Finite Group Databasebeta

← #44561921

Subgroups — #44561921

Order 32128.

478 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548746C211yesmaximal, characteristic, complemented
2C2×#22548735C211yesmaximal, characteristic, complemented
3#30632342C211yesmaximal, characteristic, complemented
4#30632082C211yesmaximal, characteristic, complemented
5#30631931C211yesmaximal, characteristic, complemented
6#30631926C211yesmaximal, characteristic, complemented
7#30631899C211yesmaximal, characteristic, complemented
8#30631897C211yesmaximal, characteristic, complemented
9#30631849C211yesmaximal, characteristic, complemented
10#30631847C211yesmaximal, characteristic, complemented
11#30631843C211yesmaximal, characteristic, complemented
12#30631830C211yesmaximal, characteristic, complemented
13#30631785C211yesmaximal, characteristic, complemented
14#30631759C211yesmaximal, characteristic, complemented
15#548×C251C211yesmaximal, characteristic, complemented
16C2×#16860810C2211yescharacteristic
17C2×#16860810C2211yescharacteristic
18#22548746—24no—
19#22548735—24no—
20C2×#16860800C2211yescharacteristic
21C2×#16860800C2211yescharacteristic
22C2×#16860800C2211yescharacteristic
23C2×#16860800C2211yescharacteristic
24C2×#16860798C2211yescharacteristic
25C2×#16860798C2211yescharacteristic
26C2×#16860797C2211yescharacteristic
27C2×#16860797C2211yescharacteristic
28C2×#16860795C2211yescharacteristic
29C2×#16860794C2211yescharacteristic
30C22×C251⋊C8C2211yescharacteristic
31#22548715C2211yescharacteristic
32#22548697—21no—
33#22548697—21no—
34#22548694C2211yescharacteristic
35#22548694C2211yescharacteristic
36#22548686—21no—
37C4×D1004C2211yescharacteristic
38C4⋊C4×D251—21no—
39#22548648C2211yescharacteristic
40C4⋊(C251⋊C8)C2211yescharacteristic
41#22548639C2211yescharacteristic
42#22548639C2211yescharacteristic
43#22548638C2211yescharacteristic
44#22548638C2211yescharacteristic
45#22548636C2211yescharacteristic
46#22548635C2211yescharacteristic
47C22⋊(C251⋊C8)C2211yescharacteristic
48C22⋊(C251⋊C8)C2211yescharacteristic
49#22548626C2211yescharacteristic
50C2×#41×C251C2211yescharacteristic
51C22×Q8×C251C2211yescharacteristic
52#124×C251—21no—
53#124×C251C2211yescharacteristic
54C22⋊Q8×C251—21no—
55C22⋊Q8×C251C2211yescharacteristic
56#121×C251—21no—
57#121×C251C2211yescharacteristic
58C4×Q8×C251C2211yescharacteristic
59C4×D4×C251—21no—
60C4×D4×C251C2211yescharacteristic
61#16860810—22no—
62#16860810—22no—
63#16860810—41no—
64#16860810—41no—
65C2×Dic502C2311yescharacteristic
66C2×Dic502C2311yescharacteristic
67C2×D1004C2311yescharacteristic
68C2×C22⋊D251—21no—
69C2×C22⋊D251—21no—
70C2×C4×D251—21no—
71C2×C4×D251—21no—
72C2×C4×D251—21no—
73#16860800—22no—
74#16860800—22no—
75#16860800—22no—
76#16860800—41no—
77#16860798—22no—
78#16860798—41no—
79#16860797—22no—
80#16860797D414yescomplemented
81#16860795—22no—
82#16860794—22no—
83C2×C251⋊C8—22no—
84C2×C251⋊C8C2311yescharacteristic
85C2×C251⋊C8C2311yescharacteristic
86C2×C251⋊C8C2311yescharacteristic
87C2×C251⋊C8C2311yescharacteristic
88C4⋊Dic251—21no—
89C4⋊Dic251—21no—
90C4⋊Dic251C2311yescharacteristic
91Dic251⋊C4—21no—
92Dic251⋊C4—21no—
93C22⋊Dic251—21no—
94C22⋊Dic251—21no—
95(C2×D251)⋊C4—21no—
96#41×C251—22no—
97#41×C251—41no—
98C2×Q8×C251—21no—
99C2×Q8×C251—22no—
100C2×Q8×C251—41no—
101C2×Q8×C251C2311yescharacteristic
102C2×Q8×C251C2311yescharacteristic
