Finite Group Databasebeta

← #27182646

Subgroups — #27182646

Order 12928.

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×#15692284C211yesmaximal, characteristic, complemented
2C22×#13108062C211yesmaximal, characteristic, complemented
3#20265840C211yesmaximal, characteristic, complemented
4#20265821C211yesmaximal, characteristic
5#20265568C211yesmaximal, characteristic, complemented
6#20265422C211yesmaximal, characteristic, complemented
7#20265416C211yesmaximal, characteristic, complemented
8#20265398C211yesmaximal, characteristic, complemented
9#20265380C211yesmaximal, characteristic, complemented
10#20265340C211yesmaximal, characteristic, complemented
11#20265335C211yesmaximal, characteristic, complemented
12#20265314C211yesmaximal, characteristic, complemented
13#20265274C211yesmaximal, characteristic, complemented
14#20265245C211yesmaximal, characteristic, complemented
15#450×C101C211yesmaximal, characteristic, complemented
16C2×#13108073C2211yescharacteristic
17C22×Dic202C2211yescharacteristic
18C2×Q8×D101C2211yescharacteristic, complemented
19#15692284—24no—
20C2×Dic404C2211yescharacteristic, complemented
21C2×Dic404C2211yescharacteristic, complemented
22C2×#13108064C2211yescharacteristic, complemented
23C2×#13108062—24no—
24C2×#13108062C2211yescharacteristic
25C2×#13108062C2211yescharacteristic
26C2×#13108062C2211yescharacteristic
27C2×#13108062C2211yescharacteristic
28C2×#13108057C2211yescharacteristic, complemented
29C22×#11717579C2211yescharacteristic
30#15692254—21no—
31#15692253C2211yescharacteristic, complemented
32#15692242C2211yescharacteristic
33#15692236—21no—
34#15692235C2211yescharacteristic, complemented
35#15692234C2211yescharacteristic
36#15692233—21no—
37#15692232—21no—
38#15692227—21no—
39#15692226C2211yescharacteristic
40#15692224—21no—
41#15692220C2211yescharacteristic, complemented
42D4×Dic101C2211yescharacteristic
43C4×C22⋊D101C2211yescharacteristic
44#15692215—21no—
45C2×C22⋊Dic101—21no—
46#15692188C2211yescharacteristic, complemented
47#15692184C2211yescharacteristic, complemented
48#15692180C2211yescharacteristic, complemented
49#15692180C2211yescharacteristic, complemented
50#15692179C2211yescharacteristic, complemented
51#15692178C2211yescharacteristic, complemented
52#15692167C2211yescharacteristic, complemented
53#15692165C2211yescharacteristic, complemented
54C22×D4×C101C2211yescharacteristic
55C2×SD16×C101C2211yescharacteristic
56C2×SD16×C101C2211yescharacteristic
57C22⋊Q8×C101C2211yescharacteristic
58#102×C101C2211yescharacteristic
59#102×C101C2211yescharacteristic
60C22⋊C8×C101C2211yescharacteristic
61#13108073—22no—
62#13108073—41no—
63C2×Dic202—22no—
64C2×Dic202—21no—
65C2×Dic202—21no—
66C2×Dic202—41no—
67C2×Dic202C2311yescharacteristic
68C2×Dic202C2311yescharacteristic
69C2×Dic202C2311yescharacteristic
70Q8×D101—22no—
71Q8×D101—41no—
72C2×C22⋊D101—21no—
73C2×C22⋊D101—21no—
74C2×C22⋊D101D411yescharacteristic
75C22×Dic101—21no—
76C22×Dic101—21no—
77C22×Dic101—21no—
78C2×C4×D101—21no—
79C2×C4×D101C2311yescharacteristic
80Dic404—22no—
81Dic404—41no—
82#13108064—22no—
83#13108062—22no—
84#13108062—22no—
85#13108062—22no—
86#13108062D414yescomplemented
87#13108057—22no—
88C2×#11717579—22no—
89C2×#11717579C2311yescharacteristic
90C2×#11717579C2311yescharacteristic
91C4⋊Dic101—21no—
92C4⋊Dic101C2311yescharacteristic
93#13108053—21no—
94#13108053—21no—
95#13108053—21no—
96#13108053—21no—
97#13108053D411yescharacteristic
98C22⋊Dic101—21no—
99C22⋊Dic101—21no—
100C22⋊Dic101—21no—
101C22⋊Dic101—21no—
102C22⋊Dic101—21no—
103C22⋊Dic101D411yescharacteristic
104#13108051—21no—
105#13108051D411yescharacteristic
