Finite Group Databasebeta

← #27182645

Subgroups — #27182645

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×#15692286C211yesmaximal, characteristic, complemented
2C2×SD16×D101C211yesmaximal, characteristic, complemented
3Q8×C22⋊D101C211yesmaximal, characteristic, complemented
4#20265824C211yesmaximal, characteristic
5#20265568C211yesmaximal, characteristic, complemented
6#20265421C211yesmaximal, characteristic, complemented
7#20265417C211yesmaximal, characteristic, complemented
8#20265397C211yesmaximal, characteristic, complemented
9#20265381C211yesmaximal, characteristic, complemented
10#20265341C211yesmaximal, characteristic, complemented
11#20265334C211yesmaximal, characteristic, complemented
12#20265326C211yesmaximal, characteristic, complemented
13#20265280C211yesmaximal, characteristic, complemented
14#20265251C211yesmaximal, characteristic, complemented
15#448×C101C211yesmaximal, characteristic, complemented
16C2×#13108074C2211yescharacteristic
17C2×Q8×D101C2211yescharacteristic
18C2×D4×D101C2211yescharacteristic
19#15692286—24no—
20C2×Dic404C2211yescharacteristic, complemented
21C2×#13108064C2211yescharacteristic, complemented
22C2×#13108064C2211yescharacteristic, complemented
23C2×#13108063C2211yescharacteristic
24C2×#13108062C2211yescharacteristic
25C2×#13108061C2211yescharacteristic
26C2×#13108061C2211yescharacteristic
27SD16×D101—24no—
28C2×#13108058C2211yescharacteristic, complemented
29C2×C8×D101C2211yescharacteristic
30#15692253—21no—
31#15692253C2211yescharacteristic, complemented
32#15692242C2211yescharacteristic
33#15692237—21no—
34#15692235C2211yescharacteristic
35#15692234—21no—
36#15692234C2211yescharacteristic, complemented
37#15692231—21no—
38#15692231C2211yescharacteristic
39#15692226—21no—
40#15692223—21no—
41Q8×Dic101C2211yescharacteristic, complemented
42#15692217C2211yescharacteristic
43C4×C22⋊D101—21no—
44C4×C22⋊D101C2211yescharacteristic
45C22⋊C4×D101—21no—
46#15692188C2211yescharacteristic, complemented
47#15692184C2211yescharacteristic, complemented
48#15692180C2211yescharacteristic, complemented
49#15692178C2211yescharacteristic, complemented
50#15692177C2211yescharacteristic
51#15692176C2211yescharacteristic
52#15692167C2211yescharacteristic
53#15692165C2211yescharacteristic, complemented
54C22×Q8×C101C2211yescharacteristic, complemented
55C2×SD16×C101C2211yescharacteristic
56C2×SD16×C101C2211yescharacteristic
57#121×C101C2211yescharacteristic
58#103×C101C2211yescharacteristic, complemented
59#103×C101C2211yescharacteristic, complemented
60C22⋊C8×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62#13108074—22no—
63#13108074—41no—
64C2×Dic202—21no—
65C2×Dic202C2311yescharacteristic
66C2×Dic202C2311yescharacteristic
67Q8×D101—22no—
68Q8×D101—41no—
69C2×D404C2311yescharacteristic
70C2×C22⋊D101—21no—
71C2×C22⋊D101—21no—
72C2×C22⋊D101D411yescharacteristic
73D4×D101—22no—
74D4×D101—41no—
75C2×C4×D101—21no—
76C2×C4×D101—21no—
77C2×C4×D101—21no—
78C2×C4×D101C2311yescharacteristic
79Dic404—41no—
80#13108064—22no—
81#13108064—22no—
82#13108063—22no—
83#13108062—22no—
84#13108061—22no—
85#13108061D414yescomplemented