103C2×D4×C251—21no—
104C2×D4×C251—21no—
105C2×D4×C251D411yescharacteristic
106C2×D4×C251C2311yescharacteristic
107C22×C4×C251—21no—
108C22×C4×C251—21no—
109C22×C4×C251C2311yescharacteristic
110C22×C4×C251C2311yescharacteristic
111C4⋊C4×C251—21no—
112C4⋊C4×C251—21no—
113C4⋊C4×C251D411yescharacteristic
114C4⋊C4×C251C2311yescharacteristic
115C22⋊C4×C251—21no—
116C22⋊C4×C251—21no—
117C22⋊C4×C251—21no—
118C22⋊C4×C251—21no—
119C22⋊C4×C251D411yescharacteristic
120C22⋊C4×C251D411yescharacteristic
121C42×C251—21no—
122C42×C251C2311yescharacteristic
123C22×D251—21no—
124Dic502—21no—
125Dic502—21no—
126Dic502—21no—
127Dic502—21no—
128D1004—21no—
129D1004C2×D412yes—
130C22⋊D251—42no—
131C22⋊D251—42no—
132C2×Dic251—21no—
133C2×Dic251—21no—
134C2×Dic251—21no—
135C4×D251—42no—
136C4×D251—42no—
137C4×D251—42no—
138C251⋊C8—21no—
139C251⋊C8—21no—
140C251⋊C8—21no—
141C251⋊C8C2×D412yes—
142C23×C251—21no—
143C23×C251C2×D411yescharacteristic
144C23×C251C2×D411yescharacteristic
145Q8×C251—21no—
146Q8×C251—21no—
147Q8×C251—21no—
148Q8×C251—42no—
149Q8×C251C2×D412yes—
150D4×C251—21no—
151D4×C251—21no—
152D4×C251—41no—
153D4×C251—42no—
154D4×C251—42no—
155C2×C4×C251—21no—
156C2×C4×C251—21no—
157C2×C4×C251—21no—
158C2×C4×C251—21no—
159C2×C4×C251—21no—
160C2×C4×C251—21no—
161C2×C4×C251—21no—
162C2×C4×C251—42no—
163C2×C4×C251—42no—
164C2×C4×C251C2×D411yescharacteristic
165C2×C4×C251C2×D411yescharacteristic
166C2×C4×C251C2×D411yescharacteristic
167C2×C4×C251C2×D411yescharacteristic
168C2×C4×C251C2411yescharacteristic
169C22×C251—21no—
170C22×C251—21no—
171C22×C251—21no—
172C22×C251—21no—
173C22×C251—21no—
174C22×C251—21no—
175C22×C251—41no—
176C22×C251—41no—
177C22×C251—41no—
178C22×C251C22×D411yescharacteristic
179C2×D251—22no—
180C2×D251—41no—
181C2×D251—41no—
182C4×C251—21no—
183C4×C251—21no—
184C4×C251—22no—
185C4×C251—21no—
186C4×C251—21no—
187C4×C251—41no—
188C4×C251—41no—
189C4×C251C22×D411yescharacteristic
190C4×C251H(5,𝔽2)11yescharacteristic
191Dic251—41no—
192Dic251—41no—
193Dic251—41no—
194C2×C251—41no—
195C2×C251—41no—
196C2×C251—81no—
197C2×C251D4211yescharacteristic
198C2×C251#57511yescharacteristic
199C2×C251#57811yescharacteristic
200D251—42no—
201C251#326911yescharacteristic, complemented, Sylow
202#3269—2511nomaximal, Sylow
203C2×#137—2511no—
204C2×#135—2511no—
205#548D25111yescharacteristic, complemented
206#547—2511no—
207#497—2511no—
208#468—2511no—
209#467—2511no—
210#463—2511no—
211#462—2511no—
212#452—2511no—
213#452—2511no—
214#451—2511no—
215#449—2511no—
216#440—2511no—
217#435—2511no—
218C2×#41—2511no—
219C2×#41—2511no—
220C2×#41C2×D25111yescharacteristic
221C22×Q8C2×D25111yescharacteristic
222#137—5024no—
223#135—5024no—
224C2×Q16—2511no—
225C2×Q16—2511no—
226C2×Q16—2511no—
227C2×Q16—2511no—
228C2×SD16—2511no—
229C2×SD16—2511no—
230C2×SD16—2511no—
231C2×SD16—2511no—
232C2×D8—2511no—
233C2×M16—2511no—
234C22×C8—2511no—
235C4⋊Q8—2511no—
236#124—21no—
237#124C2×D25111yescharacteristic
238#123—5021no—
239#123—5021no—
240C22⋊Q8—21no—
241C22⋊Q8—2511no—
242C22⋊Q8—2511no—
243C22⋊Q8C2×D25111yescharacteristic
244#121—21no—
245#121—5021no—
246#121C2×D25111yescharacteristic
247C4×Q8C2×D25111yescharacteristic
248C4×D4—21no—
249C4×D4—2511no—
250C4×D4C2×D25111yescharacteristic
251C2×C4⋊C4—5021no—
252#107—2511no—
253C4⋊C8—2511no—
254#103—2511no—
255#103—2511no—
256#103—2511no—
