Finite Group Databasebeta

← #44550307

Subgroups — #44550307

Order 32128.

618 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D251C211yesmaximal, characteristic
2C2×#22548643C211yesmaximal, characteristic, complemented
3C2×#22548642C211yesmaximal, characteristic, complemented
4C2×#22548641C211yesmaximal, characteristic, complemented
5#30631650C211yesmaximal, characteristic, complemented
6C4×(C2×D251)⋊C4C211yesmaximal, characteristic, complemented
7C2×#104×C251C211yesmaximal, characteristic, complemented
8C2×#16860811C411yescharacteristic, complemented
9C2×#16860810—21no—
10C2×#16860810C2211yescharacteristic
11C2×#16860809—21no—
12C2×#16860809C411yescharacteristic, complemented
13#41×D251—22no—
14#41×D251—22no—
15#41×D251—22no—
16#41×D251—22no—
17C2×Q8×D251C411yescharacteristic, complemented
18C2×D4×D251—21no—
19C2×D4×D251C411yescharacteristic, complemented
20C22×C4×D251—21no—
21C22×C4×D251C2211yescharacteristic
22C2×#16860794C2211yescharacteristic, complemented
23C2×(C2×D251)⋊C4—21no—
24C2×C4×Dic251C2211yescharacteristic, complemented
25#22548643—22no—
26#22548642—22no—
27#22548641—22no—
28(C2×D251)⋊C8—21no—
29(C2×D251)⋊C8—21no—
30C251⋊#95—21no—
31C2×#41×C251C2211yescharacteristic
32C2×M16×C251C2211yescharacteristic, complemented
33C2×C42×C251C2211yescharacteristic, complemented
34#104×C251—22no—
35C23×D251—21no—
36C23×D251D411yescharacteristic
37#16860811—22no—
38#16860811—22no—
39#16860810—24no—
40#16860810—24no—
41#16860810—21no—
42#16860810—21no—
43#16860810D412yes—
44#16860810D412yes—
45#16860809—24no—
46#16860809—24no—
47#16860809—22no—
48#16860809—22no—
49C2×Dic502—21no—
50C2×Dic502C2×C411yescharacteristic
51Q8×D251—22no—
52Q8×D251—22no—
53C2×D1004—21no—
54C2×D1004C2×C411yescharacteristic
55C2×C22⋊D251—21no—
56C2×C22⋊D251—21no—
57C2×C22⋊D251—21no—
58D4×D251—24no—
59D4×D251—24no—
60D4×D251—22no—
61D4×D251—22no—
62C22×Dic251—21no—
63C22×Dic251—21no—
64C22×Dic251D411yescharacteristic
65C2×C4×D251—24no—
66C2×C4×D251—24no—
67C2×C4×D251—21no—
68C2×C4×D251—21no—
69C2×C4×D251—21no—
70C2×C4×D251—21no—
71C2×C4×D251—21no—
72C2×C4×D251—21no—
73C2×C4×D251—21no—
74C2×C4×D251C2×C411yescharacteristic
75C2×C4×D251C2×C411yescharacteristic
76C2×C4×D251D411yescharacteristic
77C2×C4×D251D411yescharacteristic
78#16860794—21no—
79#16860794—21no—
80C2×C251⋊C8—21no—
81(C2×D251)⋊C4—24no—
82C4×Dic251—21no—
83C4×Dic251—21no—
84#41×C251—21no—
85#41×C251—21no—
86#41×C251D412yes—
87#41×C251D412yes—
88C2×Q8×C251C2×C411yescharacteristic
89C2×D4×C251—21no—
90C2×D4×C251C2×C411yescharacteristic
91C22×C4×C251—21no—
92C22×C4×C251—21no—
93C22×C4×C251C2311yescharacteristic
94M16×C251—21no—
95M16×C251—21no—
96C2×C8×C251—21no—
97C42×C251—21no—
98C42×C251—21no—
99C22×D251—24no—
100C22×D251—21no—
101C22×D251—21no—
102C22×D251—21no—
103C22×D251—22no—
104C22×D251—41no—
105C22×D251—42no—
106C22×D251—42no—
107C22×D251C22⋊C411yescharacteristic
108C22×D251C22⋊C411yescharacteristic
109Dic502—24no—
110Dic502—21no—
111Dic502C22⋊C412yescomplemented
112D1004—24no—
113D1004—21no—
114D1004C22⋊C412yescomplemented
115C22⋊D251—24no—
116C22⋊D251—24no—
117C22⋊D251—22no—
118C22⋊D251—22no—
119C2×Dic251—24no—
120C2×Dic251—21no—
121C2×Dic251—21no—
122C2×Dic251—21no—
123C2×Dic251—22no—
124C2×Dic251—41no—
125C2×Dic251—41no—
