Finite Group Databasebeta

← #29663291

Subgroups — #29663291

Order 15616.

646 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456757C211yesmaximal, characteristic
2C2×#16840110C211yesmaximal, characteristic
3#21454754C211yesmaximal, characteristic, complemented
4#21445361C214yesmaximal, complemented
5#21445237C214yesmaximal, complemented
6#21444565C211yesmaximal, characteristic, complemented
7#21444500C211yesmaximal, characteristic, complemented
8#21444486C211yesmaximal, characteristic
9C61×#3230C211yesmaximal, characteristic, complemented
10C2×#14519774C2211yescharacteristic
11#16840136—21no—
12#16840110—21no—
13#16840110—21no—
14#16840110—21no—
15#16840110C2212yes—
16#16840110C2212yes—
17#16839962C2214yes—
18#16839912—22no—
19#16839893—21no—
20#16839893C2212yescomplemented
21Q8×#92418—21no—
22Q8×#92418C2212yescomplemented
23#16839815C2211yescharacteristic
24#16839789—21no—
25#16839786C2211yescharacteristic
26C2×#14519697—21no—
27C2×#14519697—21no—
28#16839718—21no—
29C2×C4×#92418C2211yescharacteristic
30#16839514C2212yescomplemented
31#16839512C2212yescomplemented
32#16839511C2214yescomplemented
33#16839505C2214yes—
34C61×#533C2211yescharacteristic
35C2×C4⋊Q8×C61C2211yescharacteristic
36C61×#500—21no—
37C61×#487—21no—
38C61×#484—22no—
39C61×#477C2214yescomplemented
40C61×#431C2211yescharacteristic
41C22×D244—21no—
42C22×D244—21no—
43C22×D244—21no—
44C2×#11758201—24no—
45C2×#11758194—21no—
46C22×#92418D411yescharacteristic
47C22×#92418D411yescharacteristic
48#14519774—21no—
49#14519774—21no—
50#14519774—21no—
51#14519774C2312yes—
52C4⋊#92418—21no—
53C4⋊#92418—22no—
54C4⋊#92418—22no—
55C4⋊#92418C2312yes—
56C4⋊#92418C2312yes—
57#14519697—24no—
58#14519697—22no—
59#14519697—22no—
60#14519697—22no—
61#14519697—22no—
62C22⋊#92418—21no—
63#14519688—22no—
64C4×#92418—21no—
65C4×#92418—22no—
66C4×#92418D411yescharacteristic
67C4×#92418D411yescharacteristic
68C4×#92418C2311yescharacteristic
69C4×#92418C2311yescharacteristic
70C22×Q8×C61—21no—
71C4⋊Q8×C61—21no—
72C4⋊Q8×C61—21no—
73C4⋊Q8×C61—21no—
74C4⋊Q8×C61D411yescharacteristic
75C4⋊Q8×C61D411yescharacteristic
76C4⋊Q8×C61C2312yes—
77#125×C61—21no—
78C22⋊Q8×C61—21no—
79C22⋊Q8×C61D412yes—
80C4×Q8×C61—21no—
81C4×Q8×C61C2312yes—
82#117×C61—21no—
83C2×C4⋊C4×C61—21no—
84C2×C4⋊C4×C61—21no—
85C2×C42×C61C2311yescharacteristic
86#107×C61—24no—
87C4⋊C8×C61C2312yes—
88#103×C61—24no—
89C22⋊C8×C61—21no—
90C4×C8×C61—21no—
91C23×D61—41no—
92C2×Q8×C61—21no—
93C2×Q8×C61—22no—
94C2×Q8×C61—41no—
95C2×Q8×C61—82no—
96C2×Q8×C61C2×D412yes—
97C22×C4×C61—41no—
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61#4111yescharacteristic
101C2×D244—22no—
102C2×D244—22no—
103C2×D244—22no—
104C2×D244—42no—
105C2×D244—42no—
106C2×D244—41no—
107C2×D244—41no—
108C2×D244—41no—
109C2×D244—82no—
110C2×D244—82no—
111C2×C8×C61—22no—
112C4⋊C4×C61—21no—
113C4⋊C4×C61—21no—
114C4⋊C4×C61—21no—
115C4⋊C4×C61—22no—
116C4⋊C4×C61—22no—
117C4⋊C4×C61—22no—
118C4⋊C4×C61—22no—
119C4⋊C4×C61—41no—
120C4⋊C4×C61—41no—
121C4⋊C4×C61C2×D412yes—
122C22⋊C4×C61—22no—
123C42×C61—22no—
124C42×C61C2×D411yescharacteristic
125C42×C61C2×D411yescharacteristic
