Finite Group Databasebeta

← #29663309

Subgroups — #29663309

Order 15616.

658 conjugacy classes of subgroups.

The subgroup lattice has 400 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#21445389C214yesmaximal, complemented
5#21445265C214yesmaximal, complemented
6#21444565C211yesmaximal, characteristic, complemented
7#21444504C211yesmaximal, characteristic, complemented
8#21444486C211yesmaximal, characteristic
9C61×#3209C211yesmaximal, 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#16839508C2214yescomplemented
33#16839505C2214yes—
34C61×#533C2211yescharacteristic
35C2×C4⋊Q8×C61C2211yescharacteristic
36C61×#494—21no—
37C61×#487—21no—
38C61×#462C2214yescomplemented
39C61×#451—22no—
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—
71C2×Q16×C61—24no—
72C4⋊Q8×C61—21no—
73C4⋊Q8×C61—21no—
74C4⋊Q8×C61—21no—
75C4⋊Q8×C61D411yescharacteristic
76C4⋊Q8×C61D411yescharacteristic
77C4⋊Q8×C61C2312yes—
78#125×C61—21no—
79C22⋊Q8×C61—21no—
80C22⋊Q8×C61D412yes—
81C4×Q8×C61—21no—
82C4×Q8×C61C2312yes—
83#117×C61—21no—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61—21no—
86C2×C42×C61C2311yescharacteristic
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—
111Q16×C61—84no—
112C2×C8×C61—22no—
113C4⋊C4×C61—21no—
114C4⋊C4×C61—21no—
115C4⋊C4×C61—21no—
116C4⋊C4×C61—22no—
117C4⋊C4×C61—22no—
118C4⋊C4×C61—22no—
119C4⋊C4×C61—22no—
120C4⋊C4×C61—41no—
121C4⋊C4×C61—41no—
122C4⋊C4×C61C2×D412yes—
123C22⋊C4×C61—22no—
124C42×C61—22no—
125C42×C61C2×D411yescharacteristic
126C42×C61C2×D411yescharacteristic
127C42×C61#4111yescharacteristic
128C42×C61C2411yescharacteristic
129#11758201—84no—
130#11758194—42no—
131C2×#92418—22no—
132C2×#92418—22no—
133C2×#92418—22no—
134C2×#92418—22no—
135C2×#92418C2×D411yescharacteristic
136C2×#92418C2×D411yescharacteristic
137C2×#92418C2×D411yescharacteristic
138C2×#92418C2×D411yescharacteristic
139C23×C61#12111yescharacteristic
140C22×D61—42no—
141C22×D61—82no—
142C22×D61—82no—
143C22×D61—82no—
144Q8×C61—24no—
145Q8×C61—42no—
146Q8×C61—44no—
147Q8×C61—82no—
148C2×C4×C61—22no—
149C2×C4×C61—22no—
150C2×C4×C61—22no—
151C2×C4×C61—22no—
152C2×C4×C61—42no—
153C2×C4×C61—82no—
154C2×C4×C61#12111yescharacteristic
155C2×C4×C61#12111yescharacteristic
156C2×C4×C61#12111yescharacteristic
157C2×C4×C61#13512yes—
158C2×C4×C61#13712yes—
159C2×C4×C61C22×D411yescharacteristic
160C2×C4×C61C22×D411yescharacteristic
161C2×C4×C61C2×#4111yescharacteristic
162D244—42no—
163D244—42no—
164D244—44no—
165D244—82no—
166D244—82no—
167D244—82no—
168D244—82no—
169C8×C61—42no—
170#92418—21no—
171#92418—21no—
172#92418—21no—
173#92418—21no—
174#92418—42no—
175#92418—42no—
176#19214—611nomaximal, Sylow
177C22×C61—21no—
178C22×C61—21no—
179C22×C61#46412yes—
180C22×C61C2×#12111yescharacteristic
181C2×D61—82no—
182C2×D61—82no—
183C2×D61—82no—
184C2×D61—164no—
185C4×C61—21no—
186C4×C61—21no—
187C4×C61—42no—
188C4×C61—42no—
189C4×C61—82no—
190C4×C61#46412yes—
191C4×C61#47112yes—
192C4×C61C2×#13511yescharacteristic
193C4×C61C2×#13711yescharacteristic
194C4×C61#57612yes—
195#3209D6111yescharacteristic, complemented
196#3208—611no—
197C2×#486—611no—
198#2960—611no—
199#1702—614no—
200#1682—614no—
201#1564—611no—
202#1549—611no—
203#1545—611no—
204C2×C61—22no—
205C2×C61C2×#46411yescharacteristic
206C2×C61#305811yescharacteristic
207C2×C61C2×#47111yescharacteristic
208D61—162no—
209#533C2×D6111yescharacteristic
210C2×C4⋊Q8C2×D6111yescharacteristic
211C2×#127—611no—
212#494—21no—
213#492—1221no—
214#487—21no—
215#486—612no—
216#486—612no—
217#486—1221no—
218#486—1221no—
219#486—1221no—
220#462C2×D6114yescomplemented
221#460—614no—
222#451—22no—
223#450—1222no—
224#446—612no—
225#446—1221no—
226C8×Q8—612no—
227C8×Q8—1221no—
228#431—611no—
229#431C2×D6111yescharacteristic
230#424—1221no—
231#423—611no—
232C2×#102—1221no—
233C2×#102—1221no—
234#405—1221no—
