Finite Group Databasebeta

← #30604792

Subgroups — #30604792

Order 15616.

450 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C2×#16840657C211yesmaximal, characteristic, complemented
2#21461511C211yesmaximal, characteristic, complemented
3#21460811C212yesmaximal, complemented
4#21460809C212yesmaximal, complemented
5#21455396C211yesmaximal, characteristic
6C2×D61⋊M16—21no—
7C22×#11758187—21no—
8#16840657—21no—
9#16840657C2212yes—
10#16840638—21no—
11#16840638C2212yes—
12C22×C8×D61C2211yescharacteristic
13#16840269—21no—
14#16840000C411yescharacteristic
15#16840000C2211yescharacteristic
16#16839995C411yescharacteristic
17#16839995C2211yescharacteristic
18#16839988—21no—
19#16839988—21no—
20#16839800C411yescharacteristic
21#16839758—21no—
22C2×#135×C61C411yescharacteristic
23C2×#11758213—22no—
24D61⋊M16—24no—
25D61⋊M16—42no—
26C2×#11758217—41no—
27C2×D61⋊C8—41no—
28C2×#11758187—22no—
29C2×#11758187—42no—
30C22×C4×D61D411yescharacteristic
31C22×#92418D411yescharacteristic
32C2×C8×D61—42no—
33C2×C8×D61C2×C411yescharacteristic
34C2×C8×D61D411yescharacteristic
35C2×C8×D61D411yescharacteristic
36C2×C8×D61C2311yescharacteristic
37#14519758—41no—
38#14519753—21no—
39#14519753—21no—
40#14519748—21no—
41#14519748—21no—
42#14519745—41no—
43C4×C22⋊D61—41no—
44C4×C22⋊D61—41no—
45#14519736—21no—
46#14519711C2×C411yescharacteristic
47#14519711C2×C411yescharacteristic
48#14519709—21no—
49#14519701—21no—
50#14519701—21no—
51#14519698—21no—
52#14519698—21no—
53C2×#41×C61—21no—
54#135×C61—42no—
55C2×Q16×C61C2×C411yescharacteristic
56C2×SD16×C61—21no—
57C2×D8×C61C2×C411yescharacteristic
58C22×C8×C61C2×C411yescharacteristic
59C23×D61SD1611yescharacteristic
60#41×C61—42no—
61#41×C61—42no—
62C2×Q8×C61—21no—
63C2×D4×C61—21no—
64C2×D4×C61—41no—
65C22×C4×C61—41no—
66C22×C4×C61C22⋊C411yescharacteristic
67C2×C22⋊D61—41no—
68C2×C22⋊D61—41no—
69C22×Dic61D811yescharacteristic
70C2×C4×D61—22no—
71C2×C4×D61—22no—
72C2×C4×D61C22⋊C411yescharacteristic
73C2×C4×D61D811yescharacteristic
74C2×C4×D61SD1611yescharacteristic
75C2×C4×D61C2×D411yescharacteristic
76#11758217—44no—
77C2×#92420—42no—
78#11758213—24no—
79#11758213—42no—
80D61⋊C8—44no—
81Q16×C61—41no—
82SD16×C61—42no—
83D8×C61—41no—
84C2×C8×C61—22no—
85C2×C8×C61C22⋊C411yescharacteristic
86C2×C8×C61C22×C411yescharacteristic
87C22⋊Dic61—41no—
88C22⋊Dic61—41no—
89C22⋊Dic61—41no—
90#11758198—41no—
91#11758198—41no—
92C4⋊Dic61—21no—
93C4⋊Dic61—21no—
94#11758196—41no—
95#11758196—41no—
96C4×Dic61—41no—
97C2×#92418—22no—
98C2×#92418C22⋊C411yescharacteristic
99C2×#92418C2×D411yescharacteristic
100C8×D61—41no—
101C8×D61—41no—
102C8×D61—41no—
103C8×D61—41no—
104#11758187—42no—
105C23×C61—41no—
106C23×C61#10211yescharacteristic
107C22×D61—22no—
108C22×D61—22no—
109C22×D61—41no—
110C22×D61#10211yescharacteristic
111C22×D61C2×SD1611yescharacteristic
112Q8×C61—41no—
113Q8×C61—41no—
114D4×C61—41no—
115D4×C61—41no—
116D4×C61—82no—
117C2×C4×C61—41no—
118C2×C4×C61—82no—
119C2×C4×C61#9912yes—
120C2×C4×C61#10211yescharacteristic
121C2×C4×C61#10412yes—
122C2×C4×C61C2×C22⋊C411yescharacteristic
123C22⋊D61—82no—
124C22⋊D61—82no—
125C2×Dic61—22no—
126C2×Dic61—41no—
127C2×Dic61—41no—
128C2×Dic61#10211yescharacteristic
129C2×Dic61C2×D811yescharacteristic
130C4×D61—22no—
131C4×D61—22no—
132C4×D61—21no—
133C4×D61—21no—
134C4×D61—21no—
135C4×D61—21no—
136#92420—44no—
137C8×C61—21no—
138C8×C61—21no—
139#92418—21no—
140#92418—21no—
141#11709—611nomaximal, Sylow
142C22×C61—21no—
143C22×C61—21no—
144C22×C61—81no—
145C22×C61—81no—
146C22×C61—81no—
147C22×C61#32712yes—
148C22×C61C2×#10211yescharacteristic
149C2×D61—21no—
150C2×D61—21no—
151C2×D61—21no—
152C2×D61—21no—
153C2×D61—21no—
154C2×D61—44no—
155C2×D61#35912yes—
156C4×C61—81no—
157C4×C61#32712yes—
158C4×C61C2×#9911yescharacteristic
159C4×C61C2×#10411yescharacteristic
160Dic61—21no—
161Dic61—21no—
