Finite Group Databasebeta

← #44560346

Subgroups — #44560346

Order 32128.

398 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632267C211yesmaximal, characteristic
2#30632167C211yesmaximal, characteristic, complemented
3#30632027C211yesmaximal, characteristic, complemented
4#30632021C211yesmaximal, characteristic, complemented
5#30632020C211yesmaximal, characteristic, complemented
6#30632013C211yesmaximal, characteristic, complemented
7#30632011C211yesmaximal, characteristic, complemented
8#30632002C211yesmaximal, characteristic, complemented
9#30632001C211yesmaximal, characteristic, complemented
10#30631726C211yesmaximal, characteristic, complemented
11#107×D251C211yesmaximal, characteristic, complemented
12#30631687C211yesmaximal, characteristic, complemented
13#30631682C211yesmaximal, characteristic, complemented
14#30631652C211yesmaximal, characteristic, complemented
15#485×C251C211yesmaximal, characteristic, complemented
16C2×#16860811C2211yescharacteristic
17C2×#16860810C2211yescharacteristic
18C2×#16860793C2211yescharacteristic
19C2×C8×D251C2211yescharacteristic
20#22548693—21no—
21#22548691C2211yescharacteristic
22#22548690—21no—
23#22548690C2211yescharacteristic
24#22548689C2211yescharacteristic
25#22548688—21no—
26#22548688—21no—
27C4×Dic502—21no—
28C4×D1004—21no—
29(C22⋊D251)⋊C4—21no—
30(C22⋊D251)⋊C4—21no—
31#22548673C2211yescharacteristic
32#22548673C2211yescharacteristic
33C2×C4⋊Dic251C2211yescharacteristic
34C4⋊C4×D251C2211yescharacteristic
35C4⋊C4×D251C2211yescharacteristic
36C4⋊C4×D251C2211yescharacteristic
37#22548649C2211yescharacteristic
38#22548649C2211yescharacteristic
39#22548648C2211yescharacteristic
40#22548648C2211yescharacteristic
41#22548648C2211yescharacteristic
42#22548646C2211yescharacteristic
43#22548639C2211yescharacteristic
44#22548639C2211yescharacteristic
45#22548637C2211yescharacteristic
46#22548637C2211yescharacteristic
47#22548635C2211yescharacteristic
48#22548635C2211yescharacteristic
49C22⋊(C251⋊C8)C2211yescharacteristic
50(C2×D251)⋊C8C2211yescharacteristic
51(C2×D251)⋊C8C2211yescharacteristic
52C22⋊Q8×C251C2211yescharacteristic
53#117×C251C2211yescharacteristic
54#107×C251C2211yescharacteristic
55#106×C251C2211yescharacteristic
56#103×C251C2211yescharacteristic
57#103×C251C2211yescharacteristic
58C22⋊C8×C251C2211yescharacteristic
59#16860811—22no—
60#16860811—22no—
61#16860810—22no—
62#16860810—22no—
63C2×Dic502D411yescharacteristic
64C2×D1004—21no—
65C2×D1004—21no—
66C2×D1004D411yescharacteristic
67C2×D1004C2311yescharacteristic
68C2×C22⋊D251—21no—
69C2×C22⋊D251D411yescharacteristic
70C2×C22⋊D251D411yescharacteristic
71C22×Dic251—21no—
72C2×C4×D251—21no—
73C2×C4×D251—21no—
74C2×C4×D251—21no—
75C2×C4×D251—21no—
76C2×C4×D251C2311yescharacteristic
77C2×C4×D251C2311yescharacteristic
78#16860793—22no—
79C2×C251⋊C8C2311yescharacteristic
80C2×C251⋊C8C2311yescharacteristic
81C8×D251—22no—
82C4⋊Dic251—21no—
83C4⋊Dic251C2311yescharacteristic
84C4⋊Dic251C2311yescharacteristic
85C4⋊Dic251C2311yescharacteristic
86Dic251⋊C4—21no—
87Dic251⋊C4—21no—
88Dic251⋊C4—21no—
89Dic251⋊C4—21no—
90(C2×D251)⋊C4—21no—
