Finite Group Databasebeta

← #24919019

Subgroups — #24919019

Order 10624.

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×#14581823C211yesmaximal, characteristic, complemented
2C2×#14581812C211yesmaximal, characteristic, complemented
3#19128356C211yesmaximal, characteristic, complemented
4#19128096C211yesmaximal, characteristic, complemented
5#19127945C211yesmaximal, characteristic, complemented
6#19127940C211yesmaximal, characteristic, complemented
7#19127913C211yesmaximal, characteristic, complemented
8#19127911C211yesmaximal, characteristic, complemented
9#19127863C211yesmaximal, characteristic, complemented
10#19127861C211yesmaximal, characteristic, complemented
11#19127857C211yesmaximal, characteristic, complemented
12#19127844C211yesmaximal, characteristic, complemented
13#19127799C211yesmaximal, characteristic, complemented
14#19127773C211yesmaximal, characteristic, complemented
15#548×C83C211yesmaximal, characteristic, complemented
16C2×#13050590C2211yescharacteristic
17C2×#13050590C2211yescharacteristic
18#14581823—24no—
19#14581812—24no—
20C2×#13050580C2211yescharacteristic
21C2×#13050580C2211yescharacteristic
22C2×#13050580C2211yescharacteristic
23C2×#13050580C2211yescharacteristic
24C2×#13050578C2211yescharacteristic
25C2×#13050578C2211yescharacteristic
26C2×#13050577C2211yescharacteristic
27C2×#13050577C2211yescharacteristic
28C2×#13050575C2211yescharacteristic
29C2×#13050574C2211yescharacteristic
30C22×C83⋊C8C2211yescharacteristic
31#14581792C2211yescharacteristic
32#14581774—21no—
33#14581774—21no—
34#14581771C2211yescharacteristic
35#14581771C2211yescharacteristic
36#14581763—21no—
37C4×D332C2211yescharacteristic
38C4⋊C4×D83—21no—
39#14581725C2211yescharacteristic
40C4⋊(C83⋊C8)C2211yescharacteristic
41#14581716C2211yescharacteristic
42#14581716C2211yescharacteristic
43#14581715C2211yescharacteristic
44#14581715C2211yescharacteristic
45#14581713C2211yescharacteristic
46#14581712C2211yescharacteristic
47C22⋊(C83⋊C8)C2211yescharacteristic
48C22⋊(C83⋊C8)C2211yescharacteristic
49#14581703C2211yescharacteristic
50C2×#41×C83C2211yescharacteristic
51C22×Q8×C83C2211yescharacteristic
52#124×C83—21no—
53#124×C83C2211yescharacteristic
54C22⋊Q8×C83—21no—
55C22⋊Q8×C83C2211yescharacteristic
56#121×C83—21no—
57#121×C83C2211yescharacteristic
58C4×Q8×C83C2211yescharacteristic
59C4×D4×C83—21no—
60C4×D4×C83C2211yescharacteristic
61#13050590—22no—
62#13050590—22no—
63#13050590—41no—
64#13050590—41no—
65C2×Dic166C2311yescharacteristic
66C2×Dic166C2311yescharacteristic
67C2×D332C2311yescharacteristic
68C2×C22⋊D83—21no—
69C2×C22⋊D83—21no—
70C2×C4×D83—21no—
71C2×C4×D83—21no—
72C2×C4×D83—21no—
73#13050580—22no—
74#13050580—22no—
75#13050580—22no—
76#13050580—41no—
77#13050578—22no—
78#13050578—41no—
79#13050577—22no—
80#13050577D414yescomplemented
81#13050575—22no—
82#13050574—22no—
83C2×C83⋊C8—22no—
84C2×C83⋊C8C2311yescharacteristic
85C2×C83⋊C8C2311yescharacteristic
86C2×C83⋊C8C2311yescharacteristic
87C2×C83⋊C8C2311yescharacteristic
88C4⋊Dic83—21no—
89C4⋊Dic83—21no—
90C4⋊Dic83C2311yescharacteristic
91Dic83⋊C4—21no—
92Dic83⋊C4—21no—
93C22⋊Dic83—21no—
94C22⋊Dic83—21no—
95(C2×D83)⋊C4—21no—
96#41×C83—22no—
97#41×C83—41no—
98C2×Q8×C83—21no—
99C2×Q8×C83—22no—
100C2×Q8×C83—41no—
101C2×Q8×C83C2311yescharacteristic