235C2×C4×C8—611no—
236#335—612no—
237#334—612no—
238#332—614no—
239#331—614no—
240C61#1921411yescharacteristic, complemented, Sylow
241C22×Q8—21no—
242C22×D4—1221no—
243C22×D4—1221no—
244C22×D4—1221no—
245C2×Q16—24no—
246C2×SD16—1224no—
247C2×M16—1221no—
248C22×C8—611no—
249C22×C8—611no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8C22⋊D6111yescharacteristic
254C4⋊Q8C22⋊D6111yescharacteristic
255C4⋊Q8C22×D6112yes—
256#127—612no—
257#127—1221no—
258#127—1221no—
259#127—1221no—
260#125—21no—
261C22⋊Q8—21no—
262C22⋊Q8C22⋊D6112yes—
263C4×Q8—21no—
264C4×Q8C22×D6112yes—
265#117—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C42C22×D6111yescharacteristic
269C4⋊C8—612no—
270C4⋊C8—612no—
271C4⋊C8—1222no—
272C4⋊C8—1222no—
273C4⋊C8—1221no—
274C4⋊C8C22×D6112yes—
275#103—24no—
276#102—1222no—
277#102—1222no—
278#102—1222no—
279#102—1222no—
280#102—1224no—
281C22⋊C8—21no—
282C22⋊C8—1221no—
283#97—1222no—
284C4×C8—21no—
285C4×C8—611no—
286C4×C8—611no—
287C4×C8—611no—
288C4×C8—611no—
289C4×C8—1222no—
290C4×C8—1221no—
291C24—2441no—
292C2×Q8—21no—
293C2×Q8—22no—
294C2×Q8—41no—
295C2×Q8—82no—
296C2×Q8C2×C22⋊D6112yes—
297C2×D4—1222no—
298C2×D4—1222no—
299C2×D4—1222no—
300C2×D4—4882no—
301C2×D4—4882no—
302C2×D4—2442no—
303C2×D4—2441no—
304C2×D4—2441no—
305C2×D4—2441no—
306C2×D4—2442no—
307C22×C4—41no—
308C22×C4D4×D6111yescharacteristic
309C22×C4#1175822411yescharacteristic
310C22×C4C2×C22⋊D6111yescharacteristic
311Q16—84no—
312SD16—4884no—
313M16—2442no—
314C2×C8—22no—
315C2×C8—611no—
316C2×C8—611no—
317C2×C8—611no—
318C2×C8—611no—
319C2×C8—1222no—
320C2×C8—1222no—
321C2×C8—1222no—
322C2×C8—1222no—
323C4⋊C4—22no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4C2×C22⋊D6112yes—
333C22⋊C4—22no—
334C42—22no—
335C42D4×D6111yescharacteristic
336C42#1175822411yescharacteristic
337C42C2×C22⋊D6111yescharacteristic
338C42C23×D6111yescharacteristic
339C23—4882no—
340C23—4882no—
341C23—4882no—
342C23—2442no—
343C23#1451974911yescharacteristic
344Q8—24no—
345Q8—42no—
346Q8—44no—
347Q8—82no—
348D4—4882no—
349D4—4882no—
350D4—4882no—
351D4—4882no—
352D4—2442no—
353D4—2442no—
354D4—2444no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—42no—
360C2×C4—82no—
361C2×C4#1451974911yescharacteristic
362C2×C4#1451974911yescharacteristic
363C2×C4#1451974911yescharacteristic
364C2×C4#1451980012yes—
365C2×C4#1451980912yes—
366C2×C4C2×D4×D6111yescharacteristic
367C2×C4C22×C22⋊D6111yescharacteristic
368C2×C4C2×#1175822411yescharacteristic
369C8—42no—
370C8—1221no—
371C8—1221no—
372C8—1221no—
373C8—1221no—
374C8—2442no—
375C8—2442no—
376C22—21no—
377C22—21no—
378C22—9764no—
379C22—4882no—
380C22—4882no—
381C22—4882no—
382C22#1683998612yes—
383C22C2×#1451974911yescharacteristic
384C4—21no—
385C4—21no—
386C4—42no—
387C4—42no—
388C4—82no—
389C4#1683998612yes—
390C4#1684002712yes—
391C4C2×#1451980011yescharacteristic
392C4C2×#1451980911yescharacteristic
393C4#1684056512yes—
394C2—22no—
395C2—9762no—
396C2C2×#1683998611yescharacteristic
397C2#2145546411yescharacteristic
398C2C2×#1684002711yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3209C2
socleC22×C61C2×#121
prime radicalC2×#14519774C22
hypercenter#3209D61
hypocenterC61#19214
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
3. #21454754C2semidirect
4. #21445389C2semidirect
5. #21445265C2semidirect
6. #21444565C2semidirect
7. #21444504C2semidirect
9. C61×#3209C2semidirect
20. #16839893C22semidirect
22. Q8×#92418C22semidirect
30. #16839514C22semidirect
31. #16839512C22semidirect
32. #16839508C22semidirect
38. C61×#462C22semidirect
195. #3209D61semidirect
220. #462C2×D61semidirect
240. C61#19214semidirect

Maximal subgroups

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

subgroupindex
#214567572
C2×#168401102
#214547542
#214453892
#214452652
#214445652
#214445042
#214444862
C61×#32092
#1921461