Finite Group Databasebeta

← #44558063

Subgroups — #44558063

Order 32128.

472 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30631629C211yesmaximal, characteristic, complemented
2#30631610C211yesmaximal, characteristic, complemented
3#30631610C211yesmaximal, characteristic, complemented
4#30631608C211yesmaximal, characteristic, complemented
5#30631599C211yesmaximal, characteristic, complemented
6#30631599C211yesmaximal, characteristic, complemented
7#30631590C211yesmaximal, characteristic
8#30631574C211yesmaximal, characteristic, complemented
9#30631570C211yesmaximal, characteristic, complemented
10#30631567C211yesmaximal, characteristic
11#30631562C211yesmaximal, characteristic, complemented
12#30631560C211yesmaximal, characteristic, complemented
13#30631547C211yesmaximal, characteristic, complemented
14#30631545C211yesmaximal, characteristic, complemented
15#386×C251C211yesmaximal, characteristic
16C22×Dic502C2211yescharacteristic
17C23×Dic251C2211yescharacteristic
18C2×C4⋊Dic251C2211yescharacteristic
19C2×C4⋊Dic251C2211yescharacteristic
20C2×C4⋊Dic251C2211yescharacteristic
21C2×C4⋊Dic251C2211yescharacteristic
22C2×Dic251⋊C4C2211yescharacteristic
23C2×Dic251⋊C4C2211yescharacteristic
24C2×Dic251⋊C4C2211yescharacteristic
25C2×Dic251⋊C4C2211yescharacteristic
26C2×C22⋊Dic251C2211yescharacteristic
27C2×C22⋊Dic251C2211yescharacteristic
28C2×C22⋊Dic251C2211yescharacteristic
29C2×C22⋊Dic251C2211yescharacteristic
30C2×C22⋊Dic251C2211yescharacteristic
31C2×C22⋊Dic251C2211yescharacteristic
32C2×C22⋊Dic251C2211yescharacteristic
33C2×C4×Dic251C2211yescharacteristic
34C251⋊#95C2211yescharacteristic
35C251⋊#95C2211yescharacteristic
36C251⋊#95C2211yescharacteristic
37C251⋊#95C2211yescharacteristic
38C251⋊#95C2211yescharacteristic
39C251⋊#95C2211yescharacteristic
40C251⋊#95C2211yescharacteristic
41C251⋊#95C2211yescharacteristic
42C251⋊#95C2211yescharacteristic
43C251⋊#95C2211yescharacteristic
44C22×D4×C251C2211yescharacteristic
45C23×C4×C251C2211yescharacteristic
46C2×C22⋊C4×C251C2211yescharacteristic
47C2×C22⋊C4×C251C2211yescharacteristic
48C2×C22⋊C4×C251C2211yescharacteristic
49#95×C251C2211yescharacteristic
50#95×C251C2211yescharacteristic
51C2×Dic502—41no—
52C2×Dic502—41no—
53C2×Dic502—41no—
54C22×Dic251—22no—
55C22×Dic251—41no—
56C22×Dic251—41no—
57C22×Dic251C2311yescharacteristic
58C22×Dic251C2311yescharacteristic
59C22×Dic251C2311yescharacteristic
60C22×Dic251C2311yescharacteristic
61C22×Dic251C2311yescharacteristic
62C22×Dic251C2311yescharacteristic
63C22×Dic251C2311yescharacteristic
64C22×Dic251C2311yescharacteristic
65C4⋊Dic251—22no—
66C4⋊Dic251—41no—
67C4⋊Dic251—41no—
68C4⋊Dic251—41no—
69Dic251⋊C4—22no—
70Dic251⋊C4—41no—
71Dic251⋊C4—41no—
72Dic251⋊C4—41no—
73C22⋊Dic251—22no—
74C22⋊Dic251—41no—
75C22⋊Dic251—41no—
76C22⋊Dic251—41no—
77C22⋊Dic251—41no—
78C22⋊Dic251—41no—
79C22⋊Dic251—41no—
80C4×Dic251—41no—
81C24×C251C2311yescharacteristic
82C24×C251C2311yescharacteristic
83C2×D4×C251—41no—
84C2×D4×C251—41no—
85C2×D4×C251Q814yescomplemented
86C22×C4×C251—22no—
87C22×C4×C251—41no—
88C22×C4×C251—41no—
89C22×C4×C251C2311yescharacteristic
90C22×C4×C251C2311yescharacteristic
