Finite Group Databasebeta

← #44552964

Subgroups — #44552964

Order 32128.

512 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
1C2×C4⋊C4×D251C211yesmaximal, characteristic
2#30631574C211yesmaximal, characteristic, complemented
3#30631573C211yesmaximal, characteristic, complemented
4#30631572C211yesmaximal, characteristic, complemented
5#30631571C211yesmaximal, characteristic, complemented
6#30631570C211yesmaximal, characteristic, complemented
7#30631569C211yesmaximal, characteristic, complemented
8#30631568C211yesmaximal, characteristic, complemented
9#30631549C211yesmaximal, characteristic
10#30631549C211yesmaximal, characteristic
11#30631546C211yesmaximal, characteristic
12#30631544C211yesmaximal, characteristic
13C4×(C2×D251)⋊C4C211yesmaximal, characteristic
14#30631534C211yesmaximal, characteristic
15#387×C251C211yesmaximal, characteristic, complemented
16C22×C4×D251C2211yescharacteristic
17C22×C4×D251C2211yescharacteristic
18C22×C4×D251C2211yescharacteristic
19C2×C4⋊Dic251C2211yescharacteristic
20C2×C4⋊Dic251C2211yescharacteristic
21C2×Dic251⋊C4C2211yescharacteristic
22C2×Dic251⋊C4C2211yescharacteristic
23C2×Dic251⋊C4C2211yescharacteristic
24C2×Dic251⋊C4C2211yescharacteristic
25C4⋊C4×D251C418yescomplemented
26C2×(C2×D251)⋊C4C2211yescharacteristic
27C2×(C2×D251)⋊C4C2211yescharacteristic
28C2×(C2×D251)⋊C4C2211yescharacteristic
29C2×(C2×D251)⋊C4C2211yescharacteristic
30C2×C4×Dic251C2211yescharacteristic
31C2×C4×Dic251C2211yescharacteristic
32C2×C4×Dic251C2211yescharacteristic
33C251⋊#95C2211yescharacteristic
34C251⋊#95C2211yescharacteristic
35C251⋊#95C2211yescharacteristic
36C251⋊#95C2211yescharacteristic
37C251⋊#95C2211yescharacteristic
38C251⋊#95C2211yescharacteristic
39C251⋊#95C2211yescharacteristic
40C251⋊#95C2211yescharacteristic
41C251⋊#95C2211yescharacteristic
42C251⋊#95C2211yescharacteristic
43C251⋊#95C2211yescharacteristic
44C251⋊#95C2211yescharacteristic
45C2×C4⋊C4×C251C2211yescharacteristic
46C2×C4⋊C4×C251C2211yescharacteristic
47C2×C42×C251C2211yescharacteristic
48#95×C251C2211yescharacteristic
49#95×C251C2211yescharacteristic
50#95×C251C2211yescharacteristic
51#95×C251C2211yescharacteristic
52C23×D251C2311yescharacteristic
53C22×Dic251C2311yescharacteristic
54C22×Dic251C2311yescharacteristic
55C22×Dic251C2311yescharacteristic
56C22×Dic251C2311yescharacteristic
57C22×Dic251C2311yescharacteristic
58C22×Dic251C2311yescharacteristic
59C22×Dic251C2311yescharacteristic
60C2×C4×D251—22no—
61C2×C4×D251—22no—
62C2×C4×D251—41no—
63C2×C4×D251—41no—
64C2×C4×D251—41no—
65C2×C4×D251—41no—
66C2×C4×D251C2×C414yes—
67C2×C4×D251C2×C414yes—
68C2×C4×D251C2×C414yes—
69C4⋊Dic251—41no—
70C4⋊Dic251C2×C414yes—
71Dic251⋊C4—41no—
72Dic251⋊C4—41no—
73Dic251⋊C4C2×C414yes—
74Dic251⋊C4C2×C414yes—
75(C2×D251)⋊C4—22no—
76(C2×D251)⋊C4—22no—
77(C2×D251)⋊C4—22no—
78(C2×D251)⋊C4—22no—
79C4×Dic251—41no—
80C4×Dic251—41no—
81C4×Dic251—41no—
82C22×C4×C251C2311yescharacteristic
83C22×C4×C251C2311yescharacteristic
84C22×C4×C251C2311yescharacteristic
85C22×C4×C251C2311yescharacteristic
86C22×C4×C251C2311yescharacteristic
87C22×C4×C251C2311yescharacteristic
