Finite Group Databasebeta

← #44559201

Subgroups — #44559201

Order 32128.

470 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548744C211yesmaximal, characteristic
2#30632166C211yesmaximal, characteristic
3#30632163C211yesmaximal, characteristic, complemented
4#30631793C211yesmaximal, characteristic, complemented
5#30631792C211yesmaximal, characteristic, complemented
6#30631788C211yesmaximal, characteristic, complemented
7#30631787C211yesmaximal, characteristic, complemented
8C4×Dic1004C211yesmaximal, characteristic, complemented
9#30631779C211yesmaximal, characteristic, complemented
10#30631778C211yesmaximal, characteristic, complemented
11#30631726C211yesmaximal, characteristic, complemented
12#30631680C211yesmaximal, characteristic
13#103×D251C211yesmaximal, characteristic, complemented
14#30631641C211yesmaximal, characteristic, complemented
15C4×SD16×C251C211yesmaximal, characteristic, complemented
16C2×#16860809C2211yescharacteristic
17C2×Q8×D251C2211yescharacteristic
18#22548744C418yescomplemented
19C2×Dic1004C2211yescharacteristic
20C2×#16860800C2211yescharacteristic
21C2×#16860798C2211yescharacteristic
22C2×#16860793C2211yescharacteristic
23C2×C8×D251C2211yescharacteristic
24Q8×Dic251C2211yescharacteristic
25C4×Dic502—21no—
26C4×Dic502C2211yescharacteristic
27C4×Dic502C2211yescharacteristic
28#22548680—21no—
29#22548680C2211yescharacteristic
30#22548680C2211yescharacteristic
31D4×Dic251C2211yescharacteristic
32C4×C22⋊D251—21no—
33(C22⋊D251)⋊C4—21no—
34#22548673—21no—
35#22548673C2211yescharacteristic
36#22548672—21no—
37#22548672C2211yescharacteristic
38#22548671—21no—
39C2×Dic251⋊C4—21no—
40C4⋊C4×D251C2211yescharacteristic
41#22548649C2211yescharacteristic
42#22548648C2211yescharacteristic
43#22548646C2211yescharacteristic
44#22548640C2211yescharacteristic
45#22548640C2211yescharacteristic
46#22548639C2211yescharacteristic
47#22548638C2211yescharacteristic
48#22548638C2211yescharacteristic
49#22548636C2211yescharacteristic
50#22548626C2211yescharacteristic
51#22548625C2211yescharacteristic
52C8×Dic251C2211yescharacteristic
53C2×SD16×C251C2211yescharacteristic
54C4×Q8×C251C2211yescharacteristic
55C4×D4×C251C2211yescharacteristic
56#106×C251C2211yescharacteristic
57#103×C251C2211yescharacteristic
58#102×C251C2211yescharacteristic
59C4×C8×C251C2211yescharacteristic
60#16860809—22no—
61#16860809C2×C414yes—
62C2×Dic502—21no—
63C2×Dic502C2311yescharacteristic
64C2×Dic502C2311yescharacteristic
65Q8×D251—22no—
66Q8×D251C2×C414yescomplemented
67C2×C22⋊D251—21no—
68C22×Dic251—21no—
69C22×Dic251—21no—
70C2×C4×D251—21no—
71C2×C4×D251—21no—
72C2×C4×D251C2311yescharacteristic
73Dic1004C2×C414yescomplemented
74#16860800C2×C414yescomplemented
75#16860798C2×C414yescomplemented
76#16860793C2×C414yescomplemented
77C2×C251⋊C8C2311yescharacteristic
78C2×C251⋊C8C2311yescharacteristic
79C8×D251—22no—
80C4⋊Dic251—21no—
81C4⋊Dic251C2311yescharacteristic
82C4⋊Dic251C2311yescharacteristic
83Dic251⋊C4—21no—
84Dic251⋊C4—21no—
85Dic251⋊C4—21no—
86Dic251⋊C4—21no—
87Dic251⋊C4—21no—
88Dic251⋊C4D411yescharacteristic
89Dic251⋊C4D411yescharacteristic
90C22⋊Dic251—21no—
91(C2×D251)⋊C4—21no—
92(C2×D251)⋊C4D411yescharacteristic
93(C2×D251)⋊C4D411yescharacteristic