91(C2×D251)⋊C4—21no—
92(C2×D251)⋊C4—21no—
93(C2×D251)⋊C4—21no—
94(C2×D251)⋊C4—21no—
95C4×Dic251—21no—
96C4×Dic251—21no—
97C2×Q8×C251C2311yescharacteristic
98C22×C4×C251C2311yescharacteristic
99C2×C8×C251C2311yescharacteristic
100C2×C8×C251C2311yescharacteristic
101C4⋊C4×C251—21no—
102C4⋊C4×C251C2311yescharacteristic
103C4⋊C4×C251C2311yescharacteristic
104C4⋊C4×C251C2311yescharacteristic
105C22⋊C4×C251—21no—
106C22⋊C4×C251—21no—
107C42×C251—21no—
108C22×D251—21no—
109C22×D251C2×D411yescharacteristic
110C22×D251C2×D411yescharacteristic
111Dic502—22no—
112D1004—22no—
113D1004—21no—
114D1004—21no—
115D1004—42no—
116D1004—81no—
117C22⋊D251—22no—
118C22⋊D251—22no—
119C22⋊D251—81no—
120C2×Dic251—21no—
121C2×Dic251—21no—
122C2×Dic251—21no—
123C2×Dic251—81no—
124C2×Dic251C2×D411yescharacteristic
125C2×Dic251C2×D411yescharacteristic
126C4×D251—21no—
127C4×D251—21no—
128C4×D251—42no—
129C4×D251—81no—
130C4×D251—81no—
131C4×D251—81no—
132C4×D251#4112yes—
133C4×D251#4112yes—
134C251⋊C8—21no—
135C251⋊C8—21no—
136C23×C251C2×D411yescharacteristic
137Q8×C251—21no—
138Q8×C251—21no—
139C2×C4×C251—21no—
140C2×C4×C251—21no—
141C2×C4×C251—21no—
142C2×C4×C251—21no—
143C2×C4×C251—21no—
144C2×C4×C251—21no—
145C2×C4×C251C2×D411yescharacteristic
146C2×C4×C251C2411yescharacteristic
147C8×C251—21no—
148C8×C251—21no—
149C22×C251—21no—
150C22×C251—21no—
151C22×C251—21no—
152C22×C251C22×D411yescharacteristic
153C2×D251—21no—
154C2×D251—21no—
155C2×D251—21no—
156C2×D251—21no—
157C2×D251—21no—
158C2×D251—41no—
159C2×D251—41no—
160C2×D251—41no—
161C2×D251#12312yes—
162C4×C251—21no—
163C4×C251—41no—
164C4×C251—41no—
165C4×C251—41no—
166C4×C251—41no—
167C4×C251C2×#4111yescharacteristic
168C4×C251C2×#4111yescharacteristic
169Dic251—21no—
170Dic251—41no—
171Dic251—41no—
172Dic251—41no—
173Dic251#12312yes—
174C2×C251—41no—
175C2×C251C2×#12311yescharacteristic
176C2×C251#57611yescharacteristic
177C2×C251#57811yescharacteristic
178D251—22no—
179D251—41no—
180D251—81no—
181C251#309811yescharacteristic, complemented, Sylow
182#3098—2511nomaximal, Sylow
183#537—2511no—
184#520—2511no—
185#485—2511no—
186#485D25111yescharacteristic, complemented
187#484—2511no—
188#484—2511no—
189#482—2511no—
190#482—2511no—
191#480—2511no—
192#480—2511no—
193#428—2511no—
194C2×#107—2511no—
195#417—2511no—
196#416—2511no—
197#408—2511no—
198C2×#41—2511no—
199C2×#41—2511no—
200C2×M16—2511no—
201C22×C8—2511no—
202C22⋊Q8—5021no—
203C22⋊Q8—2511no—
204C22⋊Q8C2×D25111yescharacteristic
205#121—5021no—
206#121—5021no—
207#121—5021no—
208#121—2511no—
209#121—2511no—
210C4×Q8—5021no—
211C4×D4—5021no—
212C4×D4—5021no—
213C4×D4—5021no—
214#117—2511no—
215#117—2511no—
216#117C2×D25111yescharacteristic
217C2×C4⋊C4—2511no—
218C2×C4⋊C4—2511no—
219C2×C4⋊C4—2511no—
220C2×C4⋊C4—2511no—
221#107—2511no—
222#107—2511no—
223#107—2511no—
224#107—2511no—
225#107—2511no—
226#107C2×D25111yescharacteristic
227#106—2511no—
228#106C2×D25111yescharacteristic
229#103—2511no—
230#103—2511no—
231#103C2×D25111yescharacteristic
232#103C2×D25111yescharacteristic
233#102—2511no—
234#102—2511no—
235#102—2511no—
236#102—2511no—
237C22⋊C8—2511no—
