102C2×Q8×C283C2311yescharacteristic
103C2×Q8×C283C2311yescharacteristic
104C2×Q8×C283C2311yescharacteristic
105C2×D4×C283D411yescharacteristic
106C22×C4×C283—21no—
107C22×C4×C283—21no—
108C22×C4×C283C2311yescharacteristic
109C22×C4×C283C2311yescharacteristic
110C4⋊C4×C283—21no—
111C4⋊C4×C283—21no—
112C4⋊C4×C283—21no—
113C4⋊C4×C283—21no—
114C4⋊C4×C283—21no—
115C4⋊C4×C283D411yescharacteristic
116C4⋊C4×C283C2311yescharacteristic
117C22⋊C4×C283—21no—
118C22⋊C4×C283—21no—
119C22⋊C4×C283D411yescharacteristic
120C22⋊C4×C283D411yescharacteristic
121C42×C283—21no—
122C42×C283C2311yescharacteristic
123C22×D283—21no—
124C22×D283—21no—
125C22×D283—42no—
126C22×D283—42no—
127C22×D283—81no—
128Dic566—21no—
129Dic566—21no—
130D1132—21no—
131D1132—21no—
132D1132—21no—
133D1132—42no—
134D1132C2×D412yes—
135C22⋊D283—42no—
136C22⋊D283—81no—
137C2×Dic283—21no—
138C2×Dic283—21no—
139C4×D283—42no—
140C4×D283—81no—
141C283⋊C8—21no—
142C283⋊C8—21no—
143C283⋊C8—21no—
144C283⋊C8C2×D412yes—
145C23×C283C2×D411yescharacteristic
146C23×C283C2×D411yescharacteristic
147Q8×C283—21no—
148Q8×C283—21no—
149Q8×C283—21no—
150Q8×C283—21no—
151Q8×C283—21no—
152Q8×C283—42no—
153Q8×C283—42no—
154Q8×C283C2×D412yes—
155D4×C283—22no—
156C2×C4×C283—21no—
157C2×C4×C283—21no—
158C2×C4×C283—21no—
159C2×C4×C283—21no—
160C2×C4×C283—21no—
161C2×C4×C283—21no—
162C2×C4×C283—21no—
163C2×C4×C283—21no—
164C2×C4×C283—42no—
165C2×C4×C283—42no—
166C2×C4×C283C2×D411yescharacteristic
167C2×C4×C283C2×D411yescharacteristic
168C2×C4×C283C2×D411yescharacteristic
169C2×C4×C283C2×D411yescharacteristic
170C2×C4×C283C2411yescharacteristic
171C22×C283—21no—
172C22×C283—21no—
173C22×C283—21no—
174C22×C283—21no—
175C22×C283—21no—
176C22×C283#13712yescomplemented
177C22×C283C22×D411yescharacteristic
178C2×D283—22no—
179C2×D283—41no—
180C2×D283—41no—
181C2×D283—41no—
182C2×D283—41no—
183C2×D283—41no—
184C2×D283—84no—
185C4×C283—21no—
186C4×C283—21no—
187C4×C283—22no—
188C4×C283—21no—
189C4×C283—21no—
190C4×C283—41no—
191C4×C283—41no—
192C4×C283—41no—
193C4×C283C22×D411yescharacteristic
194C4×C283H(5,𝔽2)11yescharacteristic
195Dic283—41no—
196Dic283—41no—
197C2×C283—22no—
198C2×C283—41no—
199C2×C283D4211yescharacteristic
200C2×C283C2×#13711yescharacteristic
201C2×C283#57711yescharacteristic
202D283—42no—
203D283—81no—
204C283#326611yescharacteristic, complemented, Sylow
205#3266—2831nomaximal, Sylow
206C2×#137—2831no—
207C22×SD16—2831no—
208D4×Q8D28311yescharacteristic, complemented
209#546—2831no—
210#497—2831no—
211#468—2831no—
212#467—2831no—
213#464—2831no—
214#461—2831no—
215#452—2831no—
216#451—2831no—
217#450—2831no—
218#448—2831no—
219#440—2831no—
220#435—2831no—
221C2×#41—2831no—
222C22×Q8C2×D28311yescharacteristic
223C22×Q8C2×D28311yescharacteristic, complemented
224C22×D4—2831no—
225#137—5664no—
226C2×Q16—2831no—
227C2×Q16—2831no—
228C2×Q16—2831no—
229C2×SD16—2831no—
230C2×SD16—2831no—
231C2×SD16—2831no—
232C2×SD16—2831no—
233C2×SD16—2831no—
234C2×SD16—2831no—
235C2×SD16—5664no—
236C2×M16—2831no—
237C22×C8—2831no—
238C4⋊Q8—21no—
239C4⋊Q8C2×D28311yescharacteristic, complemented
240#124—2831no—
241#123—5661no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8—2831no—
245C22⋊Q8C2×D28311yescharacteristic
246C22⋊Q8C2×D28311yescharacteristic, complemented
247#121—2831no—
248#121—5661no—
249#120—5661no—
250C4×Q8C2×D28311yescharacteristic, complemented
251C4×D4—21no—
252C4×D4—2831no—
253C4×D4C2×D28311yescharacteristic
254C2×C22⋊C4—5661no—
255#107—2831no—
256C4⋊C8—2831no—
257#103—2831no—
258#103—2831no—
259#103—2831no—
260#103—2831no—
261#102—2831no—
262#102—2831no—
263C22⋊C8—2831no—
264C22⋊C8—2831no—
265#97—2831no—
266C24—5661no—
267#41—5662no—
268#41—11321no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—2831no—
276C2×Q8C22×D28311yescharacteristic