86#13108058—22no—
87C2×#11717579C2311yescharacteristic
88C2×#11717579C2311yescharacteristic
89C8×D101—22no—
90C4⋊Dic101—21no—
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⋊Dic101D411yescharacteristic
102#13108051—21no—
103#13108051—21no—
104#13108051—21no—
105#13108051D411yescharacteristic
106C4×Dic101—21no—
107C4×Dic101C2311yescharacteristic
108C2×Q8×C101—21no—
109C2×Q8×C101—22no—
110C2×Q8×C101—41no—
111C2×Q8×C101C2311yescharacteristic
112C2×Q8×C101C2311yescharacteristic
113C2×D4×C101—21no—
114C2×D4×C101C2311yescharacteristic
115C22×C4×C101—21no—
116C22×C4×C101C2311yescharacteristic
117SD16×C101—22no—
118SD16×C101—41no—
119C2×C8×C101C2311yescharacteristic
120C2×C8×C101C2311yescharacteristic
121C4⋊C4×C101C2311yescharacteristic
122C22⋊C4×C101—21no—
123C23×C101—21no—
124C23×C101C2×D411yescharacteristic
125C22×D101—21no—
126C22×D101—42no—
127C22×D101—42no—
128C22×D101—81no—
129C22×D101C2×D411yescharacteristic
130Q8×C101—21no—
131Q8×C101—21no—
132Q8×C101—21no—
133Q8×C101—42no—
134Q8×C101C2×D412yes—
135D4×C101—21no—
136D4×C101—21no—
137D4×C101—81no—
138C2×C4×C101—21no—
139C2×C4×C101—21no—
140C2×C4×C101—21no—
141C2×C4×C101—21no—
142C2×C4×C101—21no—
143C2×C4×C101—42no—
144C2×C4×C101C2×D411yescharacteristic
145C2×C4×C101C2411yescharacteristic
146C22⋊D101—22no—
147C22⋊D101—42no—
148C22⋊D101—42no—
149C2×Dic101—21no—
150C2×Dic101—21no—
151C2×Dic101—21no—
152C2×Dic101C2×D411yescharacteristic
153C2×Dic101C2×D411yescharacteristic
154C2×Dic101C2×D411yescharacteristic
155D404—21no—
156D404C2×D412yes—
157C4×D101—21no—
158C4×D101—21no—
159C4×D101—42no—
160C4×D101—42no—
161C4×D101—81no—
162Dic202—21no—
163Dic202—21no—
164Dic202—21no—
165Dic202—21no—
166Dic202—42no—
167C8×C101—21no—
168C8×C101—21no—
169#11717579—21no—
170#11717579C2×D412yes—
171C22×C101—21no—
172C22×C101—21no—
173C22×C101—21no—
174C22×C101—41no—
175C22×C101—41no—
176C22×C101—41no—
177C22×C101C22×D411yescharacteristic
178C2×D101—21no—
179C2×D101—21no—
180C2×D101—22no—
181C2×D101—41no—
182C2×D101—41no—
183C2×D101—84no—
184C2×D101#13712yescomplemented
185C4×C101—22no—
186C4×C101—21no—
187C4×C101—41no—
188C4×C101—41no—
189C4×C101C22×D411yescharacteristic
190C4×C101H(5,𝔽2)11yescharacteristic
191Dic101—21no—
192Dic101—21no—
193Dic101—21no—
194Dic101—41no—
195Dic101—41no—
196Dic101—41no—
197C2×C101—41no—
198C2×C101—81no—
199C2×C101D4211yescharacteristic
200C2×C101C2×#13711yescharacteristic
201C2×C101#57711yescharacteristic
202D101—22no—
203D101—42no—
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#450—1011no—
218#448D10111yescharacteristic, complemented
219#440—1011no—
220#435—1011no—
221C2×#41—1011no—
222C22×Q8—1011no—
223C22×Q8C2×D10111yescharacteristic, complemented
224C22×D4—1011no—
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—1011no—
245C22⋊Q8—2021no—
246C22⋊Q8—2021no—
247#121—2021no—
248#121C2×D10111yescharacteristic
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#103C2×D10111yescharacteristic, complemented
260#103C2×D10111yescharacteristic, complemented
261#102—1011no—
262#102—1011no—
263C22⋊C8—1011no—
264C22⋊C8C2×D10111yescharacteristic, complemented
