Finite Group Databasebeta

← #29665227

Subgroups — #29665227

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455612C211yesmaximal, characteristic
2C2×#16839964C211yesmaximal, characteristic
3#21454842C211yesmaximal, characteristic
4#21445239C214yesmaximal, complemented
5#21445191C214yesmaximal, complemented
6#21444590C211yesmaximal, characteristic, complemented
7#21444491C211yesmaximal, characteristic, complemented
8#21444476C211yesmaximal, characteristic, complemented
9C61×#3229C211yesmaximal, characteristic, complemented
10C22×#11758201—21no—
11C2×C4×D244C2211yescharacteristic
12#16840059—22no—
13#16840049C2212yes—
14#16840039C2212yes—
15#16839964—22no—
16#16839964C2214yes—
17C8×D244C2214yes—
18#16839827—22no—
19#16839815C2211yescharacteristic
20#16839805—21no—
21#16839786C2211yescharacteristic
22C2×C4⋊#92418C2211yescharacteristic
23#16839766—21no—
24C2×#14519701—21no—
25C22⋊C8×D61—22no—
26#16839513C2214yescomplemented
27#16839511C2212yescomplemented
28#16839508C2214yescomplemented
29#16839508C2212yescomplemented
30C61×#533C2211yescharacteristic
31C2×C4⋊Q8×C61C2211yescharacteristic
32C61×#498C2212yescomplemented
33C61×#487C2212yescomplemented
34C61×#483—22no—
35C61×#475—22no—
36C61×#431C2211yescharacteristic
37C22×D244D411yescharacteristic
38C22×C4×D61—21no—
39C2×#11758201—24no—
40C2×#11758201—44no—
41C2×#11758194—21no—
42C22×#92418—21no—
43C2×C8×D61—24no—
44C4×D244—21no—
45C4×D244—21no—
46C4×D244D412yes—
47C4×D244C2312yes—
48C2×C4⋊Dic61D411yescharacteristic
49C2×#11758198—21no—
50#14519710—22no—
51#14519708—22no—
52C4⋊#92418—21no—
53C4⋊#92418—21no—
54C4⋊#92418C2312yes—
55C4⋊#92418C2312yes—
56C4⋊#92418C2312yes—
57#14519701—24no—
58#14519697—22no—
59#14519697—22no—
60C22⋊#92418—21no—
61#14519689—24no—
62C4×#92418—21no—
63C22×Q8×C61—21no—
64C4⋊Q8×C61—21no—
65C4⋊Q8×C61—21no—
66C4⋊Q8×C61—21no—
67C4⋊Q8×C61C2312yes—
68C4⋊Q8×C61C2311yescharacteristic
69C4⋊Q8×C61C2311yescharacteristic
70#125×C61—21no—
71C22⋊Q8×C61—21no—
72C22⋊Q8×C61—21no—
73C4×Q8×C61—21no—
74C4×Q8×C61—21no—
75#117×C61—21no—
76C2×C4⋊C4×C61—21no—
77C2×C4⋊C4×C61—21no—
78C2×C42×C61C2311yescharacteristic
79#106×C61—24no—
80C4⋊C8×C61D412yes—
81#103×C61—24no—
82C22⋊C8×C61D412yes—
83C4×C8×C61C2312yes—
84C23×D61—21no—
85C2×Q8×C61—21no—
86C2×Q8×C61—22no—
87C2×Q8×C61—21no—
88C2×Q8×C61—41no—
89C2×Q8×C61—42no—
90C2×Q8×C61—42no—
91C22×C4×C61—41no—
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61#4111yescharacteristic
95C22×Dic61—21no—
96C2×D244—21no—
97C2×D244—22no—
98C2×D244SD1614yes—
99C2×D244C2×D412yes—
100C2×C4×D61—22no—
101C2×C4×D61—22no—
102C2×C4×D61—82no—
103C2×C8×C61C2×D414yes—
104C4⋊C4×C61—22no—
105C4⋊C4×C61—21no—
106C4⋊C4×C61—21no—
107C4⋊C4×C61—21no—
108C4⋊C4×C61—22no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—41no—
111C4⋊C4×C61—41no—
112C4⋊C4×C61—41no—
113C4⋊C4×C61—41no—
114C22⋊C4×C61—41no—
115C42×C61—41no—
116C42×C61C2×D411yescharacteristic
117C42×C61C2×D411yescharacteristic
118C42×C61#4111yescharacteristic
119C42×C61C2411yescharacteristic