126C42×C61#4111yescharacteristic
127C42×C61C2411yescharacteristic
128#11758201—84no—
129#11758194—42no—
130C2×#92418—22no—
131C2×#92418—22no—
132C2×#92418—22no—
133C2×#92418—22no—
134C2×#92418C2×D411yescharacteristic
135C2×#92418C2×D411yescharacteristic
136C2×#92418C2×D411yescharacteristic
137C2×#92418C2×D411yescharacteristic
138C23×C61#12111yescharacteristic
139C22×D61—42no—
140C22×D61—82no—
141C22×D61—82no—
142C22×D61—82no—
143Q8×C61—24no—
144Q8×C61—42no—
145Q8×C61—82no—
146Q8×C61—82no—
147C2×C4×C61—22no—
148C2×C4×C61—22no—
149C2×C4×C61—22no—
150C2×C4×C61—22no—
151C2×C4×C61—42no—
152C2×C4×C61—82no—
153C2×C4×C61#12111yescharacteristic
154C2×C4×C61#12111yescharacteristic
155C2×C4×C61#12111yescharacteristic
156C2×C4×C61#13512yes—
157C2×C4×C61#13712yes—
158C2×C4×C61C22×D411yescharacteristic
159C2×C4×C61C22×D411yescharacteristic
160C2×C4×C61C2×#4111yescharacteristic
161D244—42no—
162D244—42no—
163D244—44no—
164D244—82no—
165D244—82no—
166D244—82no—
167D244—82no—
168C8×C61—42no—
169#92418—21no—
170#92418—21no—
171#92418—21no—
172#92418—21no—
173#92418—42no—
174#92418—42no—
175#19212—611nomaximal, Sylow
176C22×C61—21no—
177C22×C61—21no—
178C22×C61#46412yes—
179C22×C61C2×#12111yescharacteristic
180C2×D61—82no—
181C2×D61—82no—
182C2×D61—82no—
183C2×D61—164no—
184C4×C61—21no—
185C4×C61—21no—
186C4×C61—42no—
187C4×C61—42no—
188C4×C61—82no—
189C4×C61#46412yes—
190C4×C61#47212yes—
191C4×C61C2×#13511yescharacteristic
192C4×C61C2×#13711yescharacteristic
193C4×C61#57712yes—
194#3230D6111yescharacteristic, complemented
195#3208—611no—
196C2×#486—611no—
197#2960—611no—
198#1698—614no—
199#1678—614no—
200#1564—611no—
201#1548—611no—
202#1545—611no—
203C2×C61—22no—
204C2×C61C2×#46411yescharacteristic
205C2×C61#305711yescharacteristic
206C2×C61C2×#47211yescharacteristic
207D61—162no—
208#533C2×D6111yescharacteristic
209C2×C4⋊Q8C2×D6111yescharacteristic
210C2×#127—611no—
211#500—21no—
212#492—1221no—
213#487—21no—
214#486—612no—
215#486—612no—
216#486—1221no—
217#486—1221no—
218#486—1221no—
219#484—22no—
220#477C2×D6114yescomplemented
221#460—614no—
222#450—1222no—
223#446—612no—
224#446—1221no—
225C8×Q8—612no—
226C8×Q8—1221no—
227#431—611no—
228#431C2×D6111yescharacteristic
229#424—1221no—
230#423—611no—
231C2×#102—1221no—
232C2×#102—1221no—
233#405—1221no—
234C2×C4×C8—611no—
235#335—612no—
236#334—612no—
237#333—614no—
238#331—614no—
239C61#1921211yescharacteristic, complemented, Sylow
240C22×Q8—21no—
241C22×D4—1221no—
242C22×D4—1221no—
243C22×D4—1221no—
244C2×SD16—1224no—
245C2×M16—1221no—
246C22×C8—611no—
247C22×C8—611no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8C22⋊D6111yescharacteristic
252C4⋊Q8C22⋊D6111yescharacteristic
253C4⋊Q8C22×D6112yes—
254#127—612no—
255#127—1221no—
256#127—1221no—
257#127—1221no—
258#125—21no—
259C22⋊Q8—21no—
260C22⋊Q8C22⋊D6112yes—
261C4×Q8—21no—
262C4×Q8C22×D6112yes—
263#117—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C42C22×D6111yescharacteristic
267#107—24no—
268C4⋊C8—612no—
269C4⋊C8—612no—
270C4⋊C8—1222no—
271C4⋊C8—1222no—
272C4⋊C8—1221no—
273C4⋊C8C22×D6112yes—
274#103—24no—
