Finite Group Databasebeta

← #38754268

Subgroups — #38754268

Order 25216.

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×#20263357C211yesmaximal, characteristic, complemented
2C2×#20263346C211yesmaximal, characteristic, complemented
3#27161734C211yesmaximal, characteristic, complemented
4#27161474C211yesmaximal, characteristic, complemented
5#27161323C211yesmaximal, characteristic, complemented
6#27161318C211yesmaximal, characteristic, complemented
7#27161291C211yesmaximal, characteristic, complemented
8#27161289C211yesmaximal, characteristic, complemented
9#27161241C211yesmaximal, characteristic, complemented
10#27161239C211yesmaximal, characteristic, complemented
11#27161235C211yesmaximal, characteristic, complemented
12#27161222C211yesmaximal, characteristic, complemented
13#27161177C211yesmaximal, characteristic, complemented
14#27161151C211yesmaximal, characteristic, complemented
15#548×C197C211yesmaximal, characteristic, complemented
16C2×#15691358C2211yescharacteristic
17C2×#15691358C2211yescharacteristic
18#20263357—24no—
19#20263346—24no—
20C2×#15691348C2211yescharacteristic
21C2×#15691348C2211yescharacteristic
22C2×#15691348C2211yescharacteristic
23C2×#15691348C2211yescharacteristic
24C2×#15691346C2211yescharacteristic
25C2×#15691346C2211yescharacteristic
26C2×#15691345C2211yescharacteristic
27C2×#15691345C2211yescharacteristic
28C2×#15691343C2211yescharacteristic
29C2×#15691342C2211yescharacteristic
30C22×#13096802C2211yescharacteristic
31#20263326C2211yescharacteristic
32#20263308—21no—
33#20263308—21no—
34#20263305C2211yescharacteristic
35#20263305C2211yescharacteristic
36#20263297—21no—
37C4×D788C2211yescharacteristic
38C4⋊C4×D197—21no—
39#20263259C2211yescharacteristic
40C4⋊#13096802C2211yescharacteristic
41#20263250C2211yescharacteristic
42#20263250C2211yescharacteristic
43#20263249C2211yescharacteristic
44#20263249C2211yescharacteristic
45#20263247C2211yescharacteristic
46#20263246C2211yescharacteristic
47C22⋊#13096802C2211yescharacteristic
48C22⋊#13096802C2211yescharacteristic
49#20263237C2211yescharacteristic
50C2×#41×C197C2211yescharacteristic
51C22×Q8×C197C2211yescharacteristic
52#124×C197—21no—
53#124×C197C2211yescharacteristic
54C22⋊Q8×C197—21no—
55C22⋊Q8×C197C2211yescharacteristic
56#121×C197—21no—
57#121×C197C2211yescharacteristic
58C4×Q8×C197C2211yescharacteristic
59C4×D4×C197—21no—
60C4×D4×C197C2211yescharacteristic
61#15691358—22no—
62#15691358—22no—
63#15691358—41no—
64#15691358—41no—
65C2×Dic394C2311yescharacteristic
66C2×Dic394C2311yescharacteristic
67C2×D788C2311yescharacteristic
68C2×C22⋊D197—21no—
69C2×C22⋊D197—21no—
70C2×C4×D197—21no—
71C2×C4×D197—21no—
72C2×C4×D197—21no—
73#15691348—22no—
74#15691348—22no—
75#15691348—22no—
76#15691348—41no—
77#15691346—22no—
78#15691346—41no—
79#15691345—22no—
80#15691345D414yescomplemented
81#15691343—22no—
82#15691342—22no—
83C2×#13096802—22no—
84C2×#13096802C2311yescharacteristic
85C2×#13096802C2311yescharacteristic
86C2×#13096802C2311yescharacteristic
87C2×#13096802C2311yescharacteristic
88C4⋊Dic197—21no—
89C4⋊Dic197—21no—
90C4⋊Dic197C2311yescharacteristic
91#15691337—21no—
92#15691337—21no—
93C22⋊Dic197—21no—
94C22⋊Dic197—21no—
95#15691335—21no—
96#41×C197—22no—
97#41×C197—41no—
98C2×Q8×C197—21no—
99C2×Q8×C197—22no—
100C2×Q8×C197—41no—
101C2×Q8×C197C2311yescharacteristic
102C2×Q8×C197C2311yescharacteristic
