Finite Group Databasebeta

← #29663156

Subgroups — #29663156

Order 15616.

714 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#21456947C211yesmaximal, characteristic, complemented
2#21456739C211yesmaximal, characteristic
3C2×#16840110C211yesmaximal, characteristic, complemented
4#21454530C211yesmaximal, characteristic, complemented
5#21445355C214yesmaximal, complemented
6#21445231C214yesmaximal, complemented
7#21444565C211yesmaximal, characteristic, complemented
8#21444488C211yesmaximal, characteristic, complemented
9C61×#1548C211yesmaximal, characteristic, complemented
10C2×#14519785—21no—
11#16840312C2211yescharacteristic
12C2×#14519774C2211yescharacteristic
13#16840158—21no—
14#16840139—21no—
15#16840110—21no—
16#16840110—21no—
17#16840110—21no—
18#16840110C2212yes—
19#16840110C2212yes—
20#16840109—21no—
21#16840080—22no—
22#16840049C2214yescomplemented
23#16839958C2214yes—
24#16839898—22no—
25#16839839—21no—
26#16839839C2212yescomplemented
27#16839827—21no—
28#16839827C2212yescomplemented
29#16839815C2211yescharacteristic
30C2×#14519697—21no—
31C2×#14519697—21no—
32#16839730—21no—
33C2×C4×#92418C2211yescharacteristic
34#16839514C2212yescomplemented
35#16839512C2212yescomplemented
36#16839505C2214yes—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C61×#431C2211yescharacteristic
39C61×#423C2211yescharacteristic
40C61×#333C2214yescomplemented
41C2×#11758222—21no—
42C22×D244—21no—
43C22×D244—21no—
44C22×D244—21no—
45C2×D488—24no—
46#14519785—44no—
47C2×#11758189—22no—
48C22×#92418D411yescharacteristic
49C22×#92418D411yescharacteristic
50C2×C8×D61—22no—
51#14519777D411yescharacteristic
52#14519774—21no—
53#14519774—21no—
54#14519774—21no—
55#14519774D411yescharacteristic
56#14519774C2312yes—
57#14519765—21no—
58#14519751—21no—
59#14519751D412yes—
60C4×D244—21no—
61C4×D244C2312yes—
62#14519708—24no—
63C4⋊#92418C2312yes—
64#14519699—24no—
65#14519697—24no—
66#14519697—22no—
67#14519697—22no—
68#14519697—22no—
69#14519697—22no—
70C22⋊#92418—21no—
71#14519689—22no—
72#14519689—22no—
73C4×#92418—21no—
74C4×#92418D411yescharacteristic
75C4×#92418D411yescharacteristic
76C4×#92418C2311yescharacteristic
77C4×#92418C2311yescharacteristic
78C22×Q8×C61—21no—
79C2×M16×C61—21no—
80C4⋊Q8×C61—21no—
81C4⋊Q8×C61—21no—
82C4⋊Q8×C61—21no—
83C4⋊Q8×C61C2312yes—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61—21no—
86C2×C42×C61C2311yescharacteristic
87C4⋊C8×C61—21no—
88C4⋊C8×C61C2312yes—
89C4⋊C8×C61C2312yes—
90C22⋊C8×C61—21no—
91C4×C8×C61—21no—
92C23×D61—41no—
93C2×Q8×C61—22no—
94C2×Q8×C61—41no—
95C2×Q8×C61—82no—
96C22×C4×C61—41no—
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61#4111yescharacteristic
100C2×C22⋊D61—22no—
101#11758222—42no—
102#11758222—42no—
103C2×D244—21no—
104C2×D244—21no—
105C2×D244—22no—
106C2×D244—22no—
107C2×D244—22no—
108C2×D244—42no—
109C2×D244—42no—
110C2×D244—41no—
111C2×D244—41no—
112C2×D244—41no—
113C2×D244—82no—
114C2×D244—82no—
115C2×D244C2×D412yes—
116C2×C4×D61—22no—
117C2×Dic122—21no—
118M16×C61—42no—
119C2×C8×C61—22no—
120C2×C8×C61—22no—
121C4⋊C4×C61—22no—
122C4⋊C4×C61—22no—
123C4⋊C4×C61—41no—
124C4⋊C4×C61—41no—
125C42×C61C2×D411yescharacteristic
126C42×C61C2×D411yescharacteristic
127C42×C61#4111yescharacteristic
128C42×C61C2411yescharacteristic
129#11758198—22no—