162Dic61—81no—
163Dic61—81no—
164#3050—611no—
165C2×C22⋊C16—611no—
166C2×#341—611no—
167#1350—612no—
168#1349—612no—
169C2×C61—22no—
170C2×C61—161no—
171C2×C61C2×#32711yescharacteristic
172C2×C61C2×#35911yescharacteristic
173C2×C61#214811yescharacteristic
174D61—22no—
175D61—41no—
176C2×#135#692311yescharacteristic
177C22×M16—1221no—
178C23×C8—611no—
179#525—1221no—
180C22×C16—1221no—
181#467—611no—
182#467—611no—
183#466—611no—
184#466—611no—
185#465—1221no—
186#465—1221no—
187#427—611no—
188#416—1221no—
189C22⋊C16—612no—
190C22⋊C16—1221no—
191#341—612no—
192#341—1221no—
193C61#1170911yescharacteristic, complemented, Sylow
194C2×#41—21no—
195C23×C4—611no—
196#135—42no—
197C2×Q16C2×#692311yescharacteristic
198C2×SD16—21no—
199C2×D8C2×#692311yescharacteristic
200C2×M16—1222no—
201C2×M16—1224no—
202C2×M16—2441no—
203C2×M16—2442no—
204C22×C8—611no—
205C22×C8—611no—
206C22×C8—611no—
207C22×C8—611no—
208C22×C8—611no—
209C22×C8—2442no—
210C22×C8—2441no—
211C22×C8C2×#692311yescharacteristic
212#123—2441no—
213C22⋊Q8—1221no—
214C22⋊Q8—1221no—
215#121—1221no—
216#121—1221no—
217#120—2441no—
218C4×D4—2441no—
219C4×D4—2441no—
220#117—1221no—
221C2×C16—1222no—
222C2×C16—2442no—
223#107—611no—
224#107—611no—
225#106—1221no—
226#103—1221no—
227#103—1221no—
228#102—1221no—
229#102—1221no—
230C24—611no—
231#41—42no—
232#41—42no—
233C2×Q8—21no—
234C2×D4—21no—
235C2×D4—41no—
236C2×D4—2441no—
237C2×D4—2441no—
238C22×C4—41no—
239C22×C4—611no—
240C22×C4—611no—
241C22×C4—611no—
242C22×C4—611no—
243C22×C4—611no—
244C22×C4—1222no—
245C22×C4—1222no—
246C22×C4C22⋊#692311yescharacteristic
247Q16—41no—
248SD16—42no—
249D8—41no—
250M16—1224no—
251M16—2442no—
252M16—2444no—
253C2×C8—22no—
254C2×C8—611no—
255C2×C8—611no—
256C2×C8—1222no—
257C2×C8—2441no—
258C2×C8—2441no—
259C2×C8—2441no—
260C2×C8—2441no—
261C2×C8—2442no—
262C2×C8—2444no—
263C2×C8C22⋊#692311yescharacteristic
264C2×C8C22×#692311yescharacteristic
265C4⋊C4—1221no—
266C4⋊C4—1221no—
267C4⋊C4—2441no—
268C4⋊C4—2441no—
269C22⋊C4—2441no—
270C22⋊C4—2441no—
271C22⋊C4—2441no—
272C22⋊C4—2441no—
273C22⋊C4—2441no—
274C42—2441no—
275C16—2442no—
276C23—41no—
277C23—611no—
278C23—611no—
279C23—1222no—
280C23—1222no—
281C23—2441no—
282C23#1451983911yescharacteristic
283Q8—41no—
284Q8—41no—
285D4—41no—
286D4—41no—
287D4—82no—
288D4—4882no—
289D4—4882no—
290C2×C4—41no—
291C2×C4—82no—
292C2×C4—611no—
293C2×C4—611no—
294C2×C4—1222no—
295C2×C4—1221no—
296C2×C4—1221no—
297C2×C4—1221no—
298C2×C4—1221no—
299C2×C4—1222no—
300C2×C4—1222no—
301C2×C4—2441no—
302C2×C4—2441no—
303C2×C4#1451983512yes—
304C2×C4#1451983911yescharacteristic
305C2×C4#1451984112yes—
306C2×C4C2×C22⋊#692311yescharacteristic
307C8—21no—
308C8—21no—
309C8—1221no—
310C8—1221no—
311C8—2444no—
312C22—21no—
313C22—21no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22—612no—
318C22—1221no—
319C22—1221no—
320C22—1221no—
321C22—1221no—
322C22—1221no—
323C22—2444no—
324C22#1684061312yes—
325C22C2×#1451983911yescharacteristic
326C4—81no—
327C4—1221no—
328C4—1221no—
329C4—4881no—
330C4—4881no—
331C4#1684061312yes—
332C4C2×#1451983511yescharacteristic
333C4C2×#1451984111yescharacteristic
334C2—22no—
335C2—161no—
336C2—1222no—
337C2—2441no—
338C2C2×#1684061311yescharacteristic
339C2D61⋊#35911yescharacteristic
340C2#2146214611yescharacteristic
subgroupG/H
centerC2×C4#14519839
commutatorC2×C8×C61C22×C4
FrattiniC2×C8C22×#6923
FittingC2×#135×C61C4
socleC22×C61C2×#102
prime radical#21455396C2
cosocleC2×C8×D61C23
hypercenterC2×#135#6923
hypocenterC61#11709
LCSG ▷ C2×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#16840657C2semidirect
2. #21461511C2semidirect
3. #21460811C2semidirect
4. #21460809C2semidirect
193. C61#11709semidirect

Maximal subgroups

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

subgroupindex
C2×#168406572
#214615112
#214608112
#214608092
#214553962
#1170961