265#97—1011no—
266C24—2021no—
267#41—4041no—
268#41—2022no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—1011no—
273C2×Q8—1011no—
274C2×Q8—4041no—
275C2×Q8—2021no—
276C2×Q8—2022no—
277C2×Q8C22×D10111yescharacteristic
278C2×Q8C22×D10111yescharacteristic
279C2×D4—21no—
280C2×D4—1011no—
281C2×D4—1011no—
282C2×D4—4041no—
283C2×D4—2021no—
284C2×D4—2022no—
285C2×D4—2021no—
286C2×D4C22×D10111yescharacteristic
287C22×C4—21no—
288C22×C4—1011no—
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—1011no—
309C4⋊C4—1011no—
310C4⋊C4—2021no—
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—8081no—
330C23—1011no—
331C23—4042no—
332C23—4042no—
333C23—2021no—
334C23D4×D10111yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—42no—
339Q8—4042no—
340Q8—2021no—
341Q8—2021no—
342Q8—2021no—
343Q8—2021no—
344Q8D4×D10112yes—
345D4—21no—
346D4—21no—
347D4—81no—
348D4—1012no—
349D4—4042no—
350D4—4042no—
351D4—2022no—
352D4—2021no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—8081no—
360C2×C4—1011no—
361C2×C4—1011no—
362C2×C4—1011no—
363C2×C4—4042no—
364C2×C4—4042no—
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—21no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—8084no—
383C22—1012no—
384C22—4041no—
385C22—4041no—
386C22—2022no—
387C22—2021no—
388C22—2021no—
389C22C2×D4×D10111yescharacteristic
390C4—21no—
391C4—22no—
392C4—41no—
393C4—41no—
394C4—4041no—
395C4—4041no—
396C4—4041no—
397C4—2021no—
398C4—2021no—
399C4—2021no—
400C4C2×D4×D10111yescharacteristic
401C4#1569230011yescharacteristic
402C2—41no—
403C2—81no—
404C2—4042no—
405C2—2022no—
406C2#2026580911yescharacteristic
407C2#137×D10111yescharacteristic
408C2#2026598311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#448×C101C2
socleC22×C101C22×D4
prime radical#20265824C2
hypercenter#448D101
hypocenterC101#3266
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#15692286C2semidirect
2. C2×SD16×D101C2semidirect
3. Q8×C22⋊D101C2semidirect
5. #20265568C2semidirect
6. #20265421C2semidirect
7. #20265417C2semidirect
8. #20265397C2semidirect
9. #20265381C2semidirect
10. #20265341C2semidirect
11. #20265334C2semidirect
12. #20265326C2semidirect
13. #20265280C2semidirect
14. #20265251C2semidirect
15. #448×C101C2semidirect
20. C2×Dic404C22semidirect
21. C2×#13108064C22semidirect
22. C2×#13108064C22semidirect
28. C2×#13108058C22semidirect
31. #15692253C22semidirect
36. #15692234C22semidirect
41. Q8×Dic101C22semidirect
46. #15692188C22semidirect
47. #15692184C22semidirect
48. #15692180C22semidirect
49. #15692178C22semidirect
53. #15692165C22semidirect
54. C22×Q8×C101C22semidirect
58. #103×C101C22semidirect
59. #103×C101C22semidirect
60. C22⋊C8×C101C22semidirect
85. #13108061D4semidirect
184. C2×D101#137semidirect
205. C101#3266semidirect
218. #448D101semidirect
223. C22×Q8C2×D101semidirect
259. #103C2×D101semidirect
260. #103C2×D101semidirect
264. C22⋊C8C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922862
C2×SD16×D1012
Q8×C22⋊D1012
#202658242
#202655682
#202654212
#202654172
#202653972
#202653812
#202653412
#202653342
#202653262
#202652802
#202652512
#448×C1012
#3266101