Finite Group Databasebeta

← #29663213

Subgroups — #29663213

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#21456757C211yesmaximal, characteristic
2C2×#16840136C211yesmaximal, characteristic
3#21454754C211yesmaximal, characteristic, complemented
4#21445293C214yesmaximal, complemented
5#21445169C214yesmaximal, complemented
6#21444565C211yesmaximal, characteristic, complemented
7#21444491C211yesmaximal, characteristic, complemented
8#21444486C211yesmaximal, characteristic
9C61×#3229C211yesmaximal, characteristic, complemented
10C22×#11758201—21no—
11C22×#11758201—21no—
12C2×#14519774C2211yescharacteristic
13#16840136—21no—
14#16840136—21no—
15#16840136—21no—
16#16840136C2212yes—
17#16840136C2212yes—
18#16840110—21no—
19#16839962C2214yes—
20#16839912—22no—
21#16839893—21no—
22#16839893C2212yescomplemented
23Q8×#92418—21no—
24Q8×#92418C2212yescomplemented
25#16839815C2211yescharacteristic
26#16839789—21no—
27#16839786C2211yescharacteristic
28#16839718—21no—
29C2×C4×#92418C2211yescharacteristic
30#16839514C2212yescomplemented
31#16839512C2212yescomplemented
32#16839508C2214yescomplemented
33#16839505C2214yes—
34C61×#533C2211yescharacteristic
35C2×C4⋊Q8×C61C2211yescharacteristic
36C61×#498—21no—
37C61×#487—21no—
38C61×#483—22no—
39C61×#475C2214yescomplemented
40C61×#431C2211yescharacteristic
41C22×D244—21no—
42C22×D244—21no—
43C22×D244—21no—
44C2×#11758201—24no—
45C2×#11758201—22no—
46C2×#11758201—22no—
47C2×#11758201—22no—
48C2×#11758201—22no—
49C2×#11758201—44no—
50C2×#11758201—44no—
51C2×#11758194—21no—
52C22×#92418D411yescharacteristic
53C22×#92418D411yescharacteristic
54#14519774—21no—
55#14519774—21no—
56#14519774—21no—
57#14519774C2312yes—
58C4⋊#92418—21no—
59C4⋊#92418—22no—
60C4⋊#92418—22no—
61C4⋊#92418C2312yes—
62C4⋊#92418C2312yes—
63#14519697—24no—
64C22⋊#92418—21no—
65#14519688—22no—
66C4×#92418—21no—
67C4×#92418—22no—
68C4×#92418D411yescharacteristic
69C4×#92418D411yescharacteristic
70C4×#92418C2311yescharacteristic
71C4×#92418C2311yescharacteristic
72C22×Q8×C61—21no—
73C4⋊Q8×C61—21no—
74C4⋊Q8×C61—21no—
75C4⋊Q8×C61—21no—
76C4⋊Q8×C61D411yescharacteristic
77C4⋊Q8×C61D411yescharacteristic
78C4⋊Q8×C61C2312yes—
79#125×C61—21no—
80C22⋊Q8×C61—21no—
81C22⋊Q8×C61D412yes—
82C4×Q8×C61—21no—
83C4×Q8×C61C2312yes—
84#117×C61—21no—
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61—21no—
87C2×C42×C61C2311yescharacteristic
88#106×C61—24no—
89C4⋊C8×C61C2312yes—
90#103×C61—24no—
91C22⋊C8×C61—21no—
92C4×C8×C61—21no—
93C23×D61—41no—
94C2×Q8×C61—21no—
95C2×Q8×C61—22no—
96C2×Q8×C61—41no—
97C2×Q8×C61—42no—
98C2×Q8×C61—42no—
99C2×Q8×C61C2×D412yes—
100C22×C4×C61—41no—
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61#4111yescharacteristic
104C2×D244—22no—
105C2×D244—22no—
106C2×D244—22no—
107C2×D244—42no—
108C2×D244—42no—
109C2×D244—41no—
110C2×D244—41no—
111C2×D244—41no—
112C2×D244—82no—
113C2×D244—82no—
114C2×C8×C61—22no—
115C4⋊C4×C61—21no—
116C4⋊C4×C61—21no—
117C4⋊C4×C61—21no—
118C4⋊C4×C61—22no—
