Finite Group Databasebeta

← #30607181

Subgroups — #30607181

Order 15616.

390 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462161C211yesmaximal, characteristic, complemented
2#21462140C211yesmaximal, characteristic
3#21462134C211yesmaximal, characteristic
4#21461739C211yesmaximal, characteristic, complemented
5#21460892C211yesmaximal, characteristic, complemented
6#21460883C211yesmaximal, characteristic, complemented
7#21454701C211yesmaximal, characteristic
8D61⋊M32C2211yescharacteristic
9D61⋊M32C2211yescharacteristic
10D61⋊#108C2211yescharacteristic
11#16840772C2211yescharacteristic
12#16840685—21no—
13#16840684—21no—
14#16840678—22no—
15#16840676—22no—
16#16840644—21no—
17#16840629—21no—
18C2×#14519796C2211yescharacteristic
19#16840241—21no—
20#16839874C411yescharacteristic, complemented
21#16839870C411yescharacteristic, complemented
22#16839868C411yescharacteristic, complemented
23#16839853—21no—
24C4×#11758200—21no—
25#106×D61C2211yescharacteristic
26#16839760—21no—
27#16839719C2211yescharacteristic
28C61×#442C411yescharacteristic, complemented
29D61⋊M16—21no—
30#14519856—21no—
31#14519852—21no—
32#14519852—21no—
33#14519851—21no—
34#14519851—21no—
35D61⋊C16—21no—
36D61⋊C16—21no—
37#14519848—21no—
38#14519847—21no—
39#14519846—21no—
40#14519845—21no—
41C2×#11758224—21no—
42#14519796—21no—
43#14519796—21no—
44#14519796D412yes—
45#14519796Q812yes—
46C2×Dic244C2×C411yescharacteristic
47C2×#11758200—21no—
48C2×#11758189D411yescharacteristic
49C2×C8×D61C2311yescharacteristic
50C4×Dic122—21no—
51#14519742—21no—
52D4×Dic61—21no—
53C4×C22⋊D61—41no—
54#14519737—41no—
55#14519734D411yescharacteristic
56#14519733—41no—
57C2×#11758196—41no—
58C4⋊C4×D61—21no—
59#14519709C2×C411yescharacteristic
60#14519708—21no—
61#14519702—21no—
62#14519700—21no—
63#14519698—21no—
64#14519687C2×C411yescharacteristic
65C4×#92418D411yescharacteristic
66C8×Dic61D411yescharacteristic
67C2×D8×C61C2×C411yescharacteristic
68C4×D4×C61—21no—
69#106×C61C2×C411yescharacteristic
70#102×C61—21no—
71#97×C61C2×C411yescharacteristic
72C2×D4×C61—21no—
73C22×C4×C61—41no—
74C2×C22⋊D61—41no—
75C22×Dic61—41no—
76C22×Dic61—41no—
77#11758224—22no—
78#11758224—22no—
79#11758224—41no—
80#11758224—41no—
81C2×C4×D61—41no—
82C2×C4×D61C2×D411yescharacteristic
83C2×Dic122—21no—
84C22⋊#6923—41no—
85#11758217—41no—
86C4⋊#6923—41no—
87C4×#6923—41no—
88#11758213—41no—
89D61⋊C8—41no—
90D8×C61—21no—
91D8×C61C4⋊C412yes—
92C2×C8×C61C22⋊C411yescharacteristic
93C2×C8×C61C22×C411yescharacteristic
94C4⋊C4×C61—21no—
95C22⋊C4×C61—41no—
96C42×C61C22⋊C411yescharacteristic
97C22⋊Dic61—41no—
98#11758200—22no—
99#11758200—22no—
100#11758198—21no—
101C4⋊Dic61—21no—
102#11758196—21no—
103#11758196—41no—
104#11758196—41no—
