Finite Group Databasebeta

← #27182672

Subgroups — #27182672

Order 12928.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692282C211yesmaximal, characteristic, complemented
2#20265846C211yesmaximal, characteristic, complemented
3#20265815C211yesmaximal, characteristic, complemented
4#20265563C211yesmaximal, characteristic, complemented
5#20265420C211yesmaximal, characteristic, complemented
6#20265413C211yesmaximal, characteristic, complemented
7#20265401C211yesmaximal, characteristic, complemented
8#20265370C211yesmaximal, characteristic, complemented
9#20265338C211yesmaximal, characteristic, complemented
10#20265322C211yesmaximal, characteristic, complemented
11#20265322C211yesmaximal, characteristic, complemented
12#20265309C211yesmaximal, characteristic, complemented
13#20265278C211yesmaximal, characteristic, complemented
14#20265248C211yesmaximal, characteristic, complemented
15C2×#135×C101C211yesmaximal, characteristic, complemented
16C2×#13108075C2211yescharacteristic
17C2×D4×D101C2211yescharacteristic
18#15692282—24no—
19C2×#13108061C2211yescharacteristic
20C2×#13108061C2211yescharacteristic, complemented
21C2×D808C2211yescharacteristic
22C2×#13108059C2211yescharacteristic
23C2×#13108059C2211yescharacteristic
24C2×#13108057C2211yescharacteristic
25#15692251—21no—
26#15692251C2211yescharacteristic, complemented
27#15692244C2211yescharacteristic
28#15692238—21no—
29#15692231C2211yescharacteristic
30#15692229—21no—
31#15692229C2211yescharacteristic
32#15692229C2211yescharacteristic, complemented
33#15692227—21no—
34#15692226—21no—
35#15692226C2211yescharacteristic
36#15692223—21no—
37Q8×Dic101C2211yescharacteristic, complemented
38D4×Dic101C2211yescharacteristic
39C4×C22⋊D101—21no—
40C4×C22⋊D101C2211yescharacteristic
41C2×#13108051—21no—
42#15692189C2211yescharacteristic, complemented
43#15692184C2211yescharacteristic, complemented
44#15692179C2211yescharacteristic, complemented
45#15692179C2211yescharacteristic, complemented
46#15692177C2211yescharacteristic
47#15692177C2211yescharacteristic, complemented
48#15692176C2211yescharacteristic
49#15692176C2211yescharacteristic, complemented
50C22⋊#11717579C2211yescharacteristic, complemented
51#15692167C2211yescharacteristic
52#15692165C2211yescharacteristic, complemented
53C2×#41×C101C2211yescharacteristic
54C2×#41×C101C2211yescharacteristic, complemented
55#135×C101—24no—
56C2×Q16×C101C2211yescharacteristic, complemented
57C2×SD16×C101C2211yescharacteristic
58C2×SD16×C101C2211yescharacteristic, complemented
59C2×D8×C101C2211yescharacteristic
60C22×C8×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62#13108075—22no—
63#13108075—41no—
64C2×D404—21no—
65C2×D404C2311yescharacteristic
66C2×D404C2311yescharacteristic
67C2×C22⋊D101—21no—
68C2×C22⋊D101—21no—
69C2×C22⋊D101—21no—
70C2×C22⋊D101—21no—