103C2×D4×C197—21no—
104C2×D4×C197—21no—
105C2×D4×C197D411yescharacteristic
106C2×D4×C197C2311yescharacteristic
107C22×C4×C197—21no—
108C22×C4×C197—21no—
109C22×C4×C197C2311yescharacteristic
110C22×C4×C197C2311yescharacteristic
111C4⋊C4×C197—21no—
112C4⋊C4×C197—21no—
113C4⋊C4×C197D411yescharacteristic
114C4⋊C4×C197C2311yescharacteristic
115C22⋊C4×C197—21no—
116C22⋊C4×C197—21no—
117C22⋊C4×C197—21no—
118C22⋊C4×C197—21no—
119C22⋊C4×C197D411yescharacteristic
120C22⋊C4×C197D411yescharacteristic
121C42×C197—21no—
122C42×C197C2311yescharacteristic
123C22×D197—21no—
124Dic394—21no—
125Dic394—21no—
126Dic394—21no—
127Dic394—21no—
128D788—21no—
129D788C2×D412yes—
130C22⋊D197—42no—
131C22⋊D197—42no—
132C2×Dic197—21no—
133C2×Dic197—21no—
134C2×Dic197—21no—
135C4×D197—42no—
136C4×D197—42no—
137C4×D197—42no—
138#13096802—21no—
139#13096802—21no—
140#13096802—21no—
141#13096802C2×D412yes—
142C23×C197—21no—
143C23×C197C2×D411yescharacteristic
144C23×C197C2×D411yescharacteristic
145Q8×C197—21no—
146Q8×C197—21no—
147Q8×C197—21no—
148Q8×C197—42no—
149Q8×C197C2×D412yes—
150D4×C197—21no—
151D4×C197—21no—
152D4×C197—41no—
153D4×C197—42no—
154D4×C197—42no—
155C2×C4×C197—21no—
156C2×C4×C197—21no—
157C2×C4×C197—21no—
158C2×C4×C197—21no—
159C2×C4×C197—21no—
160C2×C4×C197—21no—
161C2×C4×C197—21no—
162C2×C4×C197—42no—
163C2×C4×C197—42no—
164C2×C4×C197C2×D411yescharacteristic
165C2×C4×C197C2×D411yescharacteristic
166C2×C4×C197C2×D411yescharacteristic
167C2×C4×C197C2×D411yescharacteristic
168C2×C4×C197C2411yescharacteristic
169C22×C197—21no—
170C22×C197—21no—
171C22×C197—21no—
172C22×C197—21no—
173C22×C197—21no—
174C22×C197—21no—
175C22×C197—41no—
176C22×C197—41no—
177C22×C197—41no—
178C22×C197C22×D411yescharacteristic
179C2×D197—22no—
180C2×D197—41no—
181C2×D197—41no—
182C4×C197—21no—
183C4×C197—21no—
184C4×C197—22no—
185C4×C197—21no—
186C4×C197—21no—
187C4×C197—41no—
188C4×C197—41no—
189C4×C197C22×D411yescharacteristic
190C4×C197H(5,𝔽2)11yescharacteristic
191Dic197—41no—
192Dic197—41no—
193Dic197—41no—
194C2×C197—41no—
195C2×C197—41no—
196C2×C197—81no—
197C2×C197D4211yescharacteristic
198C2×C197#57511yescharacteristic
199C2×C197#57811yescharacteristic
200D197—42no—
201C197#326911yescharacteristic, complemented, Sylow
202#3269—1971nomaximal, Sylow
203C2×#137—1971no—
204C2×#135—1971no—
205#548D19711yescharacteristic, complemented
206#547—1971no—
207#497—1971no—
208#468—1971no—
209#467—1971no—
210#463—1971no—
211#462—1971no—
212#452—1971no—
213#452—1971no—
214#451—1971no—
215#449—1971no—
216#440—1971no—
217#435—1971no—
218C2×#41—1971no—
219C2×#41—1971no—
220C2×#41C2×D19711yescharacteristic
221C22×Q8C2×D19711yescharacteristic
222#137—3944no—
223#135—3944no—
224C2×Q16—1971no—
225C2×Q16—1971no—
226C2×Q16—1971no—
227C2×Q16—1971no—
228C2×SD16—1971no—
229C2×SD16—1971no—
230C2×SD16—1971no—
231C2×SD16—1971no—
232C2×D8—1971no—
233C2×M16—1971no—
234C22×C8—1971no—
235C4⋊Q8—1971no—
236#124—21no—
237#124C2×D19711yescharacteristic
238#123—3941no—
239#123—3941no—
240C22⋊Q8—21no—
241C22⋊Q8—1971no—
242C22⋊Q8—1971no—
243C22⋊Q8C2×D19711yescharacteristic
244#121—21no—
245#121—3941no—
246#121C2×D19711yescharacteristic
247C4×Q8C2×D19711yescharacteristic
248C4×D4—21no—
249C4×D4—1971no—
250C4×D4C2×D19711yescharacteristic
251C2×C4⋊C4—3941no—
252#107—1971no—
253C4⋊C8—1971no—
254#103—1971no—
255#103—1971no—
256#103—1971no—