106C4×Dic101—21no—
107C4×Dic101C2311yescharacteristic
108C24×C101—21no—
109C2×Q8×C101C2311yescharacteristic
110C2×D4×C101—22no—
111C2×D4×C101—21no—
112C2×D4×C101—41no—
113C2×D4×C101C2311yescharacteristic
114C2×D4×C101C2311yescharacteristic
115C22×C4×C101C2311yescharacteristic
116SD16×C101—22no—
117SD16×C101—41no—
118C2×C8×C101C2311yescharacteristic
119C2×C8×C101C2311yescharacteristic
120C4⋊C4×C101—21no—
121C4⋊C4×C101C2311yescharacteristic
122C22⋊C4×C101—21no—
123C23×C101—21no—
124C23×C101—21no—
125C23×C101—42no—
126C23×C101—42no—
127C23×C101—81no—
128C23×C101C2×D411yescharacteristic
129C22×D101C2×D411yescharacteristic
130Q8×C101—21no—
131Q8×C101—21no—
132D4×C101—21no—
133D4×C101—21no—
134D4×C101—21no—
135D4×C101—42no—
136D4×C101C2×D412yes—
137C2×C4×C101—21no—
138C2×C4×C101—21no—
139C2×C4×C101—21no—
140C2×C4×C101—21no—
141C2×C4×C101C2×D411yescharacteristic
142C2×C4×C101C2411yescharacteristic
143C22⋊D101—22no—
144C22⋊D101—42no—
145C22⋊D101—81no—
146C2×Dic101—21no—
147C2×Dic101—21no—
148C2×Dic101—21no—
149C2×Dic101—21no—
150C2×Dic101—42no—
151C2×Dic101—42no—
152C2×Dic101—81no—
153C2×Dic101C2×D411yescharacteristic
154C2×Dic101C2×D411yescharacteristic
155C2×Dic101C2×D411yescharacteristic
156C4×D101—21no—
157C4×D101—21no—
158C4×D101—42no—
159Dic202—21no—
160Dic202—21no—
161Dic202—21no—
162Dic202—21no—
163Dic202—21no—
164Dic202—42no—
165Dic202—42no—
166Dic202C2×D412yes—
167C8×C101—21no—
168C8×C101—21no—
169#11717579—21no—
170#11717579C2×D412yes—
171C22×C101—21no—
172C22×C101—21no—
173C22×C101—22no—
174C22×C101—41no—
175C22×C101—41no—
176C22×C101—41no—
177C22×C101—41no—
178C22×C101—41no—
179C22×C101—84no—
180C22×C101#13712yescomplemented
181C22×C101C22×D411yescharacteristic
182C2×D101—21no—
183C2×D101—21no—
184C2×D101—21no—
185C4×C101—21no—
186C4×C101—41no—
187C4×C101—41no—
188C4×C101C22×D411yescharacteristic
189C4×C101H(5,𝔽2)11yescharacteristic
190Dic101—22no—
191Dic101—21no—
192Dic101—21no—
193Dic101—21no—
194Dic101—41no—
195Dic101—41no—
196Dic101—41no—
197C2×C101—22no—
198C2×C101—42no—
199C2×C101—81no—
200C2×C101D4211yescharacteristic
201C2×C101C2×#13711yescharacteristic
202C2×C101#57711yescharacteristic
203D101—41no—
204#3266—1011nomaximal, Sylow
205C101#326611yescharacteristic, complemented, Sylow
206C2×#137—1011no—
207C22×SD16—1011no—
208D4×Q8—1011no—
209#546—1011no—
210#497—1011no—
211#468—1011no—
212#467—1011no—
213#464—1011no—
214#461—1011no—
215#452—1011no—
216#451—1011no—
217#450D10111yescharacteristic, complemented
218#448—1011no—
219#440—1011no—
220#435—1011no—
221C2×#41—1011no—
222C22×Q8—1011no—
223C22×Q8—1011no—
224C22×D4C2×D10111yescharacteristic
225#137—2024no—
226C2×Q16—1011no—
227C2×Q16—1011no—
228C2×Q16—1011no—
229C2×SD16—1011no—
230C2×SD16—1011no—
231C2×SD16—1011no—
232C2×SD16—1011no—
233C2×SD16—2024no—
234C2×SD16C2×D10111yescharacteristic
235C2×SD16C2×D10111yescharacteristic
236C2×M16—1011no—
237C22×C8—1011no—
238C4⋊Q8—1011no—
239C4⋊Q8—2021no—
240#124—1011no—
241#123—2021no—
242C22⋊Q8—1011no—
243C22⋊Q8—1011no—
244C22⋊Q8—2021no—
245C22⋊Q8—2021no—
246C22⋊Q8C2×D10111yescharacteristic
247#121—1011no—
248#121—2021no—
249#120—2021no—
250C4×Q8—1011no—
251C4×D4—1011no—
252C4×D4—1011no—
253C4×D4—2021no—
254C2×C22⋊C4—2021no—
255#107—1011no—
256C4⋊C8—1011no—
257#103—1011no—
258#103—1011no—
259#103—1011no—
260#103—1011no—
261#102C2×D10111yescharacteristic
262#102C2×D10111yescharacteristic
263C22⋊C8—1011no—
264C22⋊C8C2×D10111yescharacteristic
265#97—1011no—
266C24—21no—
267#41—4041no—
268#41—2022no—
269C2×Q8—1011no—
270C2×Q8—1011no—
271C2×Q8—1011no—
272C2×Q8—4041no—
273C2×Q8—4041no—
274C2×Q8—2021no—
275C2×Q8—2022no—
276C2×Q8—2022no—
277C2×Q8—2021no—