94C4×Dic251—21no—
95C4×Dic251—21no—
96C4×Dic251—21no—
97C4×Dic251—21no—
98C4×Dic251—21no—
99C4×Dic251—21no—
100C4×Dic251C2311yescharacteristic
101C2×Q8×C251C2311yescharacteristic
102C2×D4×C251C2311yescharacteristic
103C22×C4×C251—21no—
104SD16×C251C2×C414yescomplemented
105C2×C8×C251C2311yescharacteristic
106C2×C8×C251C2311yescharacteristic
107C4⋊C4×C251—21no—
108C4⋊C4×C251C2311yescharacteristic
109C4⋊C4×C251C2311yescharacteristic
110C22⋊C4×C251—21no—
111C42×C251—21no—
112C42×C251C2311yescharacteristic
113C22×D251C2×D411yescharacteristic
114Dic502—21no—
115Dic502—21no—
116Dic502—24no—
117Dic502C22×C412yes—
118Dic502C22×C412yes—
119C22⋊D251—24no—
120C2×Dic251—21no—
121C2×Dic251—21no—
122C2×Dic251—24no—
123C2×Dic251—21no—
124C2×Dic251—21no—
125C2×Dic251—81no—
126C2×Dic251C2×D411yescharacteristic
127C2×Dic251C2×D411yescharacteristic
128C2×Dic251C2×D411yescharacteristic
129C4×D251—24no—
130C4×D251—81no—
131C4×D251C22×C412yes—
132C4×D251#4112yes—
133C251⋊C8—21no—
134C251⋊C8C22×C412yes—
135C23×C251—21no—
136Q8×C251—21no—
137Q8×C251C22×C412yes—
138D4×C251—21no—
139D4×C251C22×C412yes—
140C2×C4×C251—21no—
141C2×C4×C251—21no—
142C2×C4×C251—21no—
143C2×C4×C251—81no—
144C2×C4×C251C2×D411yescharacteristic
145C2×C4×C251C2×D411yescharacteristic
146C2×C4×C251C2411yescharacteristic
147C8×C251—21no—
148C8×C251C22×C412yes—
149C22×C251—22no—
150C22×C251—41no—
151C22×C251—41no—
152C22×C251C22×D411yescharacteristic
153C2×D251—21no—
154C2×D251—21no—
155C2×D251C4×D412yes—
156C4×C251—22no—
157C4×C251—21no—
158C4×C251—21no—
159C4×C251—41no—
160C4×C251—41no—
161C4×C251C23×C411yescharacteristic
162C4×C251C2×#4111yescharacteristic
163Dic251—22no—
164Dic251—22no—
165Dic251—21no—
166Dic251—21no—
167Dic251—41no—
168Dic251—41no—
169Dic251C4×D412yes—
170C2×C251—42no—
171C2×C251C2×C4×D411yescharacteristic
172C2×C251#57711yescharacteristic
173C2×C251#57811yescharacteristic
174D251—22no—
175C251#294711yescharacteristic, complemented, Sylow
176#2947—2511nomaximal, Sylow
177C2×#137—2511no—
178#520—2511no—
179#519—2511no—
180#441—2511no—
181#441—2511no—
182#440—2511no—
183#440—2511no—
184C4×Q16—2511no—
185C4×Q16—2511no—
186C4×SD16—2511no—
187C4×SD16D25111yescharacteristic, complemented
188#428—2511no—
189#416—2511no—
190C2×#103—2511no—
191#405—2511no—
192C2×#41—2511no—
193C22×Q8—2511no—
194#137—2518no—
195C2×Q16—2511no—
196C2×Q16—2511no—
197C2×SD16—2511no—
198C2×SD16C2×D25111yescharacteristic
199C2×M16—2511no—
200C22×C8—2511no—
201C4×Q8—5021no—
202C4×Q8—5021no—
203C4×Q8—2511no—
204C4×Q8—2511no—
205C4×Q8—2511no—
206C4×Q8—2511no—
207C4×Q8—2511no—
208C4×Q8C2×D25111yescharacteristic
209C4×D4—5021no—
210C4×D4—5021no—
211C4×D4—2511no—
212C4×D4C2×D25111yescharacteristic
213#117—5021no—
214#117—5021no—
215#117—5021no—
216#117—2511no—
217#117—2511no—
218C2×C4⋊C4—5021no—
219C2×C4⋊C4—2511no—
220#107—2511no—
221#107—2511no—
222#106—2511no—
223#106C2×D25111yescharacteristic
224#103—2511no—
225#103—2511no—
226#103—2511no—
227#103—2511no—
228#103—2511no—
229#103C2×D25111yescharacteristic
230#102—2511no—
231#102C2×D25111yescharacteristic
232#97—2511no—
233#97—2511no—
234C4×C8—2511no—
235C4×C8C2×D25111yescharacteristic
236#41—5022no—
237#41—2514no—
238C2×Q8—5021no—
