Finite Group Databasebeta

← #29663214

Subgroups — #29663214

Order 15616.

710 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456963C211yesmaximal, characteristic, complemented
2#21456762C211yesmaximal, characteristic
3C2×#16840136C211yesmaximal, characteristic
4#21454749C211yesmaximal, characteristic, complemented
5#21445294C214yesmaximal, complemented
6#21445170C214yesmaximal, complemented
7#21444565C211yesmaximal, characteristic, complemented
8#21444488C211yesmaximal, characteristic
9C61×#1546C211yesmaximal, characteristic, complemented
10C22×#11758201—21no—
11C22×#11758201—21no—
12#16840319C2211yescharacteristic
13C2×#14519774C2211yescharacteristic
14#16840158—21no—
15#16840136—21no—
16#16840136—21no—
17#16840136C2212yes—
18#16840136C2212yes—
19#16840135—21no—
20#16840114—21no—
21#16840109—21no—
22#16840092—22no—
23#16840044C2214yescomplemented
24#16839963C2214yes—
25#16839915—22no—
26#16839889—21no—
27#16839889C2212yescomplemented
28#16839886—21no—
29#16839886C2212yescomplemented
30#16839815C2211yescharacteristic
31#16839790—21no—
32#16839720—21no—
33C2×C4×#92418C2211yescharacteristic
34#16839514C2212yescomplemented
35#16839512C2212yescomplemented
36#16839505C2214yes—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C61×#431C2211yescharacteristic
39C61×#423C2211yescharacteristic
40C61×#332C2214yescomplemented
41C22×D244—21no—
42C22×D244—21no—
43C22×D244—21no—
44C2×#11758190—24no—
45C2×#11758201—22no—
46C2×#11758201—22no—
47C2×#11758201—22no—
48C2×#11758201—22no—
49C2×#11758201—44no—
50C2×#11758201—44no—
51C22×#92418D411yescharacteristic
52C22×#92418D411yescharacteristic
53#14519777D411yescharacteristic
54#14519777D411yescharacteristic
55#14519774—21no—
56#14519774—21no—
57#14519774—21no—
58#14519774C2312yes—
59#14519769—21no—
60#14519756—21no—
61#14519756D412yes—
62C4×Dic122—21no—
63C4×Dic122C2312yes—
64#14519736—21no—
65#14519708—24no—
66C4⋊#92418C2312yes—
67#14519706—22no—
68#14519706—22no—
69#14519700—24no—
70#14519699—24no—
71C22⋊#92418—21no—
72#14519687—22no—
73C4×#92418—21no—
74C4×#92418D411yescharacteristic
75C4×#92418D411yescharacteristic
76C4×#92418C2311yescharacteristic
77C4×#92418C2311yescharacteristic
78C8×Dic61—22no—
79C22×Q8×C61—21no—
80C2×M16×C61—21no—
81C4⋊Q8×C61—21no—
82C4⋊Q8×C61—21no—
83C4⋊Q8×C61—21no—
84C4⋊Q8×C61C2312yes—
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61—21no—
87C2×C42×C61C2311yescharacteristic
88C4⋊C8×C61—21no—
89C4⋊C8×C61C2312yes—
90C4⋊C8×C61C2312yes—
91C22⋊C8×C61—21no—
92C4×C8×C61—21no—
93C23×D61—41no—
94C2×Q8×C61—22no—
95C2×Q8×C61—41no—
96C2×Q8×C61—42no—
97C2×Q8×C61—42no—
98C22×C4×C61—41no—
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61#4111yescharacteristic
102C2×D244—22no—
103C2×D244—22no—
104C2×D244—22no—
105C2×D244—42no—
106C2×D244—42no—
107C2×D244—41no—
108C2×D244—41no—
109C2×D244—41no—
110C2×D244—82no—
111C2×D244—82no—
112C2×Dic122—21no—
113C2×Dic122C2×D412yes—
114M16×C61—42no—
115C2×C8×C61—22no—
116C2×C8×C61—22no—
117C4⋊C4×C61—22no—
118C4⋊C4×C61—22no—
119C4⋊C4×C61—41no—
120C4⋊C4×C61—41no—
121C42×C61C2×D411yescharacteristic
122C42×C61C2×D411yescharacteristic
123C42×C61#4111yescharacteristic
124C42×C61C2411yescharacteristic
125C22⋊Dic61—22no—
126#11758201—44no—
127#11758201—44no—
128#11758201—44no—
