Finite Group Databasebeta

← #29663946

Subgroups — #29663946

Order 15616.

638 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#500×D61C211yesmaximal, characteristic, complemented
2#21455633C212yesmaximal
3C8×#11758226C211yesmaximal, characteristic
4#21445223C211yesmaximal, characteristic, complemented
5#21445221C212yesmaximal, complemented
6#21445219C211yesmaximal, characteristic, complemented
7#21445217C212yesmaximal, complemented
8#21445215C211yesmaximal, characteristic, complemented
9#21444576C211yesmaximal, characteristic, complemented
10#21444498C212yesmaximal, complemented
11C61×#1676C211yesmaximal, characteristic, complemented
12C4⋊Q8×D61C2212yes—
13C4×#11758226C2211yescharacteristic
14#16840167C2211yescharacteristic
15#16840166—21no—
16#16840166—21no—
17#16840165C2212yes—
18#16840057C2212yes—
19#16840053C2212yes—
20#16840049C2212yes—
21#16840045—22no—
22#16840041—22no—
23Q8×#92418C2211yescharacteristic
24C8×D244—21no—
25C8×D244C2211yescharacteristic
26#16839827—21no—
27#16839827C2211yescharacteristic
28#16839814—21no—
29#16839814C2211yescharacteristic
30#16839803—22no—
31#107×D61—21no—
32#107×D61—21no—
33#16839784C2212yes—
34#16839762—22no—
35C4×C8×D61—21no—
36C4×C8×D61C2211yescharacteristic
37#16839515C2211yescharacteristic
38#16839514C2212yes—
39#16839511C2212yes—
40#16839510C2212yes—
41#16839510C2212yes—
42C61×#500C2211yescharacteristic
43C61×#477C2212yes—
44C8×Q8×C61C2211yescharacteristic
45C61×#335C2211yescharacteristic
46C61×#333C2212yes—
47C2×#11758226D411yescharacteristic
48C2×Q8×D61—22no—
49C2×#11758189—22no—
50C2×C8×D61—21no—
51C2×C8×D61—21no—
52C2×C8×D61D411yescharacteristic
53C2×C8×D61D411yescharacteristic
54#14519777C2312yes—
55#14519776—22no—
56#14519776—22no—
57#14519775—22no—
58Q8×Dic61D411yescharacteristic
59C4×D244—21no—
60C4×D244C2311yescharacteristic
61#14519739—21no—
62#14519739D411yescharacteristic
63#14519735—21no—
64#14519735D411yescharacteristic
65C4⋊C4×D61—22no—
66C4⋊C4×D61—22no—
67C42×D61—21no—
68C42×D61C2311yescharacteristic
69#14519711—21no—
70#14519711—21no—
71#14519710—22no—
72#14519710—22no—
73#14519710—22no—
74#14519709—22no—
75#14519708—22no—
76C4⋊#92418—21no—
77C4⋊#92418C2311yescharacteristic
78C4⋊#92418C2312yes—
79#14519706—22no—
80#14519701—22no—
81#14519699—22no—
82#14519697—22no—
83#14519689—21no—
84#14519689—21no—
85#14519689—21no—
86C4×#92418—21no—
87C4×#92418C2311yescharacteristic
88C8×Dic61—21no—
89C8×Dic61D411yescharacteristic
90C8×Dic61D411yescharacteristic
91C4⋊Q8×C61C2312yes—
92C4×Q8×C61C2311yescharacteristic
93#107×C61—21no—
94#107×C61—21no—
95#107×C61—22no—
96C4⋊C8×C61—21no—
97C4⋊C8×C61C2311yescharacteristic
98C4⋊C8×C61C2312yes—
99#103×C61—22no—
100C4×C8×C61—21no—
101C4×C8×C61C2311yescharacteristic
102C2×Q8×C61—22no—
103C2×Q8×C61C2×D411yescharacteristic
104#11758226—22no—
105#11758226Q1614yes—
106Q8×D61—84no—
107C2×D244—21no—
108C2×D244C2×D411yescharacteristic
109C2×C4×D61—21no—
110C2×C4×D61—21no—
111C2×C4×D61—21no—
112C2×C4×D61—42no—
113C2×C4×D61C2×D411yescharacteristic
114C2×C4×D61C2×D411yescharacteristic
115C2×C4×D61#4111yescharacteristic
116C2×Dic122—22no—
117C2×Dic122—42no—
118C2×C8×C61—21no—
119C2×C8×C61—22no—
120C2×C8×C61C2×D411yescharacteristic
121C2×C8×C61C2×D411yescharacteristic
122C4⋊C4×C61—21no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—22no—
125C4⋊C4×C61C2×D411yescharacteristic