126C2×Dic251C22⋊C411yescharacteristic
127C2×Dic251C22⋊C411yescharacteristic
128C4×D251—24no—
129C4×D251—24no—
130C4×D251—21no—
131C4×D251—21no—
132C4×D251—21no—
133C4×D251—21no—
134C4×D251—21no—
135C4×D251—21no—
136C4×D251—22no—
137C4×D251—22no—
138C4×D251—21no—
139C4×D251—21no—
140C251⋊C8—41no—
141C23×C251—21no—
142C23×C251C2×D411yescharacteristic
143Q8×C251—21no—
144Q8×C251C22⋊C412yescomplemented
145D4×C251—22no—
146D4×C251—22no—
147D4×C251—21no—
148D4×C251C22⋊C412yescomplemented
149C2×C4×C251—21no—
150C2×C4×C251—22no—
151C2×C4×C251—22no—
152C2×C4×C251—22no—
153C2×C4×C251—41no—
154C2×C4×C251—41no—
155C2×C4×C251C22×C411yescharacteristic
156C2×C4×C251C2×D411yescharacteristic
157C2×C4×C251C2×D411yescharacteristic
158C2×C4×C251C2×D411yescharacteristic
159C2×C4×C251C2×D411yescharacteristic
160C2×C4×C251C2×D411yescharacteristic
161C8×C251—41no—
162C22×C251—21no—
163C22×C251—21no—
164C22×C251—22no—
165C22×C251—41no—
166C22×C251—41no—
167C22×C251C2×C22⋊C411yescharacteristic
168C22×C251#12011yescharacteristic
169C22×C251#12011yescharacteristic
170C2×D251—21no—
171C2×D251—21no—
172C2×D251—21no—
173C2×D251—22no—
174C2×D251—21no—
175C2×D251—21no—
176C2×D251—42no—
177C2×D251—42no—
178C2×D251—48no—
179C2×D251—41no—
180C2×D251—41no—
181C2×D251#10412yescomplemented
182C4×C251—22no—
183C4×C251—42no—
184C4×C251C2×C22⋊C411yescharacteristic
185C4×C251C2×C22⋊C411yescharacteristic
186C4×C251#12011yescharacteristic
187C4×C251#12011yescharacteristic
188Dic251—21no—
189Dic251—22no—
190Dic251—21no—
191Dic251—42no—
192C2×C251—21no—
193C2×C251—21no—
194C2×C251—42no—
195C2×C251#37911yescharacteristic
196C2×C251C2×#10411yescharacteristic
197C2×C251#42111yescharacteristic
198D251—22no—
199D251—41no—
200D251—42no—
201C251#178911yescharacteristic, complemented, Sylow
202#1789—2511nomaximal, Sylow
203C22×#41—2511no—
204C2×#104—2511no—
205C2×#104—2511no—
206C2×#104—2511no—
207C2×#104D25111yescharacteristic, complemented
208#407—2511no—
209C4×C22⋊C4—2511no—
210C2×#41—5021no—
211C2×#41—5022no—
212C2×#41—5022no—
213C2×#41—5022no—
214C2×#41—5022no—
215C2×#41—5021no—
216C2×#41—2511no—
217C2×#41—2511no—
218C2×#41—2511no—
219C2×#41C2×D25111yescharacteristic
220C22×Q8—2511no—
221C22×D4—5021no—
222C22×D4—2511no—
223C23×C4—5021no—
224C23×C4—2511no—
225C2×M16—2511no—
226C2×M16C2×D25111yescharacteristic, complemented
227C2×C22⋊C4—5021no—
228C2×C42—2511no—
229C2×C42C2×D25111yescharacteristic, complemented
230#104—22no—
231#104—5022no—
232#104—5022no—
233#104—5022no—
234C22⋊C8—5021no—
235C22⋊C8—5021no—
236#95—5021no—
237C24—5021no—
238C24—2511no—
239#41—21no—
240#41—21no—
241#41—5024no—
242#41—5024no—
243#41—5024no—
244#41—5024no—
245#41—5022no—
246#41—5022no—
247#41—5022no—
248#41—5022no—
249#41—5021no—
250#41—5021no—
251#41—2512no—
252#41—2512no—
253#41C22⋊D25112yes—
254#41D100412yes—
255C2×Q8—5021no—
256C2×Q8—5022no—
257C2×Q8—5022no—
258C2×Q8—2511no—
259C2×Q8C4×D25111yescharacteristic
260C2×D4—21no—
261C2×D4—5021no—
262C2×D4—5024no—
263C2×D4—5024no—
264C2×D4—5022no—
265C2×D4—5021no—
266C2×D4—5021no—
267C2×D4—5022no—
268C2×D4—5021no—
269C2×D4—2511no—
270C2×D4C4×D25111yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—5024no—
274C22×C4—5024no—
275C22×C4—5021no—
276C22×C4—5021no—
277C22×C4—5021no—
278C22×C4—5021no—
279C22×C4—5021no—