129#11758201—44no—
130C4⋊Dic61—21no—
131C4⋊Dic61—21no—
132C4⋊Dic61—21no—
133C4⋊Dic61C2×D412yes—
134#11758196—22no—
135#11758196—22no—
136C4×Dic61—22no—
137C2×#92418—22no—
138C2×#92418—22no—
139C2×#92418SD1614yescomplemented
140C2×#92418C2×D411yescharacteristic
141C2×#92418C2×D411yescharacteristic
142C2×#92418C2×D411yescharacteristic
143C2×#92418C2×D411yescharacteristic
144#11758190—84no—
145C23×C61#12111yescharacteristic
146C22×D61—42no—
147C22×D61—82no—
148C22×D61—82no—
149C22×D61—82no—
150Q8×C61—42no—
151Q8×C61—42no—
152Q8×C61—82no—
153C2×C4×C61—22no—
154C2×C4×C61—22no—
155C2×C4×C61—42no—
156C2×C4×C61—82no—
157C2×C4×C61#12111yescharacteristic
158C2×C4×C61#12111yescharacteristic
159C2×C4×C61#12111yescharacteristic
160C2×C4×C61C2×SD1612yes—
161C2×C4×C61AGL(1,ℤ/8)12yes—
162C2×C4×C61C22×D411yescharacteristic
163C2×C4×C61C22×D411yescharacteristic
164C2×C4×C61C2×#4111yescharacteristic
165C2×Dic61—22no—
166C2×Dic61—22no—
167D244—42no—
168D244—42no—
169D244—44no—
170D244—82no—
171D244—82no—
172D244—82no—
173D244—82no—
174Dic122—24no—
175Dic122—42no—
176C8×C61—42no—
177C8×C61—42no—
178#92418—21no—
179#92418—21no—
180#92418—42no—
181#92418C2×SD1612yes—
182#92418C2×SD1612yes—
183#19202—611nomaximal, Sylow
184C22×C61—21no—
185C22×C61—21no—
186C22×C61#46012yes—
187C22×C61C2×#12111yescharacteristic
188C2×D61—82no—
189C2×D61—82no—
190C2×D61—82no—
191C2×D61—164no—
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—82no—
195C4×C61#46012yes—
196C4×C61#47112yes—
197C4×C61C22×SD1611yescharacteristic
198C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
199C4×C61#57712yes—
200Dic61—42no—
201Dic61—42no—
202#3229—611no—
203#3208—611no—
204C2×#492—611no—
205#2960—611no—
206#1686—614no—
207#1666—614no—
208#1564—611no—
209#1546D6111yescharacteristic, complemented
210#1545—611no—
211C2×C61—22no—
212C2×C61C2×#46011yescharacteristic
213C2×C61#305711yescharacteristic
214C2×C61C2×#47111yescharacteristic
215D61—162no—
216C22×SD16—1221no—
217C22×SD16—1221no—
218#533—611no—
219C2×C4⋊Q8C2×D6111yescharacteristic
220C2×#127—611no—
221#498—1221no—
222#492—612no—
223#492—612no—
224#492—1221no—
225#492—1221no—
226#492—1221no—
227#487—1221no—
228#486—1221no—
229#483—1222no—
230#475—614no—
231#460—614no—
232#450—1222no—
233#446—612no—
234#446—1221no—
235C8×Q8—612no—
236C8×Q8—1221no—
237#431—611no—
238#431C2×D6111yescharacteristic
239#424—1221no—
240#423C2×D6111yescharacteristic
241#405—1221no—
242C2×C4×C8—611no—
243#335—612no—
244#334—612no—
245#332C2×D6114yescomplemented
246#331—614no—
247C61#1920211yescharacteristic, complemented, Sylow
248C22×Q8—21no—
249C22×D4—1221no—
250C22×D4—1221no—
251C22×D4—1221no—
252C2×SD16—1224no—
253C2×SD16—1222no—
254C2×SD16—1222no—
255C2×SD16—1222no—
256C2×SD16—1222no—
257C2×SD16—2444no—
258C2×SD16—2444no—
259C2×M16—21no—
260C22×C8—611no—
261C22×C8—611no—
262C4⋊Q8—21no—
263C4⋊Q8—21no—
264C4⋊Q8—21no—
265C4⋊Q8—611no—
266C4⋊Q8—611no—
267C4⋊Q8C22×D6112yes—
268#127—612no—
269#127—1221no—
270#127—1221no—
271#127—1221no—
272#125—1221no—
273C22⋊Q8—612no—
274C22⋊Q8—1221no—
275C4×Q8—612no—
276C4×Q8—1221no—
277#117—1221no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C42C22×D6111yescharacteristic
281#106—1224no—
282C4⋊C8—21no—
283C4⋊C8—612no—
284C4⋊C8—1222no—
285C4⋊C8—1222no—