238C22⋊C8—2511no—
239C22⋊C8—2511no—
240C22⋊C8C2×D25111yescharacteristic
241#41—5022no—
242#41—5022no—
243#41—5022no—
244#41—5022no—
245C2×Q8—2511no—
246C2×Q8C22×D25111yescharacteristic
247C2×D4—5021no—
248C2×D4—5021no—
249C2×D4—5021no—
250C2×D4—2511no—
251C2×D4—2511no—
252C2×D4—2511no—
253C2×D4—2511no—
254C22×C4—5021no—
255C22×C4—5021no—
256C22×C4—5021no—
257C22×C4—5021no—
258C22×C4—5021no—
259C22×C4—2511no—
260C22×C4—2511no—
261C22×C4C22×D25111yescharacteristic
262M16—5022no—
263C2×C8—5022no—
264C2×C8—2511no—
265C2×C8—2511no—
266C2×C8C22×D25111yescharacteristic
267C2×C8C22×D25111yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—5021no—
270C4⋊C4—5021no—
271C4⋊C4—5021no—
272C4⋊C4—5021no—
273C4⋊C4—5021no—
274C4⋊C4—2511no—
275C4⋊C4—2511no—
276C4⋊C4—2511no—
277C4⋊C4C22×D25111yescharacteristic
278C4⋊C4C22×D25111yescharacteristic
279C4⋊C4C22×D25111yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—21no—
282C22⋊C4—5021no—
283C22⋊C4—5021no—
284C22⋊C4—5021no—
285C22⋊C4—5021no—
286C22⋊C4—5021no—
287C42—21no—
288C42—5021no—
289C42—5021no—
290C23—5021no—
291C23—2511no—
292C23—2511no—
293C23D4×D25111yescharacteristic
294Q8—21no—
295Q8—21no—
296Q8—5022no—
297D4—20081no—
298D4—20081no—
299D4—10042no—
300D4—5022no—
301D4—5022no—
302D4—5022no—
303D4—5021no—
304D4—5021no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—20081no—
312C2×C4—20081no—
313C2×C4—20081no—
314C2×C4—20081no—
315C2×C4—10042no—
316C2×C4—5021no—
317C2×C4—5021no—
318C2×C4—5021no—
319C2×C4—5021no—
320C2×C4—5021no—
321C2×C4—2511no—
322C2×C4—2511no—
323C2×C4—2512no—
324C2×C4—2512no—
325C2×C4D4×D25111yescharacteristic
326C2×C4C23×D25111yescharacteristic
327C8—21no—
328C8—21no—
329C8—5021no—
330C8—5021no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—10041no—
335C22—10041no—
336C22—10041no—
337C22—5021no—
338C22—5021no—
339C22—5021no—
340C22—5021no—
341C22—5021no—
342C22—2512no—
343C22C2×D4×D25111yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—10041no—
350C4—10041no—
351C4—10041no—
352C4—5021no—
353C4—2512no—
354C4#41×D25111yescharacteristic
355C4#41×D25111yescharacteristic
356C2—41no—
357C2—20081no—
358C2—10041no—
359C2—5022no—
360C2#123×D25111yescharacteristic
361C2#3063248911yescharacteristic
362C2#3063250311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#485×C251C2
socleC22×C251C22×D4
prime radical#30632267C2
hypercenter#485D251
hypocenterC251#3098
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
2. #30632167C2semidirect
3. #30632027C2semidirect
4. #30632021C2semidirect
5. #30632020C2semidirect
6. #30632013C2semidirect
7. #30632011C2semidirect
8. #30632002C2semidirect
9. #30632001C2semidirect
10. #30631726C2semidirect
11. #107×D251C2semidirect
12. #30631687C2semidirect
13. #30631682C2semidirect
14. #30631652C2semidirect
15. #485×C251C2semidirect
181. C251#3098semidirect
186. #485D251semidirect

Maximal subgroups

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

subgroupindex
#306322672
#306321672
#306320272
#306320212
#306320202
#306320132
#306320112
#306320022
#306320012
#306317262
#107×D2512
#306316872
#306316822
#306316522
#485×C2512
#3098251