Finite Group Databasebeta

← #29663886

Subgroups — #29663886

Order 15616.

606 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456403C211yesmaximal, characteristic
2#21455951C212yesmaximal
3C8×Q8×D61C211yesmaximal, characteristic
4#21445248C212yesmaximal, complemented
5#21445247C212yesmaximal, complemented
6#21445214C211yesmaximal, characteristic, complemented
7#21445213C211yesmaximal, characteristic, complemented
8#21445186C211yesmaximal, characteristic, complemented
9#21444581C211yesmaximal, characteristic
10#21444518C212yesmaximal
11C61×#1670C211yesmaximal, characteristic, complemented
12#16840317C2212yes—
13C4×Q8×D61C2211yescharacteristic
14#16840167C2211yescharacteristic
15#16840112C2212yes—
16#16840094—22no—
17#16840044C2212yes—
18#16840000—21no—
19#16840000—21no—
20#16839973C2212yes—
21#16839972C2212yes—
22#16839919—22no—
23Q8×#92418C2211yescharacteristic
24C8×Dic122—21no—
25C8×Dic122C2211yescharacteristic
26#16839886—21no—
27#16839886C2211yescharacteristic
28#16839868—22no—
29#16839866—22no—
30Q16×Dic61—21no—
31Q16×Dic61—21no—
32#16839818C2212yes—
33C4⋊C8×D61—21no—
34C4⋊C8×D61C2211yescharacteristic
35C4×C8×D61—21no—
36C4×C8×D61C2211yescharacteristic
37#16839515C2211yescharacteristic
38#16839512C2212yes—
39#16839509C2212yes—
40#16839508C2212yes—
41#16839507C2212yes—
42C61×#494C2211yescharacteristic
43C61×#475C2212yes—
44C8×Q8×C61C2211yescharacteristic
45C61×#335C2211yescharacteristic
46C61×#332C2212yes—
47C2×Q8×D61D411yescharacteristic
48C2×Dic244—22no—
49C2×#11758202—22no—
50C2×C8×D61—21no—
51C2×C8×D61D411yescharacteristic
52C2×C8×D61D411yescharacteristic
53#14519777C2312yes—
54#14519768—22no—
55#14519757—22no—
56#14519753—22no—
57Q8×Dic61—22no—
58Q8×Dic61D411yescharacteristic
59C4×Dic122—21no—
60C4×Dic122C2311yescharacteristic
61#14519742—21no—
62#14519742—22no—
63#14519742D411yescharacteristic
64#14519735—22no—
65C4⋊C4×D61—21no—
66C4⋊C4×D61D411yescharacteristic
67C42×D61—21no—
68C42×D61C2311yescharacteristic
69#14519711—21no—
70#14519711—21no—
71#14519708—22no—
72C4⋊#92418—21no—
73C4⋊#92418C2312yes—
74C4⋊#92418C2311yescharacteristic
75#14519706—21no—
76#14519706—21no—
77#14519706—21no—
78#14519702—22no—
79#14519701—22no—
80#14519701—22no—
81#14519700—22no—
82#14519700—22no—
83#14519689—22no—
84#14519687—22no—
85C4×#92418—21no—
86C4×#92418C2311yescharacteristic
87C8×Dic61—21no—
88C8×Dic61—21no—
89C8×Dic61D411yescharacteristic
90C8×Dic61D411yescharacteristic
91C2×Q16×C61—21no—
92C2×Q16×C61—21no—
93C4⋊Q8×C61C2312yes—
94C4×Q8×C61C2311yescharacteristic
95#106×C61—22no—
96C4⋊C8×C61—21no—
97C4⋊C8×C61C2312yes—
98C4⋊C8×C61C2311yescharacteristic
99#103×C61—22no—
100C4×C8×C61—21no—
101C4×C8×C61C2311yescharacteristic
102C2×Q8×C61—22no—
103C2×Q8×C61C2×D411yescharacteristic
104Q8×D61—22no—
105Q8×D61—22no—
106C2×C4×D61—21no—
107C2×C4×D61—21no—
108C2×C4×D61C2×D411yescharacteristic
109C2×C4×D61C2×D411yescharacteristic
110C2×C4×D61#4111yescharacteristic
111C2×Dic122—21no—
112C2×Dic122—22no—
113C2×Dic122C2×D411yescharacteristic
114Q16×C61—42no—
115Q16×C61—42no—
116C2×C8×C61—21no—
117C2×C8×C61—22no—
118C2×C8×C61C2×D411yescharacteristic
119C2×C8×C61C2×D411yescharacteristic
120C4⋊C4×C61—21no—
121C4⋊C4×C61—22no—
122C4⋊C4×C61—22no—
123C4⋊C4×C61C2×D411yescharacteristic