119C4⋊C4×C61—22no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—22no—
122C4⋊C4×C61—41no—
123C4⋊C4×C61—41no—
124C4⋊C4×C61C2×D412yes—
125C22⋊C4×C61—22no—
126C42×C61—22no—
127C42×C61C2×D411yescharacteristic
128C42×C61C2×D411yescharacteristic
129C42×C61#4111yescharacteristic
130C42×C61C2411yescharacteristic
131#11758201—44no—
132#11758201—44no—
133#11758201—44no—
134#11758201—44no—
135#11758201—84no—
136#11758194—42no—
137C2×#92418—22no—
138C2×#92418—22no—
139C2×#92418—22no—
140C2×#92418SD1614yescomplemented
141C2×#92418C2×D411yescharacteristic
142C2×#92418C2×D411yescharacteristic
143C2×#92418C2×D411yescharacteristic
144C2×#92418C2×D411yescharacteristic
145C23×C61#12111yescharacteristic
146C22×D61—42no—
147C22×D61—82no—
148C22×D61—82no—
149C22×D61—82no—
150Q8×C61—24no—
151Q8×C61—42no—
152Q8×C61—42no—
153Q8×C61—42no—
154Q8×C61—82no—
155C2×C4×C61—22no—
156C2×C4×C61—22no—
157C2×C4×C61—22no—
158C2×C4×C61—22no—
159C2×C4×C61—42no—
160C2×C4×C61—82no—
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61#12111yescharacteristic
163C2×C4×C61#12111yescharacteristic
164C2×C4×C61C2×SD1612yes—
165C2×C4×C61AGL(1,ℤ/8)12yes—
166C2×C4×C61C22×D411yescharacteristic
167C2×C4×C61C22×D411yescharacteristic
168C2×C4×C61C2×#4111yescharacteristic
169D244—42no—
170D244—42no—
171D244—44no—
172D244—82no—
173D244—82no—
174D244—82no—
175D244—82no—
176C8×C61—42no—
177#92418—21no—
178#92418—21no—
179#92418—42no—
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—42no—
195C4×C61—42no—
196C4×C61—82no—
197C4×C61#46012yes—
198C4×C61#47112yes—
199C4×C61C22×SD1611yescharacteristic
200C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
201C4×C61#57712yes—
202#3229D6111yescharacteristic, complemented
203#3208—611no—
204C2×#492—611no—
205#2960—611no—
206#1686—614no—
207#1666—614no—
208#1564—611no—
209#1546—611no—
210#1545—611no—
211C2×C61—22no—
212C2×C61C2×#46011yescharacteristic
213C2×C61#305711yescharacteristic
214C2×C61C2×#47111yescharacteristic
215D61—162no—
216C22×SD16—1221no—
217C22×SD16—1221no—
218#533C2×D6111yescharacteristic
219C2×C4⋊Q8C2×D6111yescharacteristic
220C2×#127—611no—
221#498—21no—
222#492—612no—
223#492—612no—
224#492—1221no—
225#492—1221no—
226#492—1221no—
227#487—21no—
228#486—1221no—
229#483—22no—
230#475C2×D6114yescomplemented
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#423—611no—
241#405—1221no—
242C2×C4×C8—611no—
243#335—612no—
244#334—612no—
245#332—614no—
246#331—614no—
247C61#1920211yescharacteristic, complemented, Sylow
248C22×Q8—21no—
249C22×D4—1221no—
250C22×D4—1221no—
251C22×D4—1221no—
252C2×SD16—1222no—
253C2×SD16—1222no—
254C2×SD16—1222no—
255C2×SD16—1222no—
256C2×SD16—1224no—
257C2×SD16—2444no—
258C2×SD16—2444no—
259C2×M16—1221no—
260C22×C8—611no—
261C22×C8—611no—
262C4⋊Q8—21no—
263C4⋊Q8—21no—
264C4⋊Q8—21no—
265C4⋊Q8C22⋊D6111yescharacteristic
266C4⋊Q8C22⋊D6111yescharacteristic
267C4⋊Q8C22×D6112yes—
268#127—612no—
269#127—1221no—
270#127—1221no—
271#127—1221no—
272#125—21no—
273C22⋊Q8—21no—
274C22⋊Q8C22⋊D6112yes—
275C4×Q8—21no—
276C4×Q8C22×D6112yes—
