Finite Group Databasebeta

← #29834912

Subgroups — #29834912

Order 15616.

556 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455429C211yesmaximal, characteristic
2C2×#16839809C211yesmaximal, characteristic, complemented
3#21448040C214yesmaximal, complemented
4#21448035C214yesmaximal, complemented
5#21447591C211yesmaximal, characteristic, complemented
6#21447569C211yesmaximal, characteristic, complemented
7#21447566C211yesmaximal, characteristic, complemented
8C2×#16839543C211yesmaximal, characteristic, complemented
9C61×#2155C211yesmaximal, characteristic, complemented
10C2×#14519799C2211yescharacteristic
11C22×#11758189C2211yescharacteristic
12C2×M16×D61—21no—
13#16840274—21no—
14C2×#14519717C2211yescharacteristic
15#16840021C2212yes—
16#16840016C2212yes—
17#16840009—22no—
18#16839809—22no—
19#16839809C2214yes—
20C2×#14519713C2211yescharacteristic
21C2×#14519712—21no—
22#16839756—21no—
23#16839584C2214yescomplemented
24#16839579C2212yescomplemented
25#16839577C2212yescomplemented
26#16839576C2212yescomplemented
27#16839574C2212yescomplemented
28#16839543C2214yes—
29C2×M32×C61C2211yescharacteristic
30C2×#108×C61C2211yescharacteristic
31C61×#427C2211yescharacteristic
32C61×#365C2214yescomplemented
33C2×#11758222—21no—
34C22×C4×D61D411yescharacteristic
35#14519799—21no—
36#14519799—21no—
37#14519799—21no—
38#14519799—21no—
39C2×Dic244C2311yescharacteristic
40C2×#11758190—21no—
41C2×D488C2311yescharacteristic
42C2×#11758194—21no—
43C2×#11758189—21no—
44C2×#11758189—21no—
45C2×#11758189—21no—
46C2×#11758189—21no—
47C2×#11758189—41no—
48C2×#11758189D412yes—
49C2×#11758189C2312yes—
50M16×D61—24no—
51M16×D61—42no—
52C22×#92418D411yescharacteristic
53C2×C8×D61—41no—
54#14519761—41no—
55#14519757—22no—
56#14519752—22no—
57#14519747—41no—
58C4×D244—42no—
59#14519717—21no—
60#14519717C2312yes—
61C2×#11758185—21no—
62#14519713—21no—
63#14519713C2312yes—
64#14519712—24no—
65#14519700—22no—
66#14519697—22no—
67C2×M16×C61—21no—
68C22×C8×C61C2311yescharacteristic
69#117×C61—21no—
70M32×C61—21no—
71M32×C61C2312yes—
72C2×C16×C61—21no—
73#108×C61—21no—
74#108×C61C2312yes—
75#107×C61—21no—
76#106×C61C2312yes—
77C23×D61SD1611yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C2×C22⋊D61—41no—
80C22×Dic61SD1611yescharacteristic
81#11758222—41no—
82#11758222—41no—
83#11758222—41no—
84#11758222—41no—
85C2×D244—21no—
86C2×D244—42no—
87C2×C4×D61—21no—
88C2×C4×D61—21no—
89C2×C4×D61—21no—
90C2×C4×D61—21no—
91C2×C4×D61—41no—
92C2×C4×D61SD1612yes—
93C2×C4×D61C2×D412yes—
94C2×Dic122—21no—
95M16×C61—22no—
96M16×C61—41no—
97C2×C8×C61—41no—
98C2×C8×C61C2×D411yescharacteristic
99C2×C8×C61#4111yescharacteristic
100C2×C8×C61#4111yescharacteristic
101C2×C8×C61#4111yescharacteristic
102C2×C8×C61#4111yescharacteristic
103C2×C8×C61C2411yescharacteristic
104C4⋊C4×C61—22no—
105C22⋊C4×C61—41no—
106C42×C61—41no—
107#11758198—42no—
108#11758198—42no—
109C4⋊Dic61—42no—
110#11758194—22no—
111#11758194—41no—
112C2×#92418—21no—
113C2×#92418—21no—
114C2×#92418—41no—
115C2×#92418C2×D412yes—
116Dic244—21no—
117Dic244—21no—
118D488—21no—
119D488—21no—
120#11758190—41no—
121#11758190—41no—