130#11758198—22no—
131C4⋊Dic61—21no—
132C4⋊Dic61C2×D412yes—
133C2×#92418—22no—
134C2×#92418—22no—
135C2×#92418—22no—
136C2×#92418C2×D411yescharacteristic
137C2×#92418C2×D411yescharacteristic
138C2×#92418C2×D411yescharacteristic
139C2×#92418C2×D411yescharacteristic
140D488—84no—
141#11758189—44no—
142C8×D61—44no—
143C23×C61#12111yescharacteristic
144C22×D61—22no—
145C22×D61—42no—
146C22×D61—82no—
147C22×D61—82no—
148C22×D61—82no—
149Q8×C61—82no—
150Q8×C61—82no—
151C2×C4×C61—22no—
152C2×C4×C61—22no—
153C2×C4×C61—42no—
154C2×C4×C61—82no—
155C2×C4×C61#12111yescharacteristic
156C2×C4×C61#12111yescharacteristic
157C2×C4×C61#12111yescharacteristic
158C2×C4×C61#13512yes—
159C2×C4×C61AGL(1,ℤ/8)12yes—
160C2×C4×C61C22×D411yescharacteristic
161C2×C4×C61C22×D411yescharacteristic
162C2×C4×C61C2×#4111yescharacteristic
163C22⋊D61—44no—
164C2×Dic61—22no—
165D244—24no—
166D244—42no—
167D244—42no—
168D244—44no—
169D244—42no—
170D244—42no—
171D244—82no—
172D244—82no—
173D244—82no—
174D244—82no—
175C4×D61—42no—
176C4×D61—42no—
177Dic122—42no—
178C8×C61—42no—
179C8×C61—42no—
180#92418—21no—
181#92418—21no—
182#92418—21no—
183#92418—21no—
184#92418—42no—
185#19196—611nomaximal, Sylow
186C22×C61—21no—
187C22×C61—21no—
188C22×C61#46312yes—
189C22×C61C2×#12111yescharacteristic
190C2×D61—42no—
191C2×D61—42no—
192C2×D61—42no—
193C2×D61—82no—
194C2×D61—82no—
195C2×D61—82no—
196C2×D61—164no—
197C4×C61—21no—
198C4×C61—21no—
199C4×C61—82no—
200C4×C61#46312yes—
201C4×C61#47212yes—
202C4×C61C2×#13511yescharacteristic
203C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
204C4×C61#57712yes—
205Dic61—42no—
206#3227—611no—
207#3205—611no—
208C2×#486—611no—
209#2931—611no—
210#1697—614no—
211#1677—614no—
212#1564—611no—
213#1548D6111yescharacteristic, complemented
214#1545—611no—
215C2×C61—22no—
216C2×C61C2×#46311yescharacteristic
217C2×C61#305711yescharacteristic
218C2×C61C2×#47211yescharacteristic
219D61—82no—
220D61—162no—
221C2×#131—1221no—
222#532—611no—
223C2×C4⋊Q8C2×D6111yescharacteristic
224C2×#127—611no—
225#498—1221no—
226#493—1221no—
227#486—612no—
228#486—612no—
229#486—1221no—
230#486—1221no—
231#486—1221no—
232#486—1221no—
233#481—1222no—
234#476—614no—
235#459—614no—
236#447—1222no—
237#436—612no—
238#436—1221no—
239C8×D4—612no—
240C8×D4—1221no—
241#431—611no—
242#431C2×D6111yescharacteristic
243#423C2×D6111yescharacteristic
244C2×#102—1221no—
245C2×#102—1221no—
246#408—1221no—
247C2×C4×C8—611no—
248#335—612no—
249#334—612no—
250#333C2×D6114yescomplemented
251#331—614no—
252C61#1919611yescharacteristic, complemented, Sylow
253C2×#41—1221no—
254C22×Q8—21no—
255C22×D4—1221no—
256C22×D4—1221no—
257C22×D4—1221no—
258C2×D8—1224no—
259#131—2444no—
260C2×M16—21no—
261C2×M16—1222no—
262C22×C8—611no—
263C22×C8—611no—
264C22×C8—1222no—
265C4⋊Q8—21no—
266C4⋊Q8—21no—
267C4⋊Q8—21no—
268C4⋊Q8—611no—
269C4⋊Q8C22×D6112yes—
270#127—611no—
271#127—612no—
272#127—1221no—
273#127—1221no—
274#127—1221no—
275#124—1221no—
276#121—612no—
277#121—1221no—
278C4×D4—612no—
279C4×D4—1221no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C42C22×D6111yescharacteristic
283#106—1224no—
284C4⋊C8—21no—
285C4⋊C8—612no—
286C4⋊C8C22×D6112yes—
287C4⋊C8C22×D6112yes—
288#102—1222no—
289#102—1222no—
290#102—1222no—
291#102—1222no—
292#102—1224no—
293#102—1224no—