91C22×C4×C251C2311yescharacteristic
92C22×C4×C251C2311yescharacteristic
93C22×C4×C251C2311yescharacteristic
94C22⋊C4×C251—22no—
95C22⋊C4×C251—41no—
96C22⋊C4×C251—41no—
97Dic502—44no—
98C2×Dic251—21no—
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×Dic251—21no—
118C2×Dic251—21no—
119C2×Dic251—21no—
120C2×Dic251—21no—
121C2×Dic251—44no—
122C2×Dic251#4112yes—
123C23×C251—21no—
124C23×C251—21no—
125C23×C251—21no—
126C23×C251—21no—
127C23×C251—21no—
128C23×C251—21no—
129C23×C251—21no—
130C23×C251—21no—
131C23×C251—21no—
132C23×C251—21no—
133C23×C251—21no—
134C23×C251—21no—
135C23×C251C2×Q812yes—
136C23×C251C2×Q812yes—
137C23×C251C2411yescharacteristic
138D4×C251—44no—
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×C251—21no—
146C2×C4×C251—21no—
147C2×C4×C251—21no—
148C2×C4×C251—21no—
149C2×C4×C251—21no—
150C2×C4×C251—21no—
151C2×C4×C251—21no—
152C2×C4×C251—44no—
153C2×C4×C251C2×Q812yes—
154C2×C4×C251#4112yes—
155C22×C251—22no—
156C22×C251—22no—
157C22×C251—22no—
158C22×C251—22no—
159C22×C251—22no—
160C22×C251—22no—
161C22×C251—41no—
162C22×C251—41no—
163C22×C251—41no—
164C22×C251—41no—
165C22×C251—41no—
166C22×C251—41no—
167C22×C251—41no—
168C22×C251—41no—
169C22×C251C22×Q811yescharacteristic
170C22×C251C2×#4111yescharacteristic
171C22×C251C2×#4111yescharacteristic
172C22×C251H(5,𝔽2)11yescharacteristic
173C22×C251H(5,𝔽2)11yescharacteristic
174C22×C251H(5,𝔽2)11yescharacteristic
175C22×C251H(5,𝔽2)11yescharacteristic
176C4×C251—22no—
177C4×C251—22no—
178C4×C251—41no—
179C4×C251—41no—
180C4×C251—41no—
181Dic251—22no—
182Dic251—41no—
183Dic251—41no—
184Dic251—41no—
185Dic251—41no—
186Dic251—41no—
187Dic251—41no—
188Dic251—41no—
189C2×C251—42no—
190C2×C251—42no—
191C2×C251#53811yescharacteristic
192C2×C251#53811yescharacteristic
193C2×C251#54311yescharacteristic
194C2×C251#55411yescharacteristic
195C2×C251#55411yescharacteristic
196C2×C251#55511yescharacteristic
197C2×C251#56011yescharacteristic
198C251#278411yescharacteristic, complemented, Sylow
199#2784—2511nomaximal, Sylow
200#400—2511no—
201#396—2511no—
202#396—2511no—
203#395—2511no—
204#393—2511no—
205#393—2511no—
206#390—2511no—
207#387—2511no—
208#387—2511no—
209#386—2511no—
210#386D25111yescharacteristic
211#385—2511no—
212#385—2511no—
213#380—2511no—
214#380—2511no—
215C22×Q8—2511no—
216C22×D4C2×D25111yescharacteristic
217C23×C4—2511no—
218C23×C4C2×D25111yescharacteristic
219C2×C4⋊C4—2511no—
220C2×C4⋊C4—2511no—
221C2×C4⋊C4—2511no—
222C2×C4⋊C4—2511no—
223C2×C4⋊C4—2511no—
224C2×C4⋊C4—2511no—
225C2×C4⋊C4—2511no—
226C2×C4⋊C4—2511no—
227C2×C22⋊C4—2511no—
228C2×C22⋊C4—2511no—
229C2×C22⋊C4—2511no—
230C2×C22⋊C4—2511no—
231C2×C22⋊C4—2511no—
232C2×C22⋊C4—2511no—
233C2×C22⋊C4—2511no—
234C2×C22⋊C4C2×D25111yescharacteristic
235C2×C22⋊C4C2×D25111yescharacteristic
236C2×C22⋊C4C2×D25111yescharacteristic
237C2×C42—2511no—
238#95—2511no—
239#95—2511no—
240#95—2511no—
241#95—2511no—
242#95—2511no—
243#95—2511no—
244#95—2511no—
245#95—2511no—
246#95—2511no—
247#95—2511no—
248#95C2×D25111yescharacteristic
249#95C2×D25111yescharacteristic
250C24C22×D25111yescharacteristic
251C24C22×D25111yescharacteristic
252C2×Q8—10041no—