124C42×C61—21no—
125C42×C61C2411yescharacteristic
126#11758202—44no—
127#11758198—42no—
128C4⋊Dic61—21no—
129C4⋊Dic61—22no—
130C4⋊Dic61—22no—
131C4⋊Dic61—42no—
132C4⋊Dic61C2×D411yescharacteristic
133#11758196—21no—
134#11758196—21no—
135#11758196—21no—
136#11758196—42no—
137C4×Dic61—21no—
138C4×Dic61—21no—
139C4×Dic61—21no—
140C4×Dic61—42no—
141C4×Dic61C2×D411yescharacteristic
142C4×Dic61C2×D411yescharacteristic
143C4×Dic61#4111yescharacteristic
144C2×#92418—21no—
145C2×#92418—22no—
146C2×#92418C2×D411yescharacteristic
147C2×#92418C2×D411yescharacteristic
148Dic244—44no—
149C8×D61—22no—
150C8×D61—41no—
151C8×D61—42no—
152C22×D61#12111yescharacteristic
153Q8×C61—21no—
154Q8×C61—42no—
155Q8×C61—42no—
156Q8×C61#13512yes—
157C2×C4×C61—21no—
158C2×C4×C61—21no—
159C2×C4×C61—42no—
160C2×C4×C61C22×D411yescharacteristic
161C2×C4×C61C22×D411yescharacteristic
162C2×C4×C61C2×#4111yescharacteristic
163C2×Dic61—21no—
164C2×Dic61—21no—
165C2×Dic61—42no—
166C2×Dic61#12111yescharacteristic
167C2×Dic61#12111yescharacteristic
168C2×Dic61#12111yescharacteristic
169C4×D61—24no—
170C4×D61—22no—
171C4×D61—21no—
172C4×D61—21no—
173C4×D61—21no—
174C4×D61—21no—
175C4×D61—42no—
176Dic122—24no—
177Dic122—21no—
178Dic122—42no—
179Dic122—42no—
180Dic122#13512yes—
181C8×C61—21no—
182C8×C61—42no—
183C8×C61—41no—
184C8×C61#13712yes—
185#92418—21no—
186#92418—21no—
187#92418—42no—
188#92418—41no—
189#19265—611nomaximal, Sylow
190C22×C61C2×#12111yescharacteristic
191C2×D61—21no—
192C2×D61—21no—
193C2×D61#47312yes—
194C4×C61—22no—
195C4×C61—21no—
196C4×C61—41no—
197C4×C61—82no—
198C4×C61C2×#13511yescharacteristic
199C4×C61C2×#13511yescharacteristic
200C4×C61C2×#13711yescharacteristic
201C4×C61#57511yescharacteristic
202Dic61—21no—
203Dic61—21no—
204Dic61—22no—
205Dic61—21no—
206Dic61—41no—
207Dic61—82no—
208#3170—611no—
209#3113—612no—
210C2×C8×Q8—611no—
211#1680—612no—
212#1680—612no—
213#1674—611no—
214#1674—611no—
215#1670—611no—
216#1670D6111yescharacteristic, complemented
217#1568—611no—
218#1552—612no—
219C2×C61#303711yescharacteristic
220C2×C61#305011yescharacteristic
221C2×C61C2×#47311yescharacteristic
222D61—22no—
223#533—612no—
224C2×C4×Q8—611no—
225#500—611no—
226#494C2×D6111yescharacteristic
227#487—612no—
228#483—1222no—
229#475—612no—
230#475C2×D6112yes—
231#467—1221no—
232#467—1221no—
233#462—612no—
234#462—612no—
235#451—1222no—
236C8×Q8—611no—
237C8×Q8—611no—
238C8×Q8—611no—
239C8×Q8—1221no—
240C8×Q8—1221no—
241C8×Q8C2×D6111yescharacteristic
242#441—1222no—
243#441—1222no—
244C4×Q16—1221no—
245C4×Q16—1221no—
246#432—612no—
247C2×C4⋊C8—611no—
248C2×C4⋊C8—1221no—
249C2×C4×C8—611no—
250C2×C4×C8—1221no—
251#335—611no—
252#335C2×D6111yescharacteristic
253#334—612no—
254#332—612no—
255#332—612no—
256#332—612no—
257#332C2×D6112yes—
258C61#1926511yescharacteristic, complemented, Sylow
259C22×Q8—611no—
260C2×Q16—21no—
261C2×Q16—21no—
262C2×Q16—1222no—
263C2×Q16—1222no—
264C22×C8—611no—
265C22×C8—611no—
266C22×C8—1221no—
267C4⋊Q8—612no—
268C4⋊Q8C22×D6112yes—
269#125—1222no—
270C22⋊Q8—1222no—
271C22⋊Q8—1222no—
272C4×Q8—611no—
273C4×Q8—611no—
274C4×Q8—611no—
275C4×Q8—1221no—
276C4×Q8—1221no—
277C4×Q8—1222no—
278C4×Q8—1222no—
279C4×Q8C22×D6111yescharacteristic
280#117—1222no—
281C2×C4⋊C4—611no—
282C2×C4⋊C4—1221no—