278C2×Q8C22×D10111yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—1011no—
283C2×D4—2021no—
284C2×D4—2021no—
285C2×D4C22×D10111yescharacteristic
286C2×D4C22×D10111yescharacteristic
287C22×C4—1011no—
288C22×C4—2021no—
289C22×C4—2021no—
290C22×C4—2021no—
291C22×C4—2021no—
292C22×C4C22×D10111yescharacteristic
293Q16—4041no—
294Q16—2022no—
295Q16—2022no—
296SD16—22no—
297SD16—41no—
298SD16—1014no—
299SD16—2022no—
300SD16—2022no—
301SD16—2022no—
302M16—2022no—
303C2×C8—1011no—
304C2×C8—1011no—
305C2×C8—2022no—
306C2×C8C22×D10111yescharacteristic
307C2×C8C22×D10111yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—1011no—
310C4⋊C4—1011no—
311C4⋊C4—2021no—
312C4⋊C4—2021no—
313C4⋊C4—2021no—
314C4⋊C4—2021no—
315C4⋊C4—2021no—
316C4⋊C4C22×D10111yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—1011no—
319C22⋊C4—1011no—
320C22⋊C4—2021no—
321C22⋊C4—2021no—
322C22⋊C4—2021no—
323C22⋊C4—2021no—
324C22⋊C4—2021no—
325C22⋊C4—2021no—
326C42—1011no—
327C42—2021no—
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23—1011no—
334C23D4×D10111yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—1012no—
338Q8—4042no—
339Q8—4042no—
340Q8—2021no—
341Q8—2021no—
342Q8—2021no—
343Q8—2021no—
344Q8—2021no—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—8081no—
350D4—4042no—
351D4—2022no—
352D4D4×D10112yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—8081no—
358C2×C4—1011no—
359C2×C4—1011no—
360C2×C4—1011no—
361C2×C4—4042no—
362C2×C4—4042no—
363C2×C4—4042no—
364C2×C4—2021no—
365C2×C4—2021no—
366C2×C4—2021no—
367C2×C4—2021no—
368C2×C4—2021no—
369C2×C4—2021no—
370C2×C4D4×D10111yescharacteristic
371C2×C4C23×D10111yescharacteristic
372C8—21no—
373C8—21no—
374C8—1012no—
375C8—2021no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—84no—
385C22—2021no—
386C22—2021no—
387C22—2021no—
388C22#1569228412yescomplemented
389C22C2×D4×D10111yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—4041no—
394C4—4041no—
395C4—4041no—
396C4—2021no—
397C4—2021no—
398C4—2022no—
399C4—2021no—
400C4C2×D4×D10111yescharacteristic
401C4#1569230011yescharacteristic
402C2—22no—
403C2—42no—
404C2—81no—
405C2—4041no—
406C2#2026580911yescharacteristic
407C2C2×#1569228411yescharacteristic
408C2#2026598711yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#450×C101C2
socleC22×C101C22×D4
prime radical#20265821C2
hypercenter#450D101
hypocenterC101#3266
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#15692284C2semidirect
2. C22×#13108062C2semidirect
3. #20265840C2semidirect
5. #20265568C2semidirect
6. #20265422C2semidirect
7. #20265416C2semidirect
8. #20265398C2semidirect
9. #20265380C2semidirect
10. #20265340C2semidirect
11. #20265335C2semidirect
12. #20265314C2semidirect
13. #20265274C2semidirect
14. #20265245C2semidirect
15. #450×C101C2semidirect
18. C2×Q8×D101C22semidirect
20. C2×Dic404C22semidirect
21. C2×Dic404C22semidirect
22. C2×#13108064C22semidirect
28. C2×#13108057C22semidirect
31. #15692253C22semidirect
34. #15692235C22semidirect
41. #15692220C22semidirect
46. #15692188C22semidirect
47. #15692184C22semidirect
48. #15692180C22semidirect
49. #15692180C22semidirect
50. #15692179C22semidirect
51. #15692178C22semidirect
52. #15692167C22semidirect
53. #15692165C22semidirect
86. #13108062D4semidirect
180. C22×C101#137semidirect
205. C101#3266semidirect
217. #450D101semidirect
388. C22#15692284semidirect

Maximal subgroups

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

subgroupindex
C2×#156922842
C22×#131080622
#202658402
#202658212
#202655682
#202654222
#202654162
#202653982
#202653802
#202653402
#202653352
#202653142
#202652742
#202652452
#450×C1012
#3266101