286C4⋊C8C22×D6112yes—
287C4⋊C8C22×D6112yes—
288#103—1224no—
289#102—1224no—
290C22⋊C8—21no—
291C22⋊C8—1221no—
292#97—1222no—
293C4×C8—21no—
294C4×C8—611no—
295C4×C8—611no—
296C4×C8—611no—
297C4×C8—611no—
298C4×C8—1222no—
299C4×C8—1221no—
300C24—2441no—
301C2×Q8—22no—
302C2×Q8—41no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—612no—
306C2×Q8—1221no—
307C2×D4—1222no—
308C2×D4—1222no—
309C2×D4—1222no—
310C2×D4—4882no—
311C2×D4—4882no—
312C2×D4—2441no—
313C2×D4—2441no—
314C2×D4—2441no—
315C2×D4—2442no—
316C2×D4—2442no—
317C22×C4—41no—
318C22×C4#1175822211yescharacteristic
319C22×C4D4×D6111yescharacteristic
320C22×C4D4×D6111yescharacteristic
321SD16—4884no—
322SD16—2444no—
323SD16—2444no—
324SD16—2444no—
325SD16—2444no—
326M16—42no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—611no—
330C2×C8—611no—
331C2×C8—614no—
332C2×C8—611no—
333C2×C8—611no—
334C2×C8—1222no—
335C2×C8—1222no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—612no—
341C4⋊C4—1222no—
342C4⋊C4—1222no—
343C4⋊C4—1221no—
344C4⋊C4—1221no—
345C4⋊C4—1221no—
346C22⋊C4—1222no—
347C42—1222no—
348C42#1175822211yescharacteristic
349C42D4×D6111yescharacteristic
350C42D4×D6111yescharacteristic
351C42C23×D6111yescharacteristic
352C23—4882no—
353C23—4882no—
354C23—4882no—
355C23—2442no—
356C23#1451975011yescharacteristic
357Q8—42no—
358Q8—42no—
359Q8—82no—
360Q8—1224no—
361Q8—2442no—
362D4—4882no—
363D4—4882no—
364D4—4882no—
365D4—4882no—
366D4—2442no—
367D4—2442no—
368D4—2444no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—42no—
372C2×C4—82no—
373C2×C4—1222no—
374C2×C4—1222no—
375C2×C4#1451975011yescharacteristic
376C2×C4#1451975011yescharacteristic
377C2×C4#1451975011yescharacteristic
378C2×C4SD16×D6112yes—
379C2×C4#1451980112yes—
380C2×C4C2×D4×D6111yescharacteristic
381C2×C4C2×D4×D6111yescharacteristic
382C2×C4C2×#1175822211yescharacteristic
383C8—42no—
384C8—42no—
385C8—612no—
386C8—612no—
387C8—1221no—
388C8—1221no—
389C8—2442no—
390C22—21no—
391C22—21no—
392C22—9764no—
393C22—4882no—
394C22—4882no—
395C22—4882no—
396C22#1683996112yes—
397C22C2×#1451975011yescharacteristic
398C4—21no—
399C4—21no—
400C4—82no—
401C4—2442no—
402C4—2442no—
403C4#1683996112yes—
404C4#1684002812yes—
405C4C2×SD16×D6111yescharacteristic
406C4C2×#1451980111yescharacteristic
407C4#1684057112yes—
408C2—22no—
409C2—9762no—
410C2C2×#1683996111yescharacteristic
411C2#2145546111yescharacteristic
412C2C2×#1684002811yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1546C2
socleC22×C61C2×#121
prime radicalC2×#14519774C22
hypercenter#1546D61
hypocenterC61#19202
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
1. #21456963C2semidirect
4. #21454749C2semidirect
5. #21445294C2semidirect
6. #21445170C2semidirect
7. #21444565C2semidirect
9. C61×#1546C2semidirect
23. #16840044C22semidirect
27. #16839889C22semidirect
29. #16839886C22semidirect
34. #16839514C22semidirect
35. #16839512C22semidirect
40. C61×#332C22semidirect
139. C2×#92418SD16semidirect
209. #1546D61semidirect
245. #332C2×D61semidirect
247. C61#19202semidirect

Maximal subgroups

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

subgroupindex
#214569632
#214567622
C2×#168401362
#214547492
#214452942
#214451702
#214445652
#214444882
C61×#15462
#1920261