105#11758196—41no—
106C4×Dic61—41no—
107C4×Dic61—41no—
108C4×Dic61C22⋊C411yescharacteristic
109C2×#92418C22⋊C411yescharacteristic
110C2×#92418C2×D411yescharacteristic
111Dic244—21no—
112Dic244C4⋊C412yes—
113#11758189—21no—
114#11758189—21no—
115C8×D61C2×D411yescharacteristic
116C8×D61C2×Q811yescharacteristic
117C8×D61#4111yescharacteristic
118C8×D61#4111yescharacteristic
119#11758187—41no—
120#11758187—41no—
121C23×C61—41no—
122C22×D61AGL(1,ℤ/8)11yescharacteristic
123C2×#6923—81no—
124D4×C61—22no—
125D4×C61—41no—
126C2×C4×C61—21no—
127C2×C4×C61—41no—
128C2×C4×C61—161no—
129C2×C4×C61C2×C22⋊C411yescharacteristic
130C22⋊D61—42no—
131C22⋊D61—42no—
132C2×Dic61—21no—
133C2×Dic61—41no—
134C2×Dic61—42no—
135C2×Dic61—42no—
136C2×Dic61—41no—
137C2×Dic61—161no—
138C2×Dic61AGL(1,ℤ/8)11yescharacteristic
139C4×D61—161no—
140C4×D61#12111yescharacteristic
141C4×D61C22⋊Q811yescharacteristic
142C4×D61#13511yescharacteristic
143C4×D61#13511yescharacteristic
144Dic122—22no—
145Dic122—41no—
146#92420—81no—
147C8×C61—21no—
148C8×C61C2×C4⋊C411yescharacteristic
149C8×C61#11711yescharacteristic
150#92418—21no—
151#92418#12111yescharacteristic
152#92418C22⋊Q811yescharacteristic
153#12828—611nomaximal, Sylow
154C22×C61—42no—
155C22×C61—81no—
156C22×C61—81no—
157C22×C61#41811yescharacteristic
158C2×D61—41no—
159C2×D61#46311yescharacteristic
160C2×D61#47811yescharacteristic
161#6923—161no—
162C4×C61—41no—
163C4×C61—81no—
164C4×C61#38011yescharacteristic
165C4×C61#41611yescharacteristic
166Dic61—41no—
167Dic61—42no—
168Dic61—81no—
169Dic61#46311yescharacteristic
170Dic61#47811yescharacteristic
171#2952—611no—
172#2153—611no—
173#2145—611no—
174#2143—611no—
175#1833—611no—
176#1386—611no—
177#1383—611no—
178C2×C61—21no—
179C2×C61—82no—
180C2×C61#180411yescharacteristic
181D61—21no—
182D61—21no—
183C2×#135—611no—
184#520—1221no—
185C2×M32—611no—
186C2×M32—611no—
187#442—611no—
188#442#692311yescharacteristic, complemented
189#441—611no—
190#441—611no—
191C4×SD16—1221no—
192C4×SD16—1221no—
193C2×#108—611no—
194#427—611no—
195C2×#106—611no—
196#417—1221no—
197#405—611no—
198#364—1221no—
199#364—1221no—
200#360—1222no—
201#359—1222no—
202#344—1221no—
203H(3,ℤ/4)—1221no—
204C61#1282811yescharacteristic, complemented, Sylow
205C2×#41—1221no—
206#135—612no—
207#135—612no—
208#135—1221no—
209#135—1221no—
210C2×Q16—611no—
211C2×SD16—1221no—
212C2×D8C2×#692311yescharacteristic
213C2×M16—611no—
214C2×M16—1221no—
215C22×C8—611no—
216C4×Q8—1221no—
217C4×Q8—1221no—
218C4×D4—21no—
219C4×D4—1221no—
220C4×D4—2441no—
221C4×D4—2441no—
222#117—611no—
223#117—1221no—
224#117—2441no—
225C2×C4⋊C4—1221no—
226C2×C4⋊C4—2441no—
227M32—1221no—
228M32—1221no—
229M32—1221no—
230M32—1221no—
231C2×C16—1221no—
232C2×C16—1221no—