71C2×C22⋊D101—21no—
72C2×C22⋊D101—21no—
73C2×C22⋊D101D411yescharacteristic
74D4×D101—22no—
75D4×D101—41no—
76C22×Dic101—21no—
77C2×C4×D101—21no—
78C2×C4×D101C2311yescharacteristic
79#13108061—22no—
80#13108061—41no—
81D808—22no—
82#13108059—22no—
83#13108059—41no—
84#13108057—22no—
85C2×#11717579C2311yescharacteristic
86C2×#11717579C2311yescharacteristic
87C4⋊Dic101—21no—
88C4⋊Dic101C2311yescharacteristic
89C4⋊Dic101C2311yescharacteristic
90#13108053—21no—
91#13108053D411yescharacteristic
92C22⋊Dic101—21no—
93C22⋊Dic101—21no—
94C22⋊Dic101D411yescharacteristic
95#13108051—21no—
96#13108051—21no—
97#13108051—21no—
98#13108051—21no—
99#13108051—21no—
100#13108051D411yescharacteristic
101C4×Dic101—21no—
102C4×Dic101C2311yescharacteristic
103#41×C101—22no—
104#41×C101—22no—
105#41×C101—41no—
106#41×C101—41no—
107C2×Q8×C101C2311yescharacteristic
108C2×Q8×C101C2311yescharacteristic
109C2×D4×C101—21no—
110C2×D4×C101—21no—
111C2×D4×C101C2311yescharacteristic
112C2×D4×C101C2311yescharacteristic
113C22×C4×C101—21no—
114C22×C4×C101—21no—
115C22×C4×C101C2311yescharacteristic
116Q16×C101—22no—
117SD16×C101—22no—
118SD16×C101D414yescomplemented
119D8×C101—22no—
120C2×C8×C101—22no—
121C2×C8×C101C2311yescharacteristic
122C2×C8×C101C2311yescharacteristic
123C23×C101—21no—
124C23×C101—21no—
125C23×C101C2×D411yescharacteristic
126C22×D101—21no—
127C22×D101—21no—
128C22×D101—42no—
129C22×D101—42no—
130C22×D101—81no—
131C22×D101C2×D411yescharacteristic
132Q8×C101—21no—
133Q8×C101—21no—
134Q8×C101—21no—
135Q8×C101C2×D412yes—
136D4×C101—21no—
137D4×C101—21no—
138D4×C101—21no—
139D4×C101—42no—
140D4×C101—42no—
141D4×C101C2×D412yes—
142C2×C4×C101—21no—
143C2×C4×C101—21no—
144C2×C4×C101—21no—
145C2×C4×C101—21no—
146C2×C4×C101—42no—
147C2×C4×C101—42no—
148C2×C4×C101C2×D411yescharacteristic
149C2×C4×C101C2411yescharacteristic
150C22⋊D101—41no—
151C22⋊D101—42no—
152C22⋊D101—42no—
153C22⋊D101—81no—
154C22⋊D101—81no—
155C22⋊D101—81no—
156C22⋊D101—81no—
157C2×Dic101—21no—
158C2×Dic101—21no—
159C2×Dic101—81no—
160C2×Dic101C2×D411yescharacteristic
161C2×Dic101C2×D411yescharacteristic
162C2×Dic101C2×D411yescharacteristic
163D404—21no—
164D404—21no—
165D404—21no—
166D404—21no—
167D404—42no—
168C4×D101—21no—
169C4×D101—21no—
170C4×D101—42no—
171C8×C101—21no—
172C8×C101C2×D412yes—
173#11717579—21no—
174#11717579—21no—
175C22×C101—22no—
176C22×C101—21no—
177C22×C101—21no—
178C22×C101—21no—
179C22×C101—41no—
180C22×C101—41no—
181C22×C101—41no—
182C22×C101—41no—
183C22×C101—41no—
184C22×C101C22×D411yescharacteristic
185C2×D101—21no—
186C2×D101—21no—
187C2×D101—21no—
188C2×D101—41no—
189C2×D101—41no—
190C2×D101—41no—
191C2×D101—41no—
192C2×D101—41no—
193C2×D101—41no—
194C2×D101—84no—
195C4×C101—22no—
196C4×C101—21no—
197C4×C101—41no—
198C4×C101C22×D411yescharacteristic