275#102—1222no—
276#102—1222no—
277#102—1222no—
278#102—1222no—
279#102—1224no—
280C22⋊C8—21no—
281C22⋊C8—1221no—
282#97—1222no—
283C4×C8—21no—
284C4×C8—611no—
285C4×C8—611no—
286C4×C8—611no—
287C4×C8—611no—
288C4×C8—1222no—
289C4×C8—1221no—
290C24—2441no—
291C2×Q8—21no—
292C2×Q8—22no—
293C2×Q8—41no—
294C2×Q8—82no—
295C2×Q8C2×C22⋊D6112yes—
296C2×D4—1222no—
297C2×D4—1222no—
298C2×D4—1222no—
299C2×D4—4882no—
300C2×D4—4882no—
301C2×D4—2442no—
302C2×D4—2441no—
303C2×D4—2441no—
304C2×D4—2441no—
305C2×D4—2442no—
306C22×C4—41no—
307C22×C4D4×D6111yescharacteristic
308C22×C4#1175822411yescharacteristic
309C22×C4C2×C22⋊D6111yescharacteristic
310SD16—4884no—
311M16—2442no—
312C2×C8—22no—
313C2×C8—611no—
314C2×C8—611no—
315C2×C8—611no—
316C2×C8—611no—
317C2×C8—1222no—
318C2×C8—1222no—
319C2×C8—1222no—
320C2×C8—1222no—
321C4⋊C4—22no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4C2×C22⋊D6112yes—
331C22⋊C4—22no—
332C42—22no—
333C42D4×D6111yescharacteristic
334C42#1175822411yescharacteristic
335C42C2×C22⋊D6111yescharacteristic
336C42C23×D6111yescharacteristic
337C23—4882no—
338C23—4882no—
339C23—4882no—
340C23—2442no—
341C23#1451974911yescharacteristic
342Q8—24no—
343Q8—42no—
344Q8—82no—
345Q8—82no—
346D4—4882no—
347D4—4882no—
348D4—4882no—
349D4—4882no—
350D4—2442no—
351D4—2442no—
352D4—2444no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—42no—
358C2×C4—82no—
359C2×C4#1451974911yescharacteristic
360C2×C4#1451974911yescharacteristic
361C2×C4#1451974911yescharacteristic
362C2×C4#1451980012yes—
363C2×C4#1451980912yes—
364C2×C4C2×D4×D6111yescharacteristic
365C2×C4C22×C22⋊D6111yescharacteristic
366C2×C4C2×#1175822411yescharacteristic
367C8—42no—
368C8—1221no—
369C8—1221no—
370C8—1221no—
371C8—1221no—
372C8—2442no—
373C8—2442no—
374C22—21no—
375C22—21no—
376C22—9764no—
377C22—4882no—
378C22—4882no—
379C22—4882no—
380C22#1683998612yes—
381C22C2×#1451974911yescharacteristic
382C4—21no—
383C4—21no—
384C4—42no—
385C4—42no—
386C4—82no—
387C4#1683998612yes—
388C4#1684003012yes—
389C4C2×#1451980011yescharacteristic
390C4C2×#1451980911yescharacteristic
391C4#1684057012yes—
392C2—22no—
393C2—9762no—
394C2C2×#1683998611yescharacteristic
395C2#2145546011yescharacteristic
396C2C2×#1684003011yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3230C2
socleC22×C61C2×#121
prime radicalC2×#14519774C22
hypercenter#3230D61
hypocenterC61#19212
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3230

Complements

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

NH ≅ G/Ntype
3. #21454754C2semidirect
4. #21445361C2semidirect
5. #21445237C2semidirect
6. #21444565C2semidirect
7. #21444500C2semidirect
9. C61×#3230C2semidirect
20. #16839893C22semidirect
22. Q8×#92418C22semidirect
30. #16839514C22semidirect
31. #16839512C22semidirect
32. #16839511C22semidirect
39. C61×#477C22semidirect
194. #3230D61semidirect
220. #477C2×D61semidirect
239. C61#19212semidirect

Maximal subgroups

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

subgroupindex
#214567572
C2×#168401102
#214547542
#214453612
#214452372
#214445652
#214445002
#214444862
C61×#32302
#1921261