126C42×C61—21no—
127C42×C61C2411yescharacteristic
128#11758198—21no—
129#11758198—21no—
130#11758198—21no—
131C4⋊Dic61—21no—
132C4⋊Dic61—22no—
133C4⋊Dic61—22no—
134C4⋊Dic61C2×D411yescharacteristic
135#11758196—42no—
136#11758196—42no—
137C4×Dic61—21no—
138C4×Dic61—21no—
139C4×Dic61C2×D411yescharacteristic
140C4×Dic61C2×D411yescharacteristic
141C4×Dic61#4111yescharacteristic
142C2×#92418—21no—
143C2×#92418—22no—
144C2×#92418C2×D411yescharacteristic
145C2×#92418C2×D411yescharacteristic
146#11758189—44no—
147C8×D61—22no—
148C8×D61—22no—
149C8×D61—42no—
150C8×D61—42no—
151C22×D61—21no—
152C22×D61#12111yescharacteristic
153Q8×C61—21no—
154Q8×C61—82no—
155Q8×C61C2×Q1612yes—
156C2×C4×C61—21no—
157C2×C4×C61—21no—
158C2×C4×C61—42no—
159C2×C4×C61C22×D411yescharacteristic
160C2×C4×C61C22×D411yescharacteristic
161C2×C4×C61C2×#4111yescharacteristic
162C2×Dic61—21no—
163C2×Dic61—42no—
164C2×Dic61#12111yescharacteristic
165C2×Dic61#12111yescharacteristic
166C2×Dic61#12111yescharacteristic
167D244—24no—
168D244—21no—
169D244C2×Q1612yes—
170C4×D61—21no—
171C4×D61—21no—
172C4×D61—24no—
173C4×D61—22no—
174C4×D61—41no—
175C4×D61—41no—
176C4×D61—42no—
177C4×D61—84no—
178C4×D61C2×Q1612yes—
179C4×D61#13512yes—
180Dic122—84no—
181Dic122—82no—
182C8×C61—21no—
183C8×C61—42no—
184C8×C61—41no—
185C8×C61#13712yes—
186#92418—21no—
187#92418—42no—
188#92418—41no—
189#92418#13712yes—
190#19272—611nomaximal, Sylow
191C22×C61C2×#12111yescharacteristic
192C2×D61—21no—
193C2×D61—21no—
194C2×D61—22no—
195C2×D61—41no—
196C2×D61—41no—
197C2×D61#46712yes—
198C4×C61—22no—
199C4×C61—21no—
200C4×C61—41no—
201C4×C61—82no—
202C4×C61C22×Q1611yescharacteristic
203C4×C61C2×#13511yescharacteristic
204C4×C61C2×#13711yescharacteristic
205C4×C61C2×#13711yescharacteristic
206Dic61—21no—
207Dic61—41no—
208Dic61—82no—
209Dic61#46712yes—
210Dic61#47212yes—
211C2×#500—611no—
212#3079—612no—
213C8×#41—611no—
214#1676—612no—
215#1676—611no—
216#1676D6111yescharacteristic, complemented
217#1675—611no—
218#1675—611no—
219#1675—612no—
220#1567—611no—
221#1548—612no—
222C2×C61#303911yescharacteristic
223C2×C61C2×#46711yescharacteristic
224C2×C61C2×#47211yescharacteristic
225D61—22no—
226D61—42no—
227C2×C4⋊Q8—612no—
228C4×#41—611no—
229#500—612no—
230#500—611no—
231#500—1221no—
232#500—1221no—
233#500C2×D6111yescharacteristic
234#477—612no—
235#477C2×D6112yes—
236#476—612no—
237#476—612no—
238#475—1222no—
239#474—1222no—
240C8×Q8—611no—
241C8×Q8C2×D6111yescharacteristic
242C8×D4—611no—
243C8×D4—611no—
244C8×D4—1221no—
245C8×D4—1221no—
246#431—611no—
247#431—1221no—
248#428—1222no—
249C2×#107—1221no—
250C2×#107—1221no—
251#423—612no—
252#417—1222no—
253C2×C4×C8—611no—
254C2×C4×C8—1221no—
255#335—612no—
256#335—611no—
257#335C2×D6111yescharacteristic
258#333—612no—
259#333—612no—
260#333—612no—
261#333C2×D6112yes—
262C61#1927211yescharacteristic, complemented, Sylow
263C2×#41—611no—
264C22×Q8—1222no—
265C2×M16—1222no—
266C22×C8—611no—
267C22×C8—611no—
268C22×C8—1221no—
269C22×C8—1221no—
270C4⋊Q8—612no—
271C4⋊Q8—1222no—
272C4⋊Q8—1222no—
273C4⋊Q8—1222no—
274C4⋊Q8C22×D6112yes—
275C4×Q8—611no—
276C4×Q8C22×D6111yescharacteristic
277C4×D4—611no—
278C4×D4—611no—
279C4×D4—1221no—
280C4×D4—1221no—
281#117—611no—
282#117—1221no—
283C2×C4⋊C4—1222no—
284C2×C4⋊C4—1222no—
285C2×C42—611no—
286C2×C42—1221no—
287#107—21no—
288#107—21no—
289#107—22no—