120#11758201—48no—
121#11758198—22no—
122#11758198—22no—
123C4⋊Dic61—21no—
124C4⋊Dic61C2×D412yes—
125#11758194—42no—
126C2×#92418—22no—
127C2×#92418—22no—
128C2×#92418—22no—
129C2×#92418—42no—
130C8×D61—84no—
131C23×C61#12111yescharacteristic
132C22×D61—22no—
133C22×D61—24no—
134C22×D61—81no—
135C22×D61—81no—
136Q8×C61—42no—
137Q8×C61—42no—
138Q8×C61—42no—
139Q8×C61—81no—
140Q8×C61—82no—
141C2×C4×C61—22no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—42no—
146C2×C4×C61—82no—
147C2×C4×C61#12111yescharacteristic
148C2×C4×C61#12111yescharacteristic
149C2×C4×C61#12111yescharacteristic
150C2×C4×C61C2×SD1612yes—
151C2×C4×C61#13512yes—
152C2×C4×C61C22×D411yescharacteristic
153C2×C4×C61C22×D411yescharacteristic
154C2×C4×C61C2×#4111yescharacteristic
155C2×Dic61—22no—
156C2×Dic61—81no—
157D244—22no—
158D244—24no—
159D244C2×SD1614yes—
160C4×D61—44no—
161C4×D61—82no—
162C8×C61—24no—
163#92418—42no—
164#92418—42no—
165#92418—42no—
166#19390—611nomaximal, Sylow
167C22×C61—21no—
168C22×C61—21no—
169C22×C61#46512yes—
170C22×C61C2×#12111yescharacteristic
171C2×D61—24no—
172C2×D61—42no—
173C2×D61—42no—
174C2×D61—84no—
175C4×C61—21no—
176C4×C61—21no—
177C4×C61—21no—
178C4×C61—81no—
179C4×C61—82no—
180C4×C61—81no—
181C4×C61#46512yes—
182C4×C61C22×SD1611yescharacteristic
183C4×C61C2×#13511yescharacteristic
184C4×C61#57712yes—
185Dic61—42no—
186#3229D6111yescharacteristic, complemented
187#3077—611no—
188C2×#461—611no—
189#2972—611no—
190#1679—614no—
191#1671—614no—
192#1570—611no—
193#1546—611no—
194#1542—611no—
195C2×C61—22no—
196C2×C61#304211yescharacteristic
197C2×C61C2×#46511yescharacteristic
198C2×C61#306911yescharacteristic
199D61—44no—
200C22×SD16—1221no—
201#533C2×D6111yescharacteristic
202C2×C4⋊Q8C2×D6111yescharacteristic
203C2×C4×D4—611no—
204#498C2×D6112yescomplemented
205#487C2×D6112yescomplemented
206#483—22no—
207#478—1222no—
208#476—612no—
209#475—22no—
210#474—612no—
211#461—614no—
212#461—1222no—
213C8×D4—614no—
214C8×D4—1222no—
215#431—611no—
216#431C2×D6111yescharacteristic
217#428—1221no—
218#423—611no—
219C2×C4⋊C8—611no—
220#418—1221no—
221C2×#103—1221no—
222C2×C22⋊C8—1222no—
223#334—614no—
224#333—612no—
225#332—612no—
226#332—614no—
227C61#1939011yescharacteristic, complemented, Sylow
228C22×Q8—21no—
229C22×D4—611no—
230C23×C4—1221no—
231C2×SD16—1224no—
232C2×SD16—2444no—
233C2×M16—1221no—
234C22×C8—1221no—
235C22×C8—1224no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8C22×D6112yes—
240C4⋊Q8C22×D6111yescharacteristic
241C4⋊Q8C22×D6111yescharacteristic
242#125—21no—
243C22⋊Q8—21no—
244C22⋊Q8—21no—
245C4×Q8—21no—
246C4×Q8—21no—
247C4×D4—612no—
248C4×D4—612no—
249C4×D4—1221no—
250C4×D4—1221no—
251#117—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—611no—
255C2×C22⋊C4—1221no—
256C2×C42C22×D6111yescharacteristic
257#107—1222no—
258#106—24no—
259#106—1222no—
260C4⋊C8—612no—
261C4⋊C8—612no—
262C4⋊C8—612no—
263C4⋊C8—1221no—
264C4⋊C8—1221no—
265C4⋊C8C22⋊D6112yes—
266#103—24no—
267#103—1224no—
268#102—1222no—
269#102—1222no—