122#11758189—22no—
123#11758189—22no—
124#11758189—22no—
125#11758189—22no—
126#11758189—44no—
127C8×D61—44no—
128C16×C61—22no—
129#11758185—22no—
130C23×C61C2×SD1611yescharacteristic
131C22×D61—21no—
132C22×D61—21no—
133C22×D61—41no—
134C22×D61—81no—
135C22×D61C2×SD1612yes—
136C2×C4×C61—42no—
137C2×C4×C61#12311yescharacteristic
138C2×C4×C61#12311yescharacteristic
139C2×C4×C61C2×SD1611yescharacteristic
140C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C61C22×D411yescharacteristic
143C22⋊D61—81no—
144C22⋊D61—81no—
145C2×Dic61—21no—
146C2×Dic61—21no—
147C2×Dic61—41no—
148C2×Dic61C2×SD1612yes—
149D244—41no—
150D244—41no—
151D244—84no—
152C4×D61—22no—
153C4×D61—22no—
154C4×D61—22no—
155C4×D61—22no—
156C4×D61—81no—
157C4×D61—81no—
158Dic122—41no—
159Dic122—41no—
160C8×C61—42no—
161C8×C61#12311yescharacteristic
162C8×C61#12311yescharacteristic
163C8×C61C2×#4111yescharacteristic
164C8×C61C2×#4111yescharacteristic
165#92418—22no—
166#92418—42no—
167#33656—611nomaximal, Sylow
168C22×C61—41no—
169C22×C61#48111yescharacteristic
170C22×C61#48111yescharacteristic
171C22×C61C22×SD1611yescharacteristic
172C2×D61—22no—
173C2×D61—42no—
174C2×D61—42no—
175C2×D61—81no—
176C2×D61—82no—
177C2×D61—81no—
178C4×C61—82no—
179C4×C61#48111yescharacteristic
180C4×C61#48111yescharacteristic
181C4×C61C2×#12311yescharacteristic
182C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
183Dic61—22no—
184Dic61—81no—
185#3054—611no—
186C2×#430—611no—
187#2239—614no—
188#2238—614no—
189#2155—611no—
190#2155D6111yescharacteristic, complemented
191#2148—611no—
192#2148—611no—
193C2×#349—611no—
194C2×C61—21no—
195C2×C61—21no—
196C2×C61—21no—
197C2×C61C2×#48111yescharacteristic
198D61—42no—
199D61—161no—
200C2×#135—611no—
201C22×M16—611no—
202C22×M16—1221no—
203#525—1221no—
204C2×M32—611no—
205C2×M32C2×D6111yescharacteristic
206#470—612no—
207#469—612no—
208#468—1222no—
209#430—614no—
210#430—1222no—
211C2×#108—611no—
212C2×#108—1221no—
213C2×#108C2×D6111yescharacteristic
214#427C2×D6111yescharacteristic
215#416—1221no—
216#365—614no—
217#365C2×D6114yescomplemented
218#362—612no—
219#362—612no—
220#361—612no—
221#361—612no—
222#349—614no—
223C61#3365611yescharacteristic, complemented, Sylow
224C2×#41—1221no—
225C23×C4—611no—
226#135—1221no—
227#135—1221no—
228#135—1221no—
229#135—1221no—
230C2×Q16—611no—
231C2×SD16—1221no—
232C2×D8—611no—
233C2×M16—21no—
234C2×M16—612no—
235C2×M16—612no—
236C2×M16—1221no—
237C2×M16—1221no—
238C2×M16—1221no—
239C2×M16—1221no—
240C2×M16—1224no—
241C2×M16—1221no—
242C2×M16—2441no—
243C2×M16—2442no—
244C22×C8—611no—
245C22×C8—2441no—
246C22×C8C22×D6111yescharacteristic
247#123—2441no—
248C22⋊Q8—1222no—
249#121—1222no—
250#120—2441no—
251C4×D4—2442no—
252#117—21no—
253M32—21no—
254M32—612no—
255M32—1221no—
256M32C22×D6112yes—
257C2×C16—21no—
258C2×C16—1221no—
259#108—21no—
260#108—612no—
261#108—1224no—
262#108—1221no—
263#108C22×D6112yes—
264#107—21no—
265#106C22×D6112yes—
266#103—1222no—
267#102—1222no—
268C24—611no—
269#41—2441no—
270#41—2441no—
271#41—2441no—
272#41—2441no—
273C2×Q8—1221no—
274C2×D4—1221no—
275C2×D4—2442no—