277#117—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C42C22×D6111yescharacteristic
281#106—24no—
282C4⋊C8—612no—
283C4⋊C8—612no—
284C4⋊C8—1222no—
285C4⋊C8—1222no—
286C4⋊C8—1221no—
287C4⋊C8C22×D6112yes—
288#103—24no—
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—21no—
302C2×Q8—22no—
303C2×Q8—41no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8C2×C22⋊D6112yes—
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×C4D4×D6111yescharacteristic
319C22×C4#1175822411yescharacteristic
320C22×C4C2×C22⋊D6111yescharacteristic
321SD16—4884no—
322SD16—2444no—
323SD16—2444no—
324SD16—2444no—
325SD16—2444no—
326M16—2442no—
327C2×C8—22no—
328C2×C8—611no—
329C2×C8—611no—
330C2×C8—614no—
331C2×C8—611no—
332C2×C8—611no—
333C2×C8—1222no—
334C2×C8—1222no—
335C2×C8—1222no—
336C4⋊C4—22no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4C2×C22⋊D6112yes—
346C22⋊C4—22no—
347C42—22no—
348C42D4×D6111yescharacteristic
349C42#1175822411yescharacteristic
350C42C2×C22⋊D6111yescharacteristic
351C42C23×D6111yescharacteristic
352C23—4882no—
353C23—4882no—
354C23—4882no—
355C23—2442no—
356C23#1451974911yescharacteristic
357Q8—24no—
358Q8—42no—
359Q8—42no—
360Q8—42no—
361Q8—82no—
362D4—4882no—
363D4—4882no—
364D4—4882no—
365D4—4882no—
366D4—2442no—
367D4—2442no—
368D4—2444no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—42no—
374C2×C4—82no—
375C2×C4#1451974911yescharacteristic
376C2×C4#1451974911yescharacteristic
377C2×C4#1451974911yescharacteristic
378C2×C4SD16×D6112yes—
379C2×C4#1451980412yes—
380C2×C4C2×D4×D6111yescharacteristic
381C2×C4C22×C22⋊D6111yescharacteristic
382C2×C4C2×#1175822411yescharacteristic
383C8—42no—
384C8—612no—
385C8—612no—
386C8—1221no—
387C8—1221no—
388C8—2442no—
389C8—2442no—
390C22—21no—
391C22—21no—
392C22—9764no—
393C22—4882no—
394C22—4882no—
395C22—4882no—
396C22#1683996012yes—
397C22C2×#1451974911yescharacteristic
398C4—21no—
399C4—21no—
400C4—42no—
401C4—42no—
402C4—82no—
403C4#1683996012yes—
404C4#1684002712yes—
405C4C2×SD16×D6111yescharacteristic
406C4C2×#1451980411yescharacteristic
407C4#1684057012yes—
408C2—22no—
409C2—9762no—
410C2C2×#1683996011yescharacteristic
411C2#2145546011yescharacteristic
412C2C2×#1684002711yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3229C2
socleC22×C61C2×#121
prime radicalC2×#14519774C22
hypercenter#3229D61
hypocenterC61#19202
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
3. #21454754C2semidirect
4. #21445293C2semidirect
5. #21445169C2semidirect
6. #21444565C2semidirect
7. #21444491C2semidirect
9. C61×#3229C2semidirect
22. #16839893C22semidirect
24. Q8×#92418C22semidirect
30. #16839514C22semidirect
31. #16839512C22semidirect
32. #16839508C22semidirect
39. C61×#475C22semidirect
140. C2×#92418SD16semidirect
202. #3229D61semidirect
230. #475C2×D61semidirect
247. C61#19202semidirect

Maximal subgroups

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

subgroupindex
#214567572
C2×#168401362
#214547542
#214452932
#214451692
#214445652
#214444912
#214444862
C61×#32292
#1920261