Finite Group Databasebeta

← #29593736

Subgroups — #29593736

Order 15616.

296 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 69 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#21458687C211yesmaximal, characteristic
2#21448046C211yesmaximal, characteristic, complemented
3#21447861C211yesmaximal, characteristic
4#21443878C211yesmaximal, characteristic, complemented
5Q8⋊#11758185C211yesmaximal, characteristic, complemented
6#21443779C211yesmaximal, characteristic
7C61×#1670C211yesmaximal, characteristic, complemented
8#16840472—21no—
9C2×#14519720—21no—
10#16840167C2211yescharacteristic
11#16840164—21no—
12#16840063—22no—
13#16840053—21no—
14C4×D488—21no—
15C4×D488C2211yescharacteristic
16C4⋊#11758185C2211yescharacteristic
17#16839566—22no—
18#16839566—21no—
19C4×#11758185C2211yescharacteristic
20C61×#494C2211yescharacteristic
21C61×#475—21no—
22C8×Q8×C61C2211yescharacteristic
23C61×#335C2211yescharacteristic
24C61×#332—21no—
25C2×D488D411yescharacteristic
26#14519777—21no—
27#14519769—22no—
28#14519755—42no—
29#14519753—41no—
30C4×D244—22no—
31C4×D244—21no—
32C4⋊C4×D61—41no—
33#14519720—42no—
34C2×#11758185—21no—
35C2×#11758185—21no—
36#14519711—21no—
37#14519711D412yes—
38#14519711D411yescharacteristic
39#14519709—22no—
40#14519699—22no—
41#14519699—21no—
42C2×Q16×C61—21no—
43C2×Q16×C61—21no—
44C4⋊Q8×C61—21no—
45C4×Q8×C61D411yescharacteristic
46#106×C61—41no—
47C4⋊C8×C61—21no—
48C4⋊C8×C61—21no—
49C4⋊C8×C61D411yescharacteristic
50#103×C61—41no—
51C4×C8×C61—21no—
52C4×C8×C61C2311yescharacteristic
53C2×Q8×C61—41no—
54C2×Q8×C61D811yescharacteristic
55C2×D244—21no—
56C2×C4×D61—41no—
57C2×C4×D61—42no—
58C2×Dic122—41no—
59Q16×C61—42no—
60Q16×C61—41no—
61Q16×C61—41no—
62C2×C8×C61—21no—
63C2×C8×C61—41no—
64C2×C8×C61C2×D411yescharacteristic
65C2×C8×C61C2×D411yescharacteristic
66C4⋊C4×C61—21no—
67C4⋊C4×C61—41no—
68C4⋊C4×C61—41no—
69C4⋊C4×C61D811yescharacteristic
70C42×C61—21no—
71C42×C61C2×D411yescharacteristic
72#11758198—42no—
73#11758198—41no—
74C4⋊Dic61—21no—
75C4⋊Dic61—22no—
76C4⋊Dic61—41no—
77#11758196—42no—
78#11758196—42no—
79D488—21no—
80D488SD1612yes—
81#11758185—41no—
82#11758185—41no—
83C22×D61—41no—
84Q8×C61—21no—
85Q8×C61—21no—
86Q8×C61—81no—
87Q8×C61—81no—
88C2×C4×C61—21no—
89C2×C4×C61—21no—
90C2×C4×C61—81no—
91C2×C4×C61#12011yescharacteristic
92C2×C4×C61C2×D811yescharacteristic
93C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
94C2×Dic61—41no—
95C2×Dic61—42no—
96D244—22no—
97D244—41no—
98C4×D61—81no—
99C4×D61—81no—
100C4×D61—162no—
101Dic122—161no—
102C8×C61—21no—
103C8×C61—41no—
104C8×C61—81no—
105C8×C61C2×SD1611yescharacteristic
106C8×C61#13711yescharacteristic
107#12359—611nomaximal, Sylow
108C22×C61#44711yescharacteristic
109C2×D61—42no—
110C2×D61—81no—
111C2×D61—81no—
112C4×C61—21no—
113C4×C61—41no—
114C4×C61—41no—
115C4×C61—161no—
116C4×C61#44811yescharacteristic
117C4×C61#45411yescharacteristic
118C4×C61#50811yescharacteristic
119C4×C61#51011yescharacteristic
120Dic61—81no—
121Dic61—82no—
122#3380—611no—
123#2241—611no—
124#2208—611no—
125#1670D6111yescharacteristic, complemented
126#1365—611no—
127Q8⋊C16—611no—
128#1331—611no—
