Finite Group Databasebeta

← #29663254

Subgroups — #29663254

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#21456766C212yesmaximal
2#500×D61C211yesmaximal, characteristic
3#21454748C211yesmaximal, characteristic
4#21445336C212yesmaximal, complemented
5#21445335C212yesmaximal, complemented
6#21445214C211yesmaximal, characteristic, complemented
7#21445213C211yesmaximal, characteristic, complemented
8#21445210C211yesmaximal, characteristic, complemented
9#21444566C211yesmaximal, characteristic
10#21444502C212yesmaximal
11C61×#1674C211yesmaximal, characteristic, complemented
12#16840316C2211yescharacteristic
13C4⋊Q8×D61C2212yes—
14#16840167C2211yescharacteristic
15#16840166—21no—
16#16840166—21no—
17#16840165C2212yes—
18#16840140—22no—
19#16840111—22no—
20#16839973C2212yes—
21#16839972C2212yes—
22#16839969C2212yes—
23#16839919—22no—
24#16839916—22no—
25#16839893C2211yescharacteristic
26#16839891—21no—
27#16839889C2211yescharacteristic
28C8×Dic122—21no—
29C8×Dic122C2211yescharacteristic
30#16839814C2212yes—
31#107×D61—21no—
32#107×D61—21no—
33#16839787—21no—
34#16839784C2211yescharacteristic
35#16839716—21no—
36C4×C8×D61C2211yescharacteristic
37#16839515C2211yescharacteristic
38#16839512C2212yes—
39#16839509C2212yes—
40#16839508C2212yes—
41#16839507C2212yes—
42C61×#500C2211yescharacteristic
43C61×#462C2212yes—
44C8×Q8×C61C2211yescharacteristic
45C61×#335C2211yescharacteristic
46C61×#332C2212yes—
47C2×Q8×D61—22no—
48C2×Dic244—22no—
49C2×#11758202—22no—
50C2×#11758202—22no—
51C2×#11758189—21no—
52C2×C8×D61D411yescharacteristic
53C2×C8×D61D411yescharacteristic
54#14519777C2312yes—
55#14519776—22no—
56#14519776—22no—
57#14519776D411yescharacteristic
58#14519775—22no—
59#14519775D411yescharacteristic
60#14519767—21no—
61#14519757D411yescharacteristic
62#14519755—21no—
63#14519753D411yescharacteristic
64C4×Dic122—21no—
65C4×Dic122C2311yescharacteristic
66#14519734—21no—
67C4⋊C4×D61—22no—
68C4⋊C4×D61—22no—
69C42×D61C2311yescharacteristic
70#14519711—21no—
71#14519711—21no—
72#14519710—22no—
73#14519710—22no—
74C4⋊#92418—21no—
75C4⋊#92418C2312yes—
76C4⋊#92418C2311yescharacteristic
77#14519706—21no—
78#14519706—21no—
79#14519706—21no—
80#14519702—22no—
81#14519701—22no—
82#14519700—22no—
83#14519689—22no—
84#14519688—21no—
85#14519687—21no—
86C4×#92418C2311yescharacteristic
87C8×Dic61—21no—
88C8×Dic61—22no—
89C8×Dic61D411yescharacteristic
90C8×Dic61D411yescharacteristic
91C2×Q16×C61—22no—
92C4⋊Q8×C61C2312yes—
93C4×Q8×C61C2311yescharacteristic
94#107×C61—21no—
95#107×C61—21no—
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—82no—
105Q8×D61—82no—
106C2×C4×D61—42no—
107C2×C4×D61C2×D411yescharacteristic
108C2×C4×D61C2×D411yescharacteristic
109C2×C4×D61#4111yescharacteristic
110C2×Dic122—21no—
111C2×Dic122—22no—
112C2×Dic122—42no—
113C2×Dic122C2×D411yescharacteristic
114Q16×C61—82no—
115C2×C8×C61—21no—
116C2×C8×C61—22no—
117C2×C8×C61C2×D411yescharacteristic
118C2×C8×C61C2×D411yescharacteristic
119C4⋊C4×C61—21no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—22no—
122C4⋊C4×C61C2×D411yescharacteristic