102C2×Q8×C83C2311yescharacteristic
103C2×D4×C83—21no—
104C2×D4×C83—21no—
105C2×D4×C83D411yescharacteristic
106C2×D4×C83C2311yescharacteristic
107C22×C4×C83—21no—
108C22×C4×C83—21no—
109C22×C4×C83C2311yescharacteristic
110C22×C4×C83C2311yescharacteristic
111C4⋊C4×C83—21no—
112C4⋊C4×C83—21no—
113C4⋊C4×C83D411yescharacteristic
114C4⋊C4×C83C2311yescharacteristic
115C22⋊C4×C83—21no—
116C22⋊C4×C83—21no—
117C22⋊C4×C83—21no—
118C22⋊C4×C83—21no—
119C22⋊C4×C83D411yescharacteristic
120C22⋊C4×C83D411yescharacteristic
121C42×C83—21no—
122C42×C83C2311yescharacteristic
123C23×C83—21no—
124C23×C83C2×D411yescharacteristic
125C23×C83C2×D411yescharacteristic
126C22×D83—21no—
127Q8×C83—21no—
128Q8×C83—21no—
129Q8×C83—21no—
130Q8×C83—42no—
131Q8×C83C2×D412yes—
132D4×C83—21no—
133D4×C83—21no—
134D4×C83—41no—
135D4×C83—42no—
136D4×C83—42no—
137C2×C4×C83—21no—
138C2×C4×C83—21no—
139C2×C4×C83—21no—
140C2×C4×C83—21no—
141C2×C4×C83—21no—
142C2×C4×C83—21no—
143C2×C4×C83—21no—
144C2×C4×C83—42no—
145C2×C4×C83—42no—
146C2×C4×C83C2×D411yescharacteristic
147C2×C4×C83C2×D411yescharacteristic
148C2×C4×C83C2×D411yescharacteristic
149C2×C4×C83C2×D411yescharacteristic
150C2×C4×C83C2411yescharacteristic
151C22⋊D83—42no—
152C22⋊D83—42no—
153C2×Dic83—21no—
154C2×Dic83—21no—
155C2×Dic83—21no—
156D332—21no—
157D332C2×D412yes—
158C4×D83—42no—
159C4×D83—42no—
160C4×D83—42no—
161Dic166—21no—
162Dic166—21no—
163Dic166—21no—
164Dic166—21no—
165C83⋊C8—21no—
166C83⋊C8—21no—
167C83⋊C8—21no—
168C83⋊C8C2×D412yes—
169C22×C83—21no—
170C22×C83—21no—
171C22×C83—21no—
172C22×C83—21no—
173C22×C83—21no—
174C22×C83—21no—
175C22×C83—41no—
176C22×C83—41no—
177C22×C83—41no—
178C22×C83C22×D411yescharacteristic
179C2×D83—22no—
180C2×D83—41no—
181C2×D83—41no—
182C4×C83—21no—
183C4×C83—21no—
184C4×C83—22no—
185C4×C83—21no—
186C4×C83—21no—
187C4×C83—41no—
188C4×C83—41no—
189C4×C83C22×D411yescharacteristic
190C4×C83H(5,𝔽2)11yescharacteristic
191Dic83—41no—
192Dic83—41no—
193Dic83—41no—
194C2×C83—41no—
195C2×C83—41no—
196C2×C83—81no—
197C2×C83D4211yescharacteristic
198C2×C83#57511yescharacteristic
199C2×C83#57811yescharacteristic
200D83—42no—
201#3269—831nomaximal, Sylow
202C83#326911yescharacteristic, complemented, Sylow
203C2×#137—831no—
204C2×#135—831no—
205#548D8311yescharacteristic, complemented
206#547—831no—
207#497—831no—
208#468—831no—
209#467—831no—
210#463—831no—
211#462—831no—
212#452—831no—
213#452—831no—
214#451—831no—
215#449—831no—
216#440—831no—
217#435—831no—
218C2×#41—831no—
219C2×#41—831no—
220C2×#41C2×D8311yescharacteristic
221C22×Q8C2×D8311yescharacteristic
222#137—1664no—
223#135—1664no—
224C2×Q16—831no—
225C2×Q16—831no—
226C2×Q16—831no—
227C2×Q16—831no—
228C2×SD16—831no—
229C2×SD16—831no—
230C2×SD16—831no—
231C2×SD16—831no—
232C2×D8—831no—
233C2×M16—831no—
234C22×C8—831no—
235C4⋊Q8—831no—
236#124—21no—
237#124C2×D8311yescharacteristic
238#123—1661no—
239#123—1661no—
240C22⋊Q8—21no—
241C22⋊Q8—831no—
242C22⋊Q8—831no—
243C22⋊Q8C2×D8311yescharacteristic
244#121—21no—
245#121—1661no—
246#121C2×D8311yescharacteristic
247C4×Q8C2×D8311yescharacteristic
248C4×D4—21no—
249C4×D4—831no—
250C4×D4C2×D8311yescharacteristic
251C2×C4⋊C4—1661no—
252#107—831no—
253C4⋊C8—831no—
254#103—831no—
255#103—831no—
256#103—831no—
257#103—831no—
258#102—831no—
259#102—831no—