88C22×C4×C251C2311yescharacteristic
89C4⋊C4×C251—41no—
90C4⋊C4×C251C2×C414yes—
91C42×C251—41no—
92C22×D251—21no—
93C22×D251—21no—
94C22×D251—21no—
95C22×D251—21no—
96C22×D251C22×C412yes—
97C22×D251#4112yes—
98C22×D251#4112yes—
99C2×Dic251—21no—
100C2×Dic251—21no—
101C2×Dic251—21no—
102C2×Dic251—21no—
103C2×Dic251—21no—
104C2×Dic251—21no—
105C2×Dic251—21no—
106C2×Dic251—21no—
107C2×Dic251—21no—
108C2×Dic251—21no—
109C2×Dic251—21no—
110C2×Dic251—21no—
111C2×Dic251—21no—
112C2×Dic251—21no—
113C2×Dic251—21no—
114C2×Dic251—21no—
115C2×Dic251—21no—
116C2×Dic251—21no—
117C2×Dic251C22×C412yes—
118C2×Dic251C22×C412yes—
119C2×Dic251C22×C412yes—
120C4×D251—28no—
121C4×D251—44no—
122C4×D251—44no—
123C23×C251C2411yescharacteristic
124C2×C4×C251—21no—
125C2×C4×C251—21no—
126C2×C4×C251—21no—
127C2×C4×C251—21no—
128C2×C4×C251—21no—
129C2×C4×C251—21no—
130C2×C4×C251—21no—
131C2×C4×C251—21no—
132C2×C4×C251—21no—
133C2×C4×C251—21no—
134C2×C4×C251—21no—
135C2×C4×C251—21no—
136C2×C4×C251—21no—
137C2×C4×C251—21no—
138C2×C4×C251—21no—
139C2×C4×C251—21no—
140C2×C4×C251—21no—
141C2×C4×C251—21no—
142C2×C4×C251C22×C412yes—
143C2×C4×C251C22×C412yes—
144C2×C4×C251C22×C412yes—
145C22×C251C23×C411yescharacteristic
146C22×C251C2×#4111yescharacteristic
147C22×C251C2×#4111yescharacteristic
148C22×C251H(5,𝔽2)11yescharacteristic
149C22×C251H(5,𝔽2)11yescharacteristic
150C22×C2512−1+411yescharacteristic
151C22×C2512−1+411yescharacteristic
152C2×D251—22no—
153C2×D251—22no—
154C2×D251—22no—
155C2×D251—22no—
156C2×D251—22no—
157C2×D251—22no—
158C2×D251#11714yes—
159C4×C251—22no—
160C4×C251—22no—
161C4×C251—22no—
162C4×C251—41no—
163C4×C251—41no—
164C4×C251—41no—
165C4×C251—41no—
166Dic251—22no—
167Dic251—22no—
168Dic251—22no—
169Dic251—41no—
170Dic251—41no—
171Dic251—41no—
172Dic251—41no—
173C2×C251C2×#11711yescharacteristic
174C2×C251#51811yescharacteristic
175C2×C251#51911yescharacteristic
176C2×C251#53911yescharacteristic
177C2×C251#53911yescharacteristic
178C2×C251#53911yescharacteristic
179C2×C251#53911yescharacteristic
180D251—24no—
181C251#231311yescharacteristic, complemented, Sylow
182#2313—2511nomaximal, Sylow
183C22×C4⋊C4—2511no—
184#387—2511no—
185#387—2511no—
186#387—2511no—
187#387—2511no—
188#387—2511no—
189#387—2511no—
190#387—2511no—
191#387D25111yescharacteristic, complemented
192#381—2511no—
193#381—2511no—
194#380—2511no—
195#380—2511no—
196C4×C22⋊C4—2511no—
197C4×C22⋊C4—2511no—
198C23×C4—2511no—
199C23×C4—2511no—
200C23×C4—2511no—
201C2×C4⋊C4—2511no—
202C2×C4⋊C4—2511no—
203C2×C4⋊C4—2511no—
204C2×C4⋊C4—2511no—
205C2×C4⋊C4—2511no—
206C2×C4⋊C4—2511no—
207C2×C4⋊C4—2518no—
208C2×C4⋊C4C2×D25111yescharacteristic
209C2×C4⋊C4C2×D25111yescharacteristic
210C2×C22⋊C4—2511no—
211C2×C22⋊C4—2511no—
212C2×C22⋊C4—2511no—
213C2×C22⋊C4—2511no—
214C2×C42—2511no—
215C2×C42—2511no—
216C2×C42—2511no—
217C2×C42C2×D25111yescharacteristic
218#95—2511no—
219#95—2511no—
220#95—2511no—
221#95—2511no—
222#95—2511no—
223#95—2511no—
224#95—2511no—
225#95—2511no—
226#95—2511no—
227#95—2511no—
228#95—2511no—
229#95—2511no—
230#95C2×D25111yescharacteristic