123C42×C61—21no—
124C42×C61C2411yescharacteristic
125#11758202—82no—
126#11758202—82no—
127#11758198—21no—
128#11758198—21no—
129C4⋊Dic61—21no—
130C4⋊Dic61—22no—
131C4⋊Dic61—22no—
132C4⋊Dic61C2×D411yescharacteristic
133#11758196—21no—
134#11758196—21no—
135#11758196—21no—
136#11758196—21no—
137#11758196—21no—
138#11758196—42no—
139#11758196—42no—
140C4×Dic61—21no—
141C4×Dic61C2×D411yescharacteristic
142C4×Dic61C2×D411yescharacteristic
143C4×Dic61#4111yescharacteristic
144C2×#92418—21no—
145C2×#92418—22no—
146C2×#92418C2×D411yescharacteristic
147C2×#92418C2×D411yescharacteristic
148Dic244—82no—
149#11758189—42no—
150C8×D61—22no—
151C8×D61Q1614yescomplemented
152C22×D61#12111yescharacteristic
153Q8×C61—21no—
154Q8×C61—21no—
155Q8×C61—42no—
156Q8×C61—42no—
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—21no—
170C4×D61—21no—
171C4×D61—22no—
172C4×D61—84no—
173C4×D61C2×Q1612yes—
174C4×D61#13712yes—
175Dic122—21no—
176Dic122—21no—
177Dic122—42no—
178Dic122—42no—
179Dic122—42no—
180Dic122—84no—
181C8×C61—21no—
182C8×C61—41no—
183C8×C61—42no—
184C8×C61C2×Q1612yes—
185#92418—21no—
186#92418—41no—
187#92418—42no—
188#92418C2×Q1612yes—
189#19208—611nomaximal, Sylow
190C22×C61C2×#12111yescharacteristic
191C2×D61—21no—
192C2×D61—21no—
193C2×D61#46212yes—
194C4×C61—21no—
195C4×C61—41no—
196C4×C61—41no—
197C4×C61—82no—
198C4×C61C22×Q1611yescharacteristic
199C4×C61C2×#13711yescharacteristic
200C4×C61#57611yescharacteristic
201C4×C61#57611yescharacteristic
202Dic61—21no—
203Dic61—41no—
204Dic61—41no—
205Dic61—82no—
206Dic61#46212yes—
207Dic61#47312yes—
208#3209—612no—
209C2×#500—611no—
210#2960—611no—
211#1694—612no—
212#1694—612no—
213#1674—611no—
214#1674—611no—
215#1674—611no—
216#1674D6111yescharacteristic, complemented
217#1564—611no—
218#1549—612no—
219C2×C61C2×#46211yescharacteristic
220C2×C61#305811yescharacteristic
221C2×C61C2×#47311yescharacteristic
222D61—22no—
223#533—611no—
224C2×C4⋊Q8—612no—
225#500—612no—
226#500—611no—
227#500—1221no—
228#500—1221no—
229#500C2×D6111yescharacteristic
230#494—1222no—
231#487—1222no—
232#462—612no—
233#462—612no—
234#462—612no—
235#462C2×D6112yes—
236#451—1222no—
237#451—1222no—
238#446—611no—
239#446—611no—
240#446—1221no—
241C8×Q8—611no—
242C8×Q8—1221no—
243C8×Q8C2×D6111yescharacteristic
244#431—612no—
245C2×#107—1221no—
246C2×#107—1221no—
247#424—1221no—
248#423—611no—
249#405—1221no—
250C2×C4×C8—611no—
251#335—611no—
252#335C2×D6111yescharacteristic
253#334—612no—
254#332—612no—
255#332—612no—
256#332—612no—
257#332C2×D6112yes—
258C61#1920811yescharacteristic, complemented, Sylow
259C22×Q8—1222no—
260C2×Q16—22no—
261C2×Q16—1222no—
262C2×Q16—1222no—
263C2×Q16—1222no—
264C2×M16—1221no—
265C22×C8—611no—
266C22×C8—611no—
267C4⋊Q8—611no—
268C4⋊Q8—611no—
269C4⋊Q8—612no—
270C4⋊Q8—1222no—
271C4⋊Q8—1222no—
272C4⋊Q8—1222no—
273C4⋊Q8C22×D6112yes—
274#125—1221no—
275C22⋊Q8—611no—
276C22⋊Q8—611no—
277C22⋊Q8—1221no—
278C4×Q8—611no—
279C4×Q8—1221no—
280C4×Q8C22×D6111yescharacteristic
281#117—1221no—
282C2×C4⋊C4—1222no—
283C2×C4⋊C4—1222no—
284C2×C42—611no—