260C22⋊C8—831no—
261C22⋊C8—831no—
262#97—831no—
263#41—22no—
264#41—41no—
265#41—3321no—
266#41—3321no—
267#41—1662no—
268#41—1662no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—831no—
273C2×Q8—831no—
274C2×Q8C22×D8311yescharacteristic
275C2×Q8C22×D8311yescharacteristic
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—831no—
279C2×D4—1661no—
280C2×D4—1661no—
281C2×D4C22⋊D8311yescharacteristic
282C2×D4C22×D8311yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—1661no—
286C22×C4—1661no—
287C22×C4—1661no—
288C22×C4C22×D8311yescharacteristic
289C22×C4C22×D8311yescharacteristic
290Q16—3321no—
291Q16—1662no—
292Q16—1662no—
293Q16—1662no—
294SD16—3321no—
295SD16—834no—
296SD16—1662no—
297SD16—1662no—
298D8—1662no—
299M16—1662no—
300C2×C8—831no—
301C2×C8—831no—
302C2×C8—831no—
303C2×C8—831no—
304C2×C8—1662no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—831no—
308C4⋊C4—1661no—
309C4⋊C4—1661no—
310C4⋊C4—1661no—
311C4⋊C4—1661no—
312C4⋊C4C22⋊D8311yescharacteristic
313C4⋊C4C22×D8311yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—21no—
318C22⋊C4—1661no—
319C22⋊C4—1661no—
320C22⋊C4—1661no—
321C22⋊C4C22⋊D8311yescharacteristic
322C22⋊C4C22⋊D8311yescharacteristic
323C42—21no—
324C42C22×D8311yescharacteristic
325C23—21no—
326C23—1661no—
327C23C2×C22⋊D8311yescharacteristic
328C23C2×C22⋊D8311yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—42no—
333Q8—1661no—
334Q8—1661no—
335Q8—1661no—
336Q8—1661no—
337Q8D4×D8312yes—
338D4—21no—
339D4—21no—
340D4—41no—
341D4—42no—
342D4—42no—
343D4—3322no—
344D4—3322no—
345D4—832no—
346D4—1661no—
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—3322no—
357C2×C4—3322no—
358C2×C4—3322no—
359C2×C4—1661no—
360C2×C4—1661no—
361C2×C4—1661no—
362C2×C4C2×C22⋊D8311yescharacteristic
363C2×C4C2×C22⋊D8311yescharacteristic
364C2×C4C2×C22⋊D8311yescharacteristic
365C2×C4C2×C22⋊D8311yescharacteristic
366C2×C4C23×D8311yescharacteristic
367C8—832no—
368C8—1661no—
369C8—1661no—
370C8—1661no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—3321no—
381C22—3321no—
382C22—1662no—
383C22C22×C22⋊D8311yescharacteristic
384C4—21no—
385C4—21no—
386C4—22no—
387C4—21no—
388C4—21no—
389C4—41no—
390C4—41no—
391C4—3321no—
392C4—3321no—
393C4—3321no—
394C4C2×D4×D8311yescharacteristic
395C4#1458183711yescharacteristic
396C2—41no—
397C2—41no—
398C2—81no—
399C2—3322no—
400C2D4×C22⋊D8311yescharacteristic
401C2#1912849911yescharacteristic
402C2#1912851611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#548×C83C2
socleC22×C83C22×D4
hypercenter#548D83
hypocenterC83#3269
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#14581823C2semidirect
2. C2×#14581812C2semidirect
3. #19128356C2semidirect
4. #19128096C2semidirect
5. #19127945C2semidirect
6. #19127940C2semidirect
7. #19127913C2semidirect
8. #19127911C2semidirect
9. #19127863C2semidirect
10. #19127861C2semidirect
11. #19127857C2semidirect
12. #19127844C2semidirect
13. #19127799C2semidirect
14. #19127773C2semidirect
15. #548×C83C2semidirect
80. #13050577D4semidirect
202. C83#3269semidirect
205. #548D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818232
C2×#145818122
#191283562
#191280962
#191279452
#191279402
#191279132
#191279112
#191278632
#191278612
#191278572
#191278442
#191277992
#191277732
#548×C832
#326983