277C2×Q8C22×D28311yescharacteristic
278C2×Q8C22×D28311yescharacteristic
279C2×D4—2831no—
280C2×D4—2831no—
281C2×D4—5662no—
282C2×D4—5661no—
283C2×D4—5661no—
284C2×D4—5661no—
285C2×D4—11321no—
286C2×D4C22⋊D28311yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—5661no—
290C22×C4—5661no—
291C22×C4C22×D28311yescharacteristic
292C22×C4C22×D28311yescharacteristic
293Q16—5662no—
294Q16—5662no—
295Q16—11321no—
296SD16—2834no—
297SD16—5662no—
298SD16—5662no—
299SD16—5662no—
300SD16—5662no—
301SD16—11321no—
302M16—5662no—
303C2×C8—2831no—
304C2×C8—2831no—
305C2×C8—2831no—
306C2×C8—2831no—
307C2×C8—5662no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—2831no—
314C4⋊C4—5661no—
315C4⋊C4C22⋊D28311yescharacteristic
316C4⋊C4C22×D28311yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—5661no—
320C22⋊C4—5661no—
321C22⋊C4—5661no—
322C22⋊C4—5661no—
323C22⋊C4—5661no—
324C22⋊C4C22⋊D28311yescharacteristic
325C22⋊C4C22⋊D28311yescharacteristic
326C42—21no—
327C42C22×D28311yescharacteristic
328C23—5661no—
329C23—5661no—
330C23—11322no—
331C23—11322no—
332C23—22641no—
333C23C2×C22⋊D28311yescharacteristic
334C23C2×C22⋊D28311yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—42no—
342Q8—5661no—
343Q8—5661no—
344Q8D4×D28312yes—
345D4—22no—
346D4—2832no—
347D4—5661no—
348D4—5661no—
349D4—5661no—
350D4—11322no—
351D4—11322no—
352D4—22641no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—5661no—
364C2×C4—5661no—
365C2×C4—11322no—
366C2×C4—22641no—
367C2×C4C2×C22⋊D28311yescharacteristic
368C2×C4C2×C22⋊D28311yescharacteristic
369C2×C4C2×C22⋊D28311yescharacteristic
370C2×C4C2×C22⋊D28311yescharacteristic
371C2×C4C23×D28311yescharacteristic
372C8—2832no—
373C8—5661no—
374C8—5661no—
375C8—5661no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—5662no—
382C22—11321no—
383C22—11321no—
384C22—11321no—
385C22—11321no—
386C22—11321no—
387C22—22644no—
388C22#2258364112yescomplemented
389C22C22×C22⋊D28311yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—11321no—
399C4—11321no—
400C4C2×D4×D28311yescharacteristic
401C4#2258365511yescharacteristic
402C2—22no—
403C2—41no—
404C2—11322no—
405C2—22641no—
406C2D4×C22⋊D28311yescharacteristic
407C2C2×#2258364111yescharacteristic
408C2#3182727311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
FittingD4×Q8×C283C2
socleC22×C283C22×D4
prime radical#31827111C2
hypercenterD4×Q8D283
hypocenterC283#3266
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#22583641C2semidirect
2. C22×#17985379C2semidirect
4. #31826857C2semidirect
5. #31826713C2semidirect
6. #31826704C2semidirect
7. #31826687C2semidirect
8. #31826667C2semidirect
9. #31826625C2semidirect
10. #31826621C2semidirect
11. #31826615C2semidirect
12. #31826604C2semidirect
13. #31826563C2semidirect
14. #31826537C2semidirect
15. D4×Q8×C283C2semidirect
19. C2×#17985382C22semidirect
20. C2×#17985382C22semidirect
21. C2×#17985382C22semidirect
29. C2×#17985376C22semidirect
39. #22583543C22semidirect
40. C4⋊(C283⋊C8)C22semidirect
41. #22583534C22semidirect
42. #22583534C22semidirect
43. #22583534C22semidirect
44. #22583533C22semidirect
48. C22⋊(C283⋊C8)C22semidirect
49. #22583521C22semidirect
51. C22×Q8×C283C22semidirect
53. C4⋊Q8×C283C22semidirect
57. C22⋊Q8×C283C22semidirect
58. C4×Q8×C283C22semidirect
82. #17985379D4semidirect
176. C22×C283#137semidirect
204. C283#3266semidirect
208. D4×Q8D283semidirect
223. C22×Q8C2×D283semidirect
239. C4⋊Q8C2×D283semidirect
246. C22⋊Q8C2×D283semidirect
250. C4×Q8C2×D283semidirect
388. C22#22583641semidirect

Maximal subgroups

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

subgroupindex
C2×#225836412
C22×#179853792
#318271112
#318268572
#318267132
#318267042
#318266872
#318266672
#318266252
#318266212
#318266152
#318266042
#318265632
#318265372
D4×Q8×C2832
#3266283