285#107—21no—
286#107—21no—
287#107—1222no—
288#107—1222no—
289#107—1221no—
290#107—1221no—
291C4⋊C8—21no—
292C4⋊C8—611no—
293C4⋊C8—612no—
294C4⋊C8—1221no—
295C4⋊C8—1221no—
296C4⋊C8—1221no—
297C4⋊C8—1221no—
298C4⋊C8C22×D6111yescharacteristic
299C4⋊C8C22×D6112yes—
300#103—22no—
301#103—1222no—
302#103—1222no—
303#103—1222no—
304C22⋊C8—1222no—
305#97—1221no—
306#97—1221no—
307C4×C8—21no—
308C4×C8—611no—
309C4×C8—611no—
310C4×C8—611no—
311C4×C8—1222no—
312C4×C8—1221no—
313C4×C8C22×D6111yescharacteristic
314C2×Q8—22no—
315C2×Q8—611no—
316C2×Q8—1221no—
317C2×Q8—1222no—
318C2×Q8—4882no—
319C2×Q8—4882no—
320C2×Q8—2442no—
321C2×Q8D4×D6111yescharacteristic
322C22×C4—611no—
323C22×C4—611no—
324C22×C4—611no—
325C22×C4—2442no—
326Q16—82no—
327Q16—4882no—
328Q16—4882no—
329Q16—4882no—
330M16—2442no—
331C2×C8—21no—
332C2×C8—22no—
333C2×C8—611no—
334C2×C8—614no—
335C2×C8—611no—
336C2×C8—1222no—
337C2×C8—1221no—
338C2×C8—1222no—
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—1221no—
350C4⋊C4—1221no—
351C4⋊C4—1222no—
352C4⋊C4—1222no—
353C4⋊C4—2442no—
354C4⋊C4—2442no—
355C4⋊C4D4×D6111yescharacteristic
356C22⋊C4—1221no—
357C22⋊C4—1221no—
358C42—21no—
359C42—611no—
360C42—611no—
361C42—611no—
362C42—1221no—
363C42C23×D6111yescharacteristic
364C23—611no—
365Q8—21no—
366Q8—21no—
367Q8—42no—
368Q8—42no—
369Q8—1221no—
370Q8—1221no—
371Q8—4884no—
372Q8—2442no—
373Q8—2442no—
374Q8—2442no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—42no—
378C2×C4—611no—
379C2×C4—611no—
380C2×C4—612no—
381C2×C4—612no—
382C2×C4—611no—
383C2×C4—1222no—
384C2×C4—1221no—
385C2×C4—1221no—
386C2×C4—1221no—
387C2×C4—1221no—
388C2×C4—4884no—
389C2×C4—2442no—
390C2×C4C2×D4×D6111yescharacteristic
391C2×C4C2×D4×D6111yescharacteristic
392C2×C4#41×D6111yescharacteristic
393C8—21no—
394C8—41no—
395C8—42no—
396C8—612no—
397C8—1221no—
398C8—2441no—
399C8—2442no—
400C8Q16×D6112yes—
401C22—612no—
402C22—1221no—
403C22—1221no—
404C22#121×D6111yescharacteristic
405C4—21no—
406C4—41no—
407C4—41no—
408C4—82no—
409C4—612no—
410C4—612no—
411C4—1221no—
412C4—4882no—
413C4—2441no—
414C4—2441no—
415C4C2×Q16×D6111yescharacteristic
416C4#137×D6111yescharacteristic
417C4#1684056311yescharacteristic
418C4#1684056411yescharacteristic
419C2—1222no—
420C2#462×D6111yescharacteristic
421C2#2145546211yescharacteristic
422C2#473×D6111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1674C2
socleC22×C61C2×#121
prime radicalC22×D61#121
hypercenter#1674D61
hypocenterC61#19208
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1674

Complements

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

NH ≅ G/Ntype
4. #21445336C2semidirect
5. #21445335C2semidirect
6. #21445214C2semidirect
7. #21445213C2semidirect
8. #21445210C2semidirect
11. C61×#1674C2semidirect
151. C8×D61Q16semidirect
216. #1674D61semidirect
258. C61#19208semidirect

Maximal subgroups

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

subgroupindex
#214567662
#500×D612
#214547482
#214453362
#214453352
#214452142
#214452132
#214452102
#214445662
#214445022
C61×#16742
#1920861