276C2×D4—2441no—
277C22×C4—611no—
278C22×C4—612no—
279C22×C4—612no—
280C22×C4—1221no—
281C22×C4—1221no—
282C22×C4—1221no—
283C22×C4—1221no—
284C22×C4—2441no—
285C22×C4D4×D6111yescharacteristic
286Q16—1221no—
287Q16—1221no—
288SD16—2441no—
289SD16—2441no—
290D8—1221no—
291D8—1221no—
292M16—22no—
293M16—41no—
294M16—1222no—
295M16—1222no—
296M16—1222no—
297M16—1222no—
298M16—1222no—
299M16—2444no—
300M16—2441no—
301C2×C8—41no—
302C2×C8—612no—
303C2×C8—1221no—
304C2×C8—1221no—
305C2×C8—2444no—
306C2×C8—2441no—
307C2×C8#1175822211yescharacteristic
308C2×C8#1175822211yescharacteristic
309C2×C8D4×D6111yescharacteristic
310C2×C8#1175822611yescharacteristic
311C2×C8#1175822611yescharacteristic
312C2×C8C23×D6111yescharacteristic
313C4⋊C4—22no—
314C4⋊C4—2442no—
315C22⋊C4—41no—
316C22⋊C4—2442no—
317C22⋊C4—2442no—
318C42—41no—
319C16—22no—
320C16—1222no—
321C23—612no—
322C23—1221no—
323C23—1221no—
324C23—4881no—
325C23—2441no—
326C23SD16×D6111yescharacteristic
327Q8—2441no—
328Q8—2441no—
329D4—4884no—
330D4—4881no—
331D4—4881no—
332D4—2441no—
333D4—2441no—
334C2×C4—42no—
335C2×C4—612no—
336C2×C4—1221no—
337C2×C4—1221no—
338C2×C4—1222no—
339C2×C4—1222no—
340C2×C4—1222no—
341C2×C4—1222no—
342C2×C4—4881no—
343C2×C4—4881no—
344C2×C4—2441no—
345C2×C4#1451976211yescharacteristic
346C2×C4#1451976211yescharacteristic
347C2×C4SD16×D6111yescharacteristic
348C2×C4#1451980411yescharacteristic
349C2×C4#1451980411yescharacteristic
350C2×C4C2×D4×D6111yescharacteristic
351C8—42no—
352C8—1222no—
353C8—2442no—
354C8#1451976211yescharacteristic
355C8#1451976211yescharacteristic
356C8C2×#1175822211yescharacteristic
357C8C2×#1175822611yescharacteristic
358C22—41no—
359C22—1222no—
360C22—4882no—
361C22—4881no—
362C22—4881no—
363C22—2442no—
364C22—2442no—
365C22#1684008211yescharacteristic
366C22#1684008211yescharacteristic
367C22C2×SD16×D6111yescharacteristic
368C4—82no—
369C4—1222no—
370C4—4881no—
371C4#1684008211yescharacteristic
372C4#1684008211yescharacteristic
373C4C2×#1451976211yescharacteristic
374C4C2×#1451980411yescharacteristic
375C2—21no—
376C2—21no—
377C2—21no—
378C2—9761no—
379C2—2442no—
380C2C2×#1684008211yescharacteristic
subgroupG/H
centerC4#16840082
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2155C2
socleC2×C61C2×#481
prime radical#21455429C2
hypercenter#2155D61
hypocenterC61#33656
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #2155

Complements

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

NH ≅ G/Ntype
2. C2×#16839809C2semidirect
3. #21448040C2semidirect
4. #21448035C2semidirect
5. #21447591C2semidirect
6. #21447569C2semidirect
7. #21447566C2semidirect
8. C2×#16839543C2semidirect
9. C61×#2155C2semidirect
23. #16839584C22semidirect
24. #16839579C22semidirect
25. #16839577C22semidirect
26. #16839576C22semidirect
27. #16839574C22semidirect
32. C61×#365C22semidirect
190. #2155D61semidirect
217. #365C2×D61semidirect
223. C61#33656semidirect

Maximal subgroups

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

subgroupindex
#214554292
C2×#168398092
#214480402
#214480352
#214475912
#214475692
#214475662
C2×#168395432
C61×#21552
#3365661