257#103—2511no—
258#102—2511no—
259#102—2511no—
260C22⋊C8—2511no—
261C22⋊C8—2511no—
262#97—2511no—
263#41—22no—
264#41—41no—
265#41—10041no—
266#41—10041no—
267#41—5022no—
268#41—5022no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—2511no—
273C2×Q8—2511no—
274C2×Q8C22×D25111yescharacteristic
275C2×Q8C22×D25111yescharacteristic
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—5021no—
279C2×D4—5021no—
280C2×D4—2511no—
281C2×D4C22⋊D25111yescharacteristic
282C2×D4C22×D25111yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—5021no—
286C22×C4—5021no—
287C22×C4—5021no—
288C22×C4C22×D25111yescharacteristic
289C22×C4C22×D25111yescharacteristic
290Q16—10041no—
291Q16—5022no—
292Q16—5022no—
293Q16—5022no—
294SD16—10041no—
295SD16—5022no—
296SD16—5022no—
297SD16—2514no—
298D8—5022no—
299M16—5022no—
300C2×C8—5022no—
301C2×C8—2511no—
302C2×C8—2511no—
303C2×C8—2511no—
304C2×C8—2511no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—5021no—
308C4⋊C4—5021no—
309C4⋊C4—5021no—
310C4⋊C4—5021no—
311C4⋊C4—2511no—
312C4⋊C4C22⋊D25111yescharacteristic
313C4⋊C4C22×D25111yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—21no—
318C22⋊C4—5021no—
319C22⋊C4—5021no—
320C22⋊C4—5021no—
321C22⋊C4C22⋊D25111yescharacteristic
322C22⋊C4C22⋊D25111yescharacteristic
323C42—21no—
324C42C22×D25111yescharacteristic
325C23—21no—
326C23—5021no—
327C23C2×C22⋊D25111yescharacteristic
328C23C2×C22⋊D25111yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—42no—
333Q8—5021no—
334Q8—5021no—
335Q8—5021no—
336Q8—5021no—
337Q8D4×D25112yes—
338D4—21no—
339D4—21no—
340D4—41no—
341D4—42no—
342D4—42no—
343D4—10042no—
344D4—10042no—
345D4—5021no—
346D4—2512no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—10042no—
357C2×C4—10042no—
358C2×C4—10042no—
359C2×C4—5021no—
360C2×C4—5021no—
361C2×C4—5021no—
362C2×C4C2×C22⋊D25111yescharacteristic
363C2×C4C2×C22⋊D25111yescharacteristic
364C2×C4C2×C22⋊D25111yescharacteristic
365C2×C4C2×C22⋊D25111yescharacteristic
366C2×C4C23×D25111yescharacteristic
367C8—5021no—
368C8—5021no—
369C8—5021no—
370C8—2512no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—10041no—
381C22—10041no—
382C22—5022no—
383C22C22×C22⋊D25111yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—22no—
389C4—41no—
390C4—41no—
391C4—10041no—
392C4—10041no—
393C4—10041no—
394C4C2×D4×D25111yescharacteristic
395C4#2254876011yescharacteristic
396C2—41no—
397C2—41no—
398C2—81no—
399C2—10042no—
400C2D4×C22⋊D25111yescharacteristic
401C2#3063248511yescharacteristic
402C2#3063250211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#548×C251C2
socleC22×C251C22×D4
hypercenter#548D251
hypocenterC251#3269
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#22548746C2semidirect
2. C2×#22548735C2semidirect
3. #30632342C2semidirect
4. #30632082C2semidirect
5. #30631931C2semidirect
6. #30631926C2semidirect
7. #30631899C2semidirect
8. #30631897C2semidirect
9. #30631849C2semidirect
10. #30631847C2semidirect
11. #30631843C2semidirect
12. #30631830C2semidirect
13. #30631785C2semidirect
14. #30631759C2semidirect
15. #548×C251C2semidirect
80. #16860797D4semidirect
201. C251#3269semidirect
205. #548D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487462
C2×#225487352
#306323422
#306320822
#306319312
#306319262
#306318992
#306318972
#306318492
#306318472
#306318432
#306318302
#306317852
#306317592
#548×C2512
#3269251