233#108—1221no—
234#108—1221no—
235#107—1221no—
236#106—611no—
237#106—1221no—
238#106—1221no—
239#106C2×#692311yescharacteristic
240#103—1221no—
241#103—1221no—
242#102—21no—
243#102—1221no—
244#97—611no—
245#97C2×#692311yescharacteristic
246C4×C8—611no—
247C4×C8—611no—
248#41—1222no—
249#41—1222no—
250#41—2441no—
251#41—2441no—
252C2×Q8—1221no—
253C2×D4—21no—
254C2×D4—2441no—
255C22×C4—41no—
256C22×C4—611no—
257C22×C4—2441no—
258C22×C4—2441no—
259C22×C4—2441no—
260Q16—612no—
261Q16—1221no—
262SD16—1222no—
263SD16—1222no—
264D8—21no—
265D8C4⋊#692312yes—
266M16—1221no—
267M16—1221no—
268M16—2441no—
269M16—2441no—
270C2×C8—611no—
271C2×C8—611no—
272C2×C8—611no—
273C2×C8—611no—
274C2×C8—611no—
275C2×C8—611no—
276C2×C8—2441no—
277C2×C8C22⋊#692311yescharacteristic
278C2×C8C22×#692311yescharacteristic
279C4⋊C4—21no—
280C4⋊C4—1221no—
281C4⋊C4—1221no—
282C4⋊C4—2441no—
283C4⋊C4—2441no—
284C4⋊C4—2441no—
285C4⋊C4—2441no—
286C22⋊C4—41no—
287C22⋊C4—1221no—
288C22⋊C4—2441no—
289C22⋊C4—2441no—
290C42—611no—
291C42—2441no—
292C42—2441no—
293C42—2441no—
294C42C22⋊#692311yescharacteristic
295C16—2441no—
296C16—2441no—
297C23—41no—
298C23—611no—
299Q8—1222no—
300Q8—2441no—
301D4—22no—
302D4—41no—
303D4—2442no—
304D4—2442no—
305C2×C4—21no—
306C2×C4—41no—
307C2×C4—161no—
308C2×C4—611no—
309C2×C4—611no—
310C2×C4—611no—
311C2×C4—611no—
312C2×C4—611no—
313C2×C4—1221no—
314C2×C4—9761no—
315C2×C4—9761no—
316C2×C4—4881no—
317C2×C4—2442no—
318C2×C4—2442no—
319C2×C4—2441no—
320C2×C4—2441no—
321C2×C4C2×C22⋊#692311yescharacteristic
322C8—21no—
323C8—611no—
324C8—611no—
325C8—1221no—
326C8—4881no—
327C8C2×C4⋊#692311yescharacteristic
328C8#1451985611yescharacteristic
329C22—42no—
330C22—81no—
331C22—81no—
332C22—611no—
333C22—611no—
334C22—2441no—
335C22#1684075411yescharacteristic
336C4—41no—
337C4—81no—
338C4—611no—
339C4—611no—
340C4—9761no—
341C4—4881no—
342C4—2441no—
343C4—2442no—
344C4#1684070411yescharacteristic
345C4#1684075211yescharacteristic
346C2—21no—
347C2—82no—
348C2—1221no—
349C2—1221no—
350C2#2146168911yescharacteristic
subgroupG/H
centerC2#21461689
commutatorC2×C8×C61C22×C4
FrattiniC2×C8C22×#6923
FittingC61×#442C4
socleC2×C61#1804
prime radicalC2×#14519796C22
cosocleC2×C8×D61C23
hypercenter#442#6923
hypocenterC61#12828
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #21462161C2semidirect
4. #21461739C2semidirect
5. #21460892C2semidirect
6. #21460883C2semidirect
20. #16839874C4semidirect
21. #16839870C4semidirect
22. #16839868C4semidirect
28. C61×#442C4semidirect
188. #442#6923semidirect
204. C61#12828semidirect

Maximal subgroups

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

subgroupindex
#214621612
#214621402
#214621342
#214617392
#214608922
#214608832
#214547012
#1282861