280C22×C4—5021no—
281C22×C4—5021no—
282C22×C4—5021no—
283C22×C4—5021no—
284C22×C4—2511no—
285C22×C4—2511no—
286C22×C4—2511no—
287C22×C4—2511no—
288C22×C4—2511no—
289C22×C4C22×D25111yescharacteristic
290M16—21no—
291M16—21no—
292M16—5021no—
293M16—5021no—
294C2×C8—21no—
295C2×C8—5021no—
296C22⋊C4—5024no—
297C42—21no—
298C42—21no—
299C42—5021no—
300C42—5021no—
301C23—21no—
302C23—10041no—
303C23—10042no—
304C23—10042no—
305C23—5024no—
306C23—5022no—
307C23—5021no—
308C23—5021no—
309C23—5021no—
310C23—2511no—
311C23—2511no—
312C23D4×D25111yescharacteristic
313Q8—21no—
314Q8—5024no—
315Q8—5021no—
316Q8—2512no—
317Q8(C2×D251)⋊C412yescomplemented
318D4—22no—
319D4—22no—
320D4—21no—
321D4—5024no—
322D4—5024no—
323D4—5024no—
324D4—5022no—
325D4—5022no—
326D4—5021no—
327D4—2512no—
328D4(C2×D251)⋊C412yescomplemented
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—10041no—
336C2×C4—10041no—
337C2×C4—5024no—
338C2×C4—5024no—
339C2×C4—5022no—
340C2×C4—5024no—
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—5022no—
350C2×C4—5021no—
351C2×C4—5021no—
352C2×C4—5022no—
353C2×C4—5021no—
354C2×C4—2511no—
355C2×C4—2511no—
356C2×C4C2×C4×D25111yescharacteristic
357C2×C4D4×D25111yescharacteristic
358C2×C4D4×D25111yescharacteristic
359C2×C4D4×D25111yescharacteristic
360C2×C4C2×C22⋊D25111yescharacteristic
361C2×C4C2×D100411yescharacteristic
362C8—41no—
363C8—10041no—
364C22—21no—
365C22—21no—
366C22—22no—
367C22—41no—
368C22—41no—
369C22—10042no—
370C22—10042no—
371C22—10048no—
372C22—10041no—
373C22—10041no—
374C22—5021no—
375C22—5021no—
376C22—5021no—
377C22—5022no—
378C22—5021no—
379C22—5021no—
380C22—2512no—
381C22C22⋊C4×D25111yescharacteristic
382C22#2254868311yescharacteristic
383C22#2254868511yescharacteristic
384C4—22no—
385C4—42no—
386C4—10042no—
387C4—5021no—
388C4—5022no—
389C4—5021no—
390C4C22⋊C4×D25111yescharacteristic
391C4C2×(C2×D251)⋊C411yescharacteristic
392C4#2254868311yescharacteristic
393C4#2254868511yescharacteristic
394C2—21no—
395C2—21no—
396C2—42no—
397C2—10042no—
398C2—10041no—
399C2—5022no—
400C2#3063154211yescharacteristic
401C2#104×D25111yescharacteristic
402C2#3063170211yescharacteristic
subgroupG/H
centerC2×C4D4×D251
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
FittingC2×#104×C251C2
socleC22×C251C2×C22⋊C4
prime radicalC2×#41×D251C2
cosocleC22×C4×C251C23
hypercenterC2×#104D251
hypocenterC251#1789
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×#22548643C2semidirect
3. C2×#22548642C2semidirect
4. C2×#22548641C2semidirect
5. #30631650C2semidirect
6. C4×(C2×D251)⋊C4C2semidirect
7. C2×#104×C251C2semidirect
8. C2×#16860811C4semidirect
12. C2×#16860809C4semidirect
17. C2×Q8×D251C4semidirect
19. C2×D4×D251C4semidirect
22. C2×#16860794C22semidirect
24. C2×C4×Dic251C22semidirect
32. C2×M16×C251C22semidirect
33. C2×C42×C251C22semidirect
111. Dic502C22⋊C4semidirect
114. D1004C22⋊C4semidirect
144. Q8×C251C22⋊C4semidirect
148. D4×C251C22⋊C4semidirect
181. C2×D251#104semidirect
201. C251#1789semidirect
207. C2×#104D251semidirect
226. C2×M16C2×D251semidirect
229. C2×C42C2×D251semidirect
317. Q8(C2×D251)⋊C4semidirect
328. D4(C2×D251)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D2512
C2×#225486432
C2×#225486422
C2×#225486412
#306316502
C4×(C2×D251)⋊C42
C2×#104×C2512
#1789251