199C4×C101H(5,𝔽2)11yescharacteristic
200Dic101—21no—
201Dic101—21no—
202Dic101—21no—
203Dic101—41no—
204Dic101—41no—
205C2×C101—42no—
206C2×C101—41no—
207C2×C101—81no—
208C2×C101D4211yescharacteristic
209C2×C101#57511yescharacteristic
210C2×C101#57611yescharacteristic
211D101—41no—
212D101—81no—
213D101—81no—
214#3268—1011nomaximal, Sylow
215C101#326811yescharacteristic, complemented, Sylow
216C2×AGL(1,ℤ/8)—1011no—
217C2×#135D10111yescharacteristic, complemented
218#550—1011no—
219#546—1011no—
220#496—1011no—
221#468—1011no—
222#466—1011no—
223#464—1011no—
224#459—1011no—
225#452—1011no—
226#449—1011no—
227#449—1011no—
228#447—1011no—
229#440—1011no—
230#435—1011no—
231C2×#41—1011no—
232C2×#41C2×D10111yescharacteristic
233C2×#41C2×D10111yescharacteristic, complemented
234C22×D4—1011no—
235AGL(1,ℤ/8)—2024no—
236#135—24no—
237C2×Q16C2×D10111yescharacteristic, complemented
238C2×SD16—1011no—
239C2×SD16—1011no—
240C2×SD16C2×D10111yescharacteristic
241C2×SD16C2×D10111yescharacteristic, complemented
242C2×D8—1011no—
243C2×D8—1011no—
244C2×D8—1011no—
245C2×D8C2×D10111yescharacteristic
246C2×M16—1011no—
247C22×C8C2×D10111yescharacteristic, complemented
248#127—1011no—
249#127—2021no—
250#124—1011no—
251#123—2021no—
252C22⋊Q8—1011no—
253#121—1011no—
254#121—1011no—
255#121—1011no—
256#121—2021no—
257#121—2021no—
258#121—2021no—
259#120—2021no—
260C4×Q8—1011no—
261C4×D4—1011no—
262C4×D4—1011no—
263C4×D4—2021no—
264C2×C22⋊C4—2021no—
265#107—1011no—
266C4⋊C8—1011no—
267#103—1011no—
268#103—1011no—
269#102—1011no—
270#102—1011no—
271#102—1011no—
272#102—1011no—
273C22⋊C8—1011no—
274C22⋊C8—1011no—
275#97—1011no—
276C24—2021no—
277#41—22no—
278#41—22no—
279#41—41no—
280#41—41no—
281#41—4041no—
282#41—2022no—
283C2×Q8C22×D10111yescharacteristic
284C2×Q8C22×D10111yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—1011no—
288C2×D4—1011no—
289C2×D4—1011no—
290C2×D4—4041no—
291C2×D4—2021no—
292C2×D4—2021no—
293C2×D4—2021no—
294C2×D4—2021no—
295C2×D4—2021no—
296C2×D4—2021no—
297C2×D4—2021no—
298C2×D4—2022no—
299C2×D4C22×D10111yescharacteristic
300C2×D4C22×D10111yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—1011no—
304C22×C4—2021no—
305C22×C4—2021no—
306C22×C4C22×D10111yescharacteristic
307Q16—22no—
308SD16—22no—
309SD16—4041no—
310SD16—2022no—
311SD16C22⋊D10114yescomplemented
312D8—22no—
313D8—4041no—
314D8—2022no—
315D8—2022no—
316M16—2022no—
317C2×C8—22no—
318C2×C8—1011no—
319C2×C8—1011no—
320C2×C8C22×D10111yescharacteristic
321C2×C8C22×D10111yescharacteristic
322C4⋊C4—1011no—
323C4⋊C4—1011no—
324C4⋊C4—1011no—
325C4⋊C4—2021no—
326C4⋊C4—2021no—
327C22⋊C4—1011no—
328C22⋊C4—1011no—
329C22⋊C4—2021no—
330C22⋊C4—2021no—
331C22⋊C4—2021no—
332C22⋊C4—2021no—
333C22⋊C4—2021no—
334C22⋊C4—2021no—
335C22⋊C4—2021no—
336C42—1011no—
337C42—2021no—
338C23—21no—
339C23—21no—
340C23—8081no—