253C2×Q8—10041no—
254C2×Q8—10041no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4Dic50214yescomplemented
258C22×C4—22no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—10041no—
262C22×C4—10041no—
263C22×C4—5022no—
264C22×C4—2511no—
265C22×C4—2511no—
266C22×C4—2511no—
267C22×C4—2511no—
268C22×C4—2511no—
269C22×C4—2511no—
270C22×C4—2511no—
271C22×C4—2511no—
272C22×C4C22×D25111yescharacteristic
273C22×C4C22×D25111yescharacteristic
274C22×C4C22×D25111yescharacteristic
275C22×C4C22×D25111yescharacteristic
276C22×C4C22×D25111yescharacteristic
277C4⋊C4—10041no—
278C4⋊C4—10041no—
279C4⋊C4—10041no—
280C4⋊C4—10041no—
281C4⋊C4—10041no—
282C4⋊C4—10041no—
283C4⋊C4—5022no—
284C4⋊C4—5022no—
285C22⋊C4—22no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—10041no—
289C22⋊C4—10041no—
290C22⋊C4—10041no—
291C22⋊C4—10041no—
292C22⋊C4—10041no—
293C22⋊C4—10041no—
294C22⋊C4—5022no—
295C42—10041no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23C2×Dic50212yes—
309C23C2×Dic50212yes—
310C23C23×D25111yescharacteristic
311Q8—10044no—
312D4—44no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—44no—
327C2×C4—10044no—
328C2×C4—5021no—
329C2×C4—5021no—
330C2×C4—5021no—
331C2×C4—5021no—
332C2×C4—5021no—
333C2×C4—5021no—
334C2×C4—5021no—
335C2×C4—5021no—
336C2×C4—5021no—
337C2×C4—5021no—
338C2×C4—5021no—
339C2×C4—5021no—
340C2×C4—5021no—
341C2×C4—5021no—
342C2×C4—5021no—
343C2×C4—5021no—
344C2×C4—5021no—
345C2×C4—5021no—
346C2×C4—5021no—
347C2×C4—5021no—
348C2×C4—5021no—
349C2×C4—5021no—
350C2×C4—5021no—
351C2×C4—2512no—
352C2×C4C2×Dic50212yes—
353C2×C4#1686080912yes—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22C22×Dic50211yescharacteristic
369C22#41×D25111yescharacteristic
370C22C2×#1686080911yescharacteristic
371C22#2254876011yescharacteristic
372C22#2254876011yescharacteristic
373C22#2254876011yescharacteristic
374C22#2254876111yescharacteristic
375C4—22no—
376C4—22no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—10041no—
381C4—10041no—
382C4—10041no—
383C4—10041no—
384C4—10041no—
385C4—10041no—
386C4—10041no—
387C4—5022no—
388C2—42no—
389C2—42no—
390C2#3063227211yescharacteristic
391C2#3063227311yescharacteristic
392C2#3063231311yescharacteristic
393C2#3063239411yescharacteristic
394C2#3063239611yescharacteristic
395C2#3063240111yescharacteristic
396C2#3063242511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#386×C251C2
prime radicalC22×D4×C251C22
hypercenter#386D251
hypocenterC251#2784
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. #30631629C2semidirect
2. #30631610C2semidirect
3. #30631610C2semidirect
4. #30631608C2semidirect
5. #30631599C2semidirect
6. #30631599C2semidirect
8. #30631574C2semidirect
9. #30631570C2semidirect
11. #30631562C2semidirect
12. #30631560C2semidirect
13. #30631547C2semidirect
14. #30631545C2semidirect
85. C2×D4×C251Q8semidirect
198. C251#2784semidirect
257. C2×D4Dic502semidirect

Maximal subgroups

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

subgroupindex
#306316292
#306316102
#306316082
#306315992
#306315902
#306315742
#306315702
#306315672
#306315622
#306315602
#306315472
#306315452
#386×C2512
#2784251