270C22⋊C8—1221no—
271C22⋊C8—1224no—
272C22⋊C8C22⋊D6112yes—
273C4×C8—1221no—
274C4×C8C22×D6112yes—
275C24—1221no—
276C2×Q8—22no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—41no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×D4—612no—
283C2×D4—614no—
284C2×D4—1221no—
285C2×D4—1222no—
286C22×C4—41no—
287C22×C4—1221no—
288C22×C4—1222no—
289C22×C4—1222no—
290C22×C4—4882no—
291C22×C4D4×D6111yescharacteristic
292C22×C4#1175822411yescharacteristic
293C22×C4C2×C22⋊D6111yescharacteristic
294SD16—2448no—
295M16—2442no—
296C2×C8—1222no—
297C2×C8—1222no—
298C2×C8—1222no—
299C2×C8—4884no—
300C2×C8—2442no—
301C2×C8C2×C22⋊D6114yes—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—21no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—612no—
313C4⋊C4—1221no—
314C22⋊C4—41no—
315C22⋊C4—1222no—
316C22⋊C4—1222no—
317C42—41no—
318C42D4×D6111yescharacteristic
319C42#1175822411yescharacteristic
320C42C2×C22⋊D6111yescharacteristic
321C42C23×D6111yescharacteristic
322C23—1224no—
323C23—1222no—
324C23—4881no—
325C23—4881no—
326C23#1451974811yescharacteristic
327Q8—42no—
328Q8—42no—
329Q8—42no—
330Q8—81no—
331Q8—82no—
332D4—614no—
333D4—1224no—
334D4—1222no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—82no—
341C2×C4—1222no—
342C2×C4—4882no—
343C2×C4—4881no—
344C2×C4—2444no—
345C2×C4#1451974811yescharacteristic
346C2×C4#1451974811yescharacteristic
347C2×C4#1451974811yescharacteristic
348C2×C4SD16×D6112yes—
349C2×C4#1451979812yes—
350C2×C4C2×D4×D6111yescharacteristic
351C2×C4C22×C22⋊D6111yescharacteristic
352C2×C4C2×#1175822411yescharacteristic
353C8—24no—
354C8—2442no—
355C8—2442no—
356C8—2442no—
357C22—21no—
358C22—21no—
359C22—1224no—
360C22—4884no—
361C22—2442no—
362C22—2442no—
363C22#1683998812yes—
364C22C2×#1451974811yescharacteristic
365C4—21no—
366C4—21no—
367C4—21no—
368C4—82no—
369C4—81no—
370C4—81no—
371C4—2442no—
372C4#1683998812yes—
373C4C2×SD16×D6111yescharacteristic
374C4C2×#1451979811yescharacteristic
375C4#1684057012yes—
376C2—22no—
377C2—2444no—
378C2#2145532711yescharacteristic
379C2C2×#1683998811yescharacteristic
380C2#2145555211yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3229C2
socleC22×C61C2×#121
prime radicalC22×D244D4
hypercenter#3229D61
hypocenterC61#19390
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3229

Complements

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

NH ≅ G/Ntype
4. #21445239C2semidirect
5. #21445191C2semidirect
6. #21444590C2semidirect
7. #21444491C2semidirect
8. #21444476C2semidirect
9. C61×#3229C2semidirect
26. #16839513C22semidirect
27. #16839511C22semidirect
28. #16839508C22semidirect
29. #16839508C22semidirect
32. C61×#498C22semidirect
33. C61×#487C22semidirect
186. #3229D61semidirect
204. #498C2×D61semidirect
205. #487C2×D61semidirect
227. C61#19390semidirect

Maximal subgroups

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

subgroupindex
#214556122
C2×#168399642
#214548422
#214452392
#214451912
#214445902
#214444912
#214444762
C61×#32292
#1939061