290#107—1222no—
291#107—1222no—
292#107—1221no—
293#107—1221no—
294#107—1222no—
295#106—1222no—
296#106—1222no—
297C4⋊C8—21no—
298C4⋊C8—611no—
299C4⋊C8—612no—
300C4⋊C8—1221no—
301C4⋊C8—1222no—
302C4⋊C8C22×D6111yescharacteristic
303C4⋊C8C22×D6112yes—
304#103—22no—
305#103—1222no—
306#102—1222no—
307#102—1222no—
308C22⋊C8—1221no—
309C22⋊C8—1221no—
310C22⋊C8—1221no—
311C4×C8—21no—
312C4×C8—611no—
313C4×C8—611no—
314C4×C8—611no—
315C4×C8—1221no—
316C4×C8—1221no—
317C4×C8C22×D6111yescharacteristic
318#41—614no—
319#41—1222no—
320C2×Q8—22no—
321C2×Q8—1222no—
322C2×Q8—4884no—
323C2×Q8—2442no—
324C2×Q8D4×D6111yescharacteristic
325C2×D4—611no—
326C2×D4—1221no—
327C22×C4—611no—
328C22×C4—611no—
329C22×C4—611no—
330C22×C4—1221no—
331C22×C4—1221no—
332C22×C4—1221no—
333C22×C4—2442no—
334M16—2444no—
335C2×C8—21no—
336C2×C8—22no—
337C2×C8—611no—
338C2×C8—611no—
339C2×C8—1222no—
340C2×C8—1222no—
341C2×C8—1221no—
342C2×C8—1222no—
343C2×C8—2442no—
344C2×C8—2442no—
345C2×C8D4×D6111yescharacteristic
346C2×C8D4×D6111yescharacteristic
347C4⋊C4—21no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—611no—
351C4⋊C4—1221no—
352C4⋊C4—1222no—
353C4⋊C4—1222no—
354C4⋊C4—2442no—
355C4⋊C4—2442no—
356C4⋊C4D4×D6111yescharacteristic
357C22⋊C4—1221no—
358C22⋊C4—1221no—
359C22⋊C4—1221no—
360C42—21no—
361C42—611no—
362C42—611no—
363C42—611no—
364C42—1221no—
365C42—1221no—
366C42C23×D6111yescharacteristic
367C23—611no—
368C23—1221no—
369Q8—21no—
370Q8—82no—
371Q8—4882no—
372Q8—4884no—
373Q8Q16×D6112yes—
374D4—612no—
375D4—1224no—
376D4—1221no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—42no—
380C2×C4—611no—
381C2×C4—611no—
382C2×C4—612no—
383C2×C4—612no—
384C2×C4—611no—
385C2×C4—1222no—
386C2×C4—1224no—
387C2×C4—1221no—
388C2×C4—1221no—
389C2×C4—1221no—
390C2×C4—4884no—
391C2×C4—2442no—
392C2×C4—2442no—
393C2×C4—2441no—
394C2×C4—2441no—
395C2×C4C2×D4×D6111yescharacteristic
396C2×C4C2×D4×D6111yescharacteristic
397C2×C4#41×D6111yescharacteristic
398C8—21no—
399C8—42no—
400C8—41no—
401C8—612no—
402C8—1221no—
403C8—2442no—
404C8—2441no—
405C8#1451980912yes—
406C22—612no—
407C22—1221no—
408C22—1221no—
409C22—1222no—
410C22—2441no—
411C22—2441no—
412C22#121×D6111yescharacteristic
413C4—22no—
414C4—21no—
415C4—41no—
416C4—82no—
417C4—612no—
418C4—612no—
419C4—1221no—
420C4—4882no—
421C4—2441no—
422C4C2×Q16×D6111yescharacteristic
423C4#135×D6111yescharacteristic
424C4#137×D6111yescharacteristic
425C4C2×#1451980911yescharacteristic
426C2—1222no—
427C2—2442no—
428C2#2145530311yescharacteristic
429C2#467×D6111yescharacteristic
430C2#472×D6111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1676C2
socleC22×C61C2×#121
prime radicalC2×#11758226D4
hypercenter#1676D61
hypocenterC61#19272
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1676

Complements

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

NH ≅ G/Ntype
1. #500×D61C2semidirect
4. #21445223C2semidirect
5. #21445221C2semidirect
6. #21445219C2semidirect
7. #21445217C2semidirect
8. #21445215C2semidirect
9. #21444576C2semidirect
10. #21444498C2semidirect
11. C61×#1676C2semidirect
216. #1676D61semidirect
262. C61#19272semidirect

Maximal subgroups

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

subgroupindex
#500×D612
#214556332
C8×#117582262
#214452232
#214452212
#214452192
#214452172
#214452152
#214445762
#214444982
C61×#16762
#1927261