283C2×C42—611no—
284C2×C42—1221no—
285#107—1221no—
286#107—1221no—
287#106—22no—
288#106—1222no—
289C4⋊C8—21no—
290C4⋊C8—611no—
291C4⋊C8—612no—
292C4⋊C8—1221no—
293C4⋊C8—1221no—
294C4⋊C8—1221no—
295C4⋊C8—1221no—
296C4⋊C8C22×D6111yescharacteristic
297C4⋊C8C22×D6112yes—
298#103—22no—
299#103—1222no—
300#103—1222no—
301#103—1222no—
302#103—1222no—
303#103—1222no—
304C22⋊C8—1222no—
305#97—1222no—
306C4×C8—21no—
307C4×C8—611no—
308C4×C8—611no—
309C4×C8—611no—
310C4×C8—1221no—
311C4×C8—1221no—
312C4×C8—1221no—
313C4×C8C22×D6111yescharacteristic
314C2×Q8—22no—
315C2×Q8—611no—
316C2×Q8—1221no—
317C2×Q8—1222no—
318C2×Q8—1222no—
319C2×Q8—1222no—
320C2×Q8D4×D6111yescharacteristic
321C22×C4—611no—
322C22×C4—611no—
323C22×C4—611no—
324C22×C4—1221no—
325C22×C4—1221no—
326Q16—42no—
327Q16—42no—
328Q16—2444no—
329Q16—2444no—
330C2×C8—21no—
331C2×C8—22no—
332C2×C8—611no—
333C2×C8—611no—
334C2×C8—1222no—
335C2×C8—1221no—
336C2×C8—1222no—
337C2×C8—2441no—
338C2×C8—2442no—
339C2×C8D4×D6111yescharacteristic
340C2×C8D4×D6111yescharacteristic
341C4⋊C4—21no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—611no—
345C4⋊C4—1221no—
346C4⋊C4—1221no—
347C4⋊C4—1221no—
348C4⋊C4—1221no—
349C4⋊C4—1222no—
350C4⋊C4—1222no—
351C4⋊C4—2442no—
352C4⋊C4—2442no—
353C4⋊C4D4×D6111yescharacteristic
354C22⋊C4—2442no—
355C42—21no—
356C42—611no—
357C42—611no—
358C42—611no—
359C42—1221no—
360C42—1221no—
361C42—1221no—
362C42—2442no—
363C42C23×D6111yescharacteristic
364C23—611no—
365Q8—21no—
366Q8—42no—
367Q8—42no—
368Q8—612no—
369Q8—1224no—
370Q8—1221no—
371Q8—2442no—
372Q8—2442no—
373Q8#1451980012yes—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—611no—
378C2×C4—611no—
379C2×C4—611no—
380C2×C4—1222no—
381C2×C4—1224no—
382C2×C4—1221no—
383C2×C4—1221no—
384C2×C4—1221no—
385C2×C4—1221no—
386C2×C4—1221no—
387C2×C4—1221no—
388C2×C4—2442no—
389C2×C4—2442no—
390C2×C4C2×D4×D6111yescharacteristic
391C2×C4C2×D4×D6111yescharacteristic
392C2×C4#41×D6111yescharacteristic
393C8—21no—
394C8—42no—
395C8—41no—
396C8—1221no—
397C8—1221no—
398C8—2442no—
399C8—2441no—
400C8#1451980612yes—
401C22—612no—
402C22—1221no—
403C22—1221no—
404C22#121×D6111yescharacteristic
405C4—22no—
406C4—21no—
407C4—41no—
408C4—82no—
409C4—1221no—
410C4—1222no—
411C4—1221no—
412C4—1221no—
413C4—4882no—
414C4—2441no—
415C4#135×D6111yescharacteristic
416C4C2×#1451980011yescharacteristic
417C4C2×#1451980611yescharacteristic
418C4#1684055811yescharacteristic
419C2—1222no—
420C2#2145528611yescharacteristic
421C2#2145540211yescharacteristic
422C2#473×D6111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1670C2
socleC22×C61C2×#121
prime radicalC22×D61#121
hypercenter#1670D61
hypocenterC61#19265
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1670

Complements

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

NH ≅ G/Ntype
4. #21445248C2semidirect
5. #21445247C2semidirect
6. #21445214C2semidirect
7. #21445213C2semidirect
8. #21445186C2semidirect
11. C61×#1670C2semidirect
216. #1670D61semidirect
258. C61#19265semidirect

Maximal subgroups

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

subgroupindex
#214564032
#214559512
C8×Q8×D612
#214452482
#214452472
#214452142
#214452132
#214451862
#214445812
#214445182
C61×#16702
#1926561