239C2×Q8—5022no—
240C2×Q8—2514no—
241C2×Q8—2511no—
242C2×Q8—2511no—
243C2×Q8C22×D25111yescharacteristic
244C2×D4—5021no—
245C2×D4C22×D25111yescharacteristic
246C22×C4—21no—
247C22×C4—5021no—
248C22×C4—5021no—
249C22×C4—5021no—
250C22×C4—5021no—
251C22×C4—2511no—
252Q16—2514no—
253Q16—2514no—
254SD16—2514no—
255SD16C4×D25114yescomplemented
256M16—2514no—
257C2×C8—5022no—
258C2×C8—2511no—
259C2×C8—2511no—
260C2×C8C22×D25111yescharacteristic
261C2×C8C22×D25111yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—5021no—
264C4⋊C4—5021no—
265C4⋊C4—5021no—
266C4⋊C4—5021no—
267C4⋊C4—5021no—
268C4⋊C4—5021no—
269C4⋊C4—2511no—
270C4⋊C4—2511no—
271C4⋊C4—2511no—
272C4⋊C4—2511no—
273C4⋊C4C22×D25111yescharacteristic
274C4⋊C4C22×D25111yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—5021no—
277C22⋊C4—5021no—
278C22⋊C4—2511no—
279C22⋊C4—2511no—
280C42—21no—
281C42—5021no—
282C42—5021no—
283C42—5021no—
284C42—5021no—
285C42—5021no—
286C42—5021no—
287C42—2511no—
288C42C22×D25111yescharacteristic
289C23—21no—
290C23—2511no—
291Q8—21no—
292Q8—5024no—
293Q8—5021no—
294Q8—5021no—
295Q8—2512no—
296Q8—2512no—
297Q8C2×C4×D25112yes—
298D4—21no—
299D4—5024no—
300D4C2×C4×D25112yes—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—81no—
305C2×C4—20081no—
306C2×C4—20081no—
307C2×C4—5024no—
308C2×C4—5021no—
309C2×C4—5021no—
310C2×C4—5021no—
311C2×C4—5021no—
312C2×C4—5024no—
313C2×C4—2511no—
314C2×C4—2511no—
315C2×C4—2511no—
316C2×C4—2512no—
317C2×C4—2512no—
318C2×C4D4×D25111yescharacteristic
319C2×C4D4×D25111yescharacteristic
320C2×C4C23×D25111yescharacteristic
321C8—21no—
322C8—5021no—
323C8—2512no—
324C8C2×C4×D25112yes—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—5021no—
329C22—5021no—
330C22—2512no—
331C22C2×D4×D25111yescharacteristic
332C4—21no—
333C4—21no—
334C4—22no—
335C4—41no—
336C4—41no—
337C4—10041no—
338C4—10041no—
339C4—5021no—
340C4—5021no—
341C4—5022no—
342C4—5022no—
343C4—2512no—
344C4C22×C4×D25111yescharacteristic
345C4#41×D25111yescharacteristic
346C2—42no—
347C2—5022no—
348C2C4×D4×D25111yescharacteristic
349C2#3063249311yescharacteristic
350C2#3063250111yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutatorC4×C251C23×C4
FrattiniC2×C4C23×D251
FittingC4×SD16×C251C2
socleC22×C251C22×D4
prime radicalC2×#16860809C22
cosocleC2×C4×C251C24
hypercenterC4×SD16D251
hypocenterC251#2947
LCSG ▷ C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
3. #30632163C2semidirect
4. #30631793C2semidirect
5. #30631792C2semidirect
6. #30631788C2semidirect
7. #30631787C2semidirect
8. C4×Dic1004C2semidirect
9. #30631779C2semidirect
10. #30631778C2semidirect
11. #30631726C2semidirect
13. #103×D251C2semidirect
14. #30631641C2semidirect
15. C4×SD16×C251C2semidirect
18. #22548744C4semidirect
66. Q8×D251C2×C4semidirect
73. Dic1004C2×C4semidirect
74. #16860800C2×C4semidirect
75. #16860798C2×C4semidirect
76. #16860793C2×C4semidirect
104. SD16×C251C2×C4semidirect
175. C251#2947semidirect
187. C4×SD16D251semidirect
255. SD16C4×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487442
#306321662
#306321632
#306317932
#306317922
#306317882
#306317872
C4×Dic10042
#306317792
#306317782
#306317262
#306316802
#103×D2512
#306316412
C4×SD16×C2512
#2947251