294C22⋊C8—21no—
295C22⋊C8—1222no—
296C22⋊C8—1222no—
297C22⋊C8—1221no—
298C4×C8—21no—
299C4×C8—611no—
300C4×C8—611no—
301C4×C8—611no—
302C4×C8—611no—
303C4×C8—1221no—
304C24—2441no—
305#41—2442no—
306#41—2442no—
307C2×Q8—22no—
308C2×Q8—41no—
309C2×Q8—82no—
310C2×Q8—1221no—
311C2×D4—612no—
312C2×D4—1222no—
313C2×D4—1222no—
314C2×D4—1222no—
315C2×D4—1222no—
316C2×D4—1221no—
317C2×D4—1221no—
318C2×D4—4882no—
319C2×D4—4882no—
320C2×D4—2441no—
321C2×D4—2441no—
322C2×D4—2441no—
323C2×D4—2442no—
324C2×D4—2442no—
325C22×C4—41no—
326C22×C4—1222no—
327C22×C4#1175822211yescharacteristic
328C22×C4D4×D6111yescharacteristic
329C22×C4D4×D6111yescharacteristic
330D8—4884no—
331M16—42no—
332M16—2444no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—611no—
336C2×C8—611no—
337C2×C8—611no—
338C2×C8—611no—
339C2×C8—1222no—
340C2×C8—1222no—
341C2×C8—1222no—
342C2×C8—2444no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—612no—
348C4⋊C4—1221no—
349C22⋊C4—1222no—
350C22⋊C4—1222no—
351C42#1175822211yescharacteristic
352C42D4×D6111yescharacteristic
353C42D4×D6111yescharacteristic
354C42C23×D6111yescharacteristic
355C23—1222no—
356C23—4882no—
357C23—4882no—
358C23—4882no—
359C23—2442no—
360C23#1451975011yescharacteristic
361Q8—82no—
362Q8—82no—
363Q8—2442no—
364D4—1224no—
365D4—4882no—
366D4—4882no—
367D4—4882no—
368D4—4882no—
369D4—2444no—
370D4—2442no—
371D4—2442no—
372D4—2442no—
373D4—2442no—
374D4—2444no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—42no—
378C2×C4—82no—
379C2×C4—1222no—
380C2×C4—2442no—
381C2×C4—2442no—
382C2×C4#1451975011yescharacteristic
383C2×C4#1451975011yescharacteristic
384C2×C4#1451975011yescharacteristic
385C2×C4#1451980012yes—
386C2×C4#1451980412yes—
387C2×C4C2×D4×D6111yescharacteristic
388C2×C4C2×D4×D6111yescharacteristic
389C2×C4C2×#1175822211yescharacteristic
390C8—42no—
391C8—42no—
392C8—1221no—
393C8—1221no—
394C8—1221no—
395C8—1221no—
396C8—2442no—
397C22—21no—
398C22—21no—
399C22—9764no—
400C22—4882no—
401C22—4882no—
402C22—4882no—
403C22—2442no—
404C22—2442no—
405C22—2442no—
406C22#1683998012yes—
407C22C2×#1451975011yescharacteristic
408C4—21no—
409C4—21no—
410C4—82no—
411C4—2442no—
412C4#1683998012yes—
413C4#1684003112yes—
414C4C2×#1451980011yescharacteristic
415C4C2×#1451980411yescharacteristic
416C4#1684057112yes—
417C2—22no—
418C2—9762no—
419C2—4882no—
420C2C2×#1683998011yescharacteristic
421C2#2145546111yescharacteristic
422C2C2×#1684003111yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1548C2
socleC22×C61C2×#121
prime radical#21456739C2
hypercenter#1548D61
hypocenterC61#19196
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
1. #21456947C2semidirect
3. C2×#16840110C2semidirect
4. #21454530C2semidirect
5. #21445355C2semidirect
6. #21445231C2semidirect
7. #21444565C2semidirect
8. #21444488C2semidirect
9. C61×#1548C2semidirect
22. #16840049C22semidirect
26. #16839839C22semidirect
28. #16839827C22semidirect
34. #16839514C22semidirect
35. #16839512C22semidirect
40. C61×#333C22semidirect
213. #1548D61semidirect
250. #333C2×D61semidirect
252. C61#19196semidirect

Maximal subgroups

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

subgroupindex
#214569472
#214567392
C2×#168401102
#214545302
#214453552
#214452312
#214445652
#214444882
C61×#15482
#1919661