231#95C2×D25111yescharacteristic
232#95C2×D25111yescharacteristic
233#95C2×D25111yescharacteristic
234C24—2511no—
235C22×C4—10041no—
236C22×C4—10041no—
237C22×C4—10041no—
238C22×C4—10041no—
239C22×C4—5022no—
240C22×C4—5022no—
241C22×C4—2514no—
242C22×C4—2514no—
243C22×C4—2511no—
244C22×C4—2511no—
245C22×C4—2511no—
246C22×C4—2511no—
247C22×C4—2511no—
248C22×C4—2511no—
249C22×C4—2511no—
250C22×C4—2514no—
251C22×C4C22×D25111yescharacteristic
252C22×C4C22×D25111yescharacteristic
253C22×C4C22×D25111yescharacteristic
254C22×C4C22×D25111yescharacteristic
255C22×C4C22×D25111yescharacteristic
256C22×C4C22×D25111yescharacteristic
257C22×C4C22×D25111yescharacteristic
258C4⋊C4—41no—
259C4⋊C4—10041no—
260C4⋊C4—10041no—
261C4⋊C4—10041no—
262C4⋊C4—2514no—
263C4⋊C4—2514no—
264C4⋊C4—2514no—
265C4⋊C4C4×D25114yes—
266C22⋊C4—5022no—
267C22⋊C4—5022no—
268C22⋊C4—5022no—
269C22⋊C4—5022no—
270C42—41no—
271C42—10041no—
272C42—10041no—
273C42—10041no—
274C23—5021no—
275C23—5021no—
276C23—5021no—
277C23—5021no—
278C23—2512no—
279C23—2512no—
280C23—2512no—
281C23C23×D25111yescharacteristic
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—10044no—
301C2×C4—10044no—
302C2×C4—5028no—
303C2×C4—5021no—
304C2×C4—5021no—
305C2×C4—5021no—
306C2×C4—5021no—
307C2×C4—5021no—
308C2×C4—5021no—
309C2×C4—5021no—
310C2×C4—5021no—
311C2×C4—5021no—
312C2×C4—5021no—
313C2×C4—5021no—
314C2×C4—5021no—
315C2×C4—5021no—
316C2×C4—5021no—
317C2×C4—5021no—
318C2×C4—5021no—
319C2×C4—5021no—
320C2×C4—5021no—
321C2×C4—2512no—
322C2×C4—2512no—
323C2×C4—2512no—
324C2×C4C2×C4×D25112yes—
325C2×C4C2×C4×D25112yes—
326C2×C4C2×C4×D25112yes—
327C22—5022no—
328C22—5022no—
329C22—5022no—
330C22—5022no—
331C22—5022no—
332C22—5022no—
333C22—2514no—
334C22C22×C4×D25111yescharacteristic
335C22#41×D25111yescharacteristic
336C22#41×D25111yescharacteristic
337C22#2254876011yescharacteristic
338C22#2254876111yescharacteristic
339C22#2254876211yescharacteristic
340C22#2254876311yescharacteristic
341C4—22no—
342C4—22no—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—10041no—
349C4—10041no—
350C4—10041no—
351C4—10041no—
352C4—5022no—
353C4—5022no—
354C4—5022no—
355C2—5024no—
356C2#117×D25111yescharacteristic
357C2#3063215911yescharacteristic
358C2#3063216311yescharacteristic
359C2#3063227811yescharacteristic
360C2#3063228011yescharacteristic
361C2#3063228311yescharacteristic
362C2#3063228511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutatorC22×C251C23×C4
Fitting#387×C251C2
socle, cosocleC23×C251C24
prime radicalC23×D251C23
hypercenter#387D251
hypocenterC251#2313
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. #30631574C2semidirect
3. #30631573C2semidirect
4. #30631572C2semidirect
5. #30631571C2semidirect
6. #30631570C2semidirect
7. #30631569C2semidirect
8. #30631568C2semidirect
15. #387×C251C2semidirect
25. C4⋊C4×D251C4semidirect
181. C251#2313semidirect
191. #387D251semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D2512
#306315742
#306315732
#306315722
#306315712
#306315702
#306315692
#306315682
#306315492
#306315462
#306315442
C4×(C2×D251)⋊C42
#306315342
#387×C2512
#2313251