129C2×C61#162711yescharacteristic
130C2×C61#218911yescharacteristic
131C2×C61#219411yescharacteristic
132D61—82no—
133#554—1221no—
134C2×SD32—1221no—
135#500—611no—
136#499—1221no—
137#494C2×D6111yescharacteristic
138#478—1222no—
139#476—1221no—
140#475—21no—
141C8×Q8C2×D6111yescharacteristic
142C4×D8—611no—
143C4×D8—1221no—
144C4⋊C16—611no—
145#358—1221no—
146#358—1222no—
147C4×C16—611no—
148#335C2×D6111yescharacteristic
149#332—21no—
150C61#1235911yescharacteristic, complemented, Sylow
151C2×Q16—21no—
152C2×Q16—21no—
153C2×D8—611no—
154C4⋊Q8—21no—
155C4⋊Q8—1221no—
156#125—1222no—
157C22⋊Q8—2442no—
158C22⋊Q8—2441no—
159C4×Q8C22⋊D6111yescharacteristic
160C4×D4—1222no—
161C4×D4—1221no—
162C2×C4⋊C4—2441no—
163SD32—2442no—
164C2×C16—1221no—
165C2×C16—1221no—
166#107—611no—
167#107—612no—
168#107—1221no—
169#106—41no—
170#106—1222no—
171C4⋊C8—21no—
172C4⋊C8—21no—
173C4⋊C8C22⋊D6111yescharacteristic
174#103—41no—
175#102—1222no—
176#102—1221no—
177C4×C8—21no—
178C4×C8C22×D6111yescharacteristic
179C2×Q8—41no—
180C2×Q8—2441no—
181C2×Q8#1175819911yescharacteristic
182C2×D4—1221no—
183C22×C4—2442no—
184C22×C4—2441no—
185Q16—42no—
186Q16—41no—
187Q16—41no—
188D8—612no—
189D8—1221no—
190C2×C8—21no—
191C2×C8—41no—
192C2×C8D4×D6111yescharacteristic
193C2×C8D4×D6111yescharacteristic
194C4⋊C4—21no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—1221no—
198C4⋊C4—1222no—
199C4⋊C4—2442no—
200C4⋊C4—2442no—
201C4⋊C4—2441no—
202C4⋊C4#1175819911yescharacteristic
203C22⋊C4—2442no—
204C22⋊C4—2441no—
205C42—21no—
206C42C2×C22⋊D6111yescharacteristic
207C16—2441no—
208C16—2441no—
209C23—2441no—
210Q8—21no—
211Q8—21no—
212Q8—81no—
213Q8—81no—
214Q8—9761no—
215D4—1222no—
216D4—2441no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—81no—
220C2×C4—9762no—
221C2×C4—4881no—
222C2×C4—4881no—
223C2×C4—2441no—
224C2×C4—2442no—
225C2×C4#1451974511yescharacteristic
226C2×C4C2×#1175819911yescharacteristic
227C2×C4#1451980211yescharacteristic
228C8—21no—
229C8—41no—
230C8—81no—
231C8SD16×D6111yescharacteristic
232C8#1451980611yescharacteristic
233C22—4881no—
234C22—4881no—
235C22—2442no—
236C22#1683989511yescharacteristic
237C4—21no—
238C4—41no—
239C4—41no—
240C4—161no—
241C4—4881no—
242C4—4882no—
243C4#1683989911yescharacteristic
244C4#1683993311yescharacteristic
245C4#1684019211yescharacteristic
246C4#1684020311yescharacteristic
247C2—4882no—
248C2#2144492011yescharacteristic
249C2#2144775711yescharacteristic
250C2#2144778811yescharacteristic
subgroupG/H
centerC22#16839895
commutator, cosocleC4×C8×C61C23
FrattiniC4×C8C22×D61
FittingC61×#1670C2
socleC22×C61#447
prime radicalC2×D488D4
hypercenter#1670D61
hypocenterC61#12359
LCSG ▷ C4×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #1670

Complements

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

NH ≅ G/Ntype
2. #21448046C2semidirect
4. #21443878C2semidirect
5. Q8⋊#11758185C2semidirect
7. C61×#1670C2semidirect
125. #1670D61semidirect
150. C61#12359semidirect

Maximal subgroups

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

subgroupindex
#214586872
#214480462
#214478612
#214438782
Q8⋊#117581852
#214437792
C61×#16702
#1235961