257#103—1971no—
258#102—1971no—
259#102—1971no—
260C22⋊C8—1971no—
261C22⋊C8—1971no—
262#97—1971no—
263#41—22no—
264#41—41no—
265#41—7881no—
266#41—7881no—
267#41—3942no—
268#41—3942no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—1971no—
273C2×Q8—1971no—
274C2×Q8C22×D19711yescharacteristic
275C2×Q8C22×D19711yescharacteristic
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—3941no—
279C2×D4—3941no—
280C2×D4—1971no—
281C2×D4C22⋊D19711yescharacteristic
282C2×D4C22×D19711yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—3941no—
286C22×C4—3941no—
287C22×C4—3941no—
288C22×C4C22×D19711yescharacteristic
289C22×C4C22×D19711yescharacteristic
290Q16—7881no—
291Q16—3942no—
292Q16—3942no—
293Q16—3942no—
294SD16—7881no—
295SD16—3942no—
296SD16—3942no—
297SD16—1974no—
298D8—3942no—
299M16—3942no—
300C2×C8—3942no—
301C2×C8—1971no—
302C2×C8—1971no—
303C2×C8—1971no—
304C2×C8—1971no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—3941no—
308C4⋊C4—3941no—
309C4⋊C4—3941no—
310C4⋊C4—3941no—
311C4⋊C4—1971no—
312C4⋊C4C22⋊D19711yescharacteristic
313C4⋊C4C22×D19711yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—21no—
318C22⋊C4—3941no—
319C22⋊C4—3941no—
320C22⋊C4—3941no—
321C22⋊C4C22⋊D19711yescharacteristic
322C22⋊C4C22⋊D19711yescharacteristic
323C42—21no—
324C42C22×D19711yescharacteristic
325C23—21no—
326C23—3941no—
327C23C2×C22⋊D19711yescharacteristic
328C23C2×C22⋊D19711yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—42no—
333Q8—3941no—
334Q8—3941no—
335Q8—3941no—
336Q8—3941no—
337Q8D4×D19712yes—
338D4—21no—
339D4—21no—
340D4—41no—
341D4—42no—
342D4—42no—
343D4—7882no—
344D4—7882no—
345D4—3941no—
346D4—1972no—
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—7882no—
357C2×C4—7882no—
358C2×C4—7882no—
359C2×C4—3941no—
360C2×C4—3941no—
361C2×C4—3941no—
362C2×C4C2×C22⋊D19711yescharacteristic
363C2×C4C2×C22⋊D19711yescharacteristic
364C2×C4C2×C22⋊D19711yescharacteristic
365C2×C4C2×C22⋊D19711yescharacteristic
366C2×C4C23×D19711yescharacteristic
367C8—3941no—
368C8—3941no—
369C8—3941no—
370C8—1972no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—7881no—
381C22—7881no—
382C22—3942no—
383C22C22×C22⋊D19711yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—22no—
389C4—41no—
390C4—41no—
391C4—7881no—
392C4—7881no—
393C4—7881no—
394C4C2×D4×D19711yescharacteristic
395C4#2026337111yescharacteristic
396C2—41no—
397C2—41no—
398C2—81no—
399C2—7882no—
400C2D4×C22⋊D19711yescharacteristic
401C2#2716187711yescharacteristic
402C2#2716189411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#548×C197C2
socleC22×C197C22×D4
hypercenter#548D197
hypocenterC197#3269
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#20263357C2semidirect
2. C2×#20263346C2semidirect
3. #27161734C2semidirect
4. #27161474C2semidirect
5. #27161323C2semidirect
6. #27161318C2semidirect
7. #27161291C2semidirect
8. #27161289C2semidirect
9. #27161241C2semidirect
10. #27161239C2semidirect
11. #27161235C2semidirect
12. #27161222C2semidirect
13. #27161177C2semidirect
14. #27161151C2semidirect
15. #548×C197C2semidirect
80. #15691345D4semidirect
201. C197#3269semidirect
205. #548D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633572
C2×#202633462
#271617342
#271614742
#271613232
#271613182
#271612912
#271612892
#271612412
#271612392
#271612352
#271612222
#271611772
#271611512
#548×C1972
#3269197