341C23—1011no—
342C23—4042no—
343C23—4042no—
344C23—2021no—
345C23—2021no—
346C23D4×D10111yescharacteristic
347Q8—21no—
348Q8—21no—
349Q8—21no—
350Q8C2×C22⋊D10112yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—42no—
355D4—42no—
356D4—8081no—
357D4—8081no—
358D4—8081no—
359D4—8081no—
360D4—4042no—
361D4—4042no—
362D4—4041no—
363D4—4042no—
364D4—2021no—
365D4—2021no—
366D4—2021no—
367D4—2021no—
368D4C2×C22⋊D10112yes—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—8081no—
376C2×C4—1011no—
377C2×C4—1011no—
378C2×C4—1011no—
379C2×C4—4042no—
380C2×C4—2021no—
381C2×C4—2021no—
382C2×C4—2021no—
383C2×C4—2021no—
384C2×C4D4×D10111yescharacteristic
385C2×C4C23×D10111yescharacteristic
386C8—21no—
387C8—2021no—
388C8—2021no—
389C8C2×C22⋊D10112yes—
390C22—21no—
391C22—22no—
392C22—21no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—8084no—
400C22—4041no—
401C22—4041no—
402C22—4041no—
403C22—4041no—
404C22—4041no—
405C22—4041no—
406C22—2021no—
407C22—2021no—
408C22—2021no—
409C22C2×D4×D10111yescharacteristic
410C4—21no—
411C4—22no—
412C4—41no—
413C4—4041no—
414C4—4041no—
415C4—2021no—
416C4—2021no—
417C4—2021no—
418C4C22×C22⋊D10111yescharacteristic
419C4#1569230011yescharacteristic
420C2—41no—
421C2—42no—
422C2—81no—
423C2—8081no—
424C2—8081no—
425C2—4041no—
426C2D4×C22⋊D10111yescharacteristic
427C2#2026597411yescharacteristic
428C2#2026597911yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×#135×C101C2
socleC22×C101C22×D4
hypercenterC2×#135D101
hypocenterC101#3268
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#15692282C2semidirect
2. #20265846C2semidirect
3. #20265815C2semidirect
4. #20265563C2semidirect
5. #20265420C2semidirect
6. #20265413C2semidirect
7. #20265401C2semidirect
8. #20265370C2semidirect
9. #20265338C2semidirect
10. #20265322C2semidirect
11. #20265322C2semidirect
12. #20265309C2semidirect
13. #20265278C2semidirect
14. #20265248C2semidirect
15. C2×#135×C101C2semidirect
20. C2×#13108061C22semidirect
26. #15692251C22semidirect
32. #15692229C22semidirect
37. Q8×Dic101C22semidirect
42. #15692189C22semidirect
43. #15692184C22semidirect
44. #15692179C22semidirect
45. #15692179C22semidirect
47. #15692177C22semidirect
49. #15692176C22semidirect
50. C22⋊#11717579C22semidirect
52. #15692165C22semidirect
54. C2×#41×C101C22semidirect
56. C2×Q16×C101C22semidirect
58. C2×SD16×C101C22semidirect
60. C22×C8×C101C22semidirect
118. SD16×C101D4semidirect
215. C101#3268semidirect
217. C2×#135D101semidirect
233. C2×#41C2×D101semidirect
237. C2×Q16C2×D101semidirect
241. C2×SD16C2×D101semidirect
247. C22×C8C2×D101semidirect
311. SD16C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922822
#202658462
#202658152
#202655632
#202654202
#202654132
#202654012
#202653702
#202653382
#202653222
#202653092
#202652782
#202652482
C2×#135×C1012
#3268101