Finite Group Databasebeta

← #29581007

Subgroups — #29581007

Order 15616.

476 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 99 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#21456341C211yesmaximal, characteristic
2#21455859C211yesmaximal, characteristic
3#21455854C211yesmaximal, characteristic
4#21446758C211yesmaximal, characteristic
5#21444221C211yesmaximal, characteristic
6#21443649C211yesmaximal, characteristic
7C61×#1481C211yesmaximal, characteristic
8#16840319C2211yescharacteristic
9C2×C4×Dic122C2211yescharacteristic
10#16840172—21no—
11#16840172—21no—
12#16840168—21no—
13#16840168—21no—
14#16840098—21no—
15#16840095—21no—
16#16840058—21no—
17#16840054—21no—
18#16839973—21no—
19#16839969—21no—
20#16839920—21no—
21#16839917—21no—
22C4×Dic244—22no—
23C4×#11758202—22no—
24#16839815C2211yescharacteristic
25#16839798—22no—
26C2×#14519688C2211yescharacteristic
27#16839516—22no—
28#16839492—22no—
29C61×#533C2211yescharacteristic
30C61×#431C2211yescharacteristic
31C2×C4⋊C8×C61C2211yescharacteristic
32C61×Q8⋊C8—22no—
33C22×Dic122D411yescharacteristic
34C2×Dic244—22no—
35C2×#11758202—22no—
36C22×#92418D412yes—
37#14519777D411yescharacteristic
38#14519777D411yescharacteristic
39#14519769—21no—
40#14519756—21no—
41#14519756—21no—
42C4×Dic122—21no—
43C4×Dic122—21no—
44C4×Dic122—21no—
45C4×Dic122—21no—
46C4×Dic122—21no—
47C4×Dic122—21no—
48#14519736—21no—
49C2×C4⋊Dic61D411yescharacteristic
50C2×#11758196—21no—
51C2×C4×Dic61—21no—
52#14519711—22no—
53#14519710—22no—
54#14519710—22no—
55#14519710—22no—
56#14519708—22no—
57#14519708—22no—
58C4⋊#92418—21no—
59#14519702—22no—
60#14519702—22no—
61#14519701—22no—
62#14519700—22no—
63C22⋊#92418—21no—
64#14519688—21no—
65#14519688—21no—
66C4×#92418D412yes—
67C22×C8×C61—21no—
68C4⋊Q8×C61D411yescharacteristic
69C4⋊Q8×C61D411yescharacteristic
70#125×C61—21no—
71C22⋊Q8×C61—21no—
72C22⋊Q8×C61—21no—
73C4×Q8×C61—21no—
74C4×Q8×C61—21no—
75#117×C61—21no—
76C2×C42×C61C2311yescharacteristic
77C4⋊C8×C61—21no—
78C4⋊C8×C61—21no—
79C4⋊C8×C61—21no—
80C22⋊C8×C61—21no—
81C4×C8×C61D412yes—
82C2×Q8×C61—21no—
83C2×Q8×C61—21no—
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61#4111yescharacteristic
87C22×Dic61—21no—
88C22×Dic61—21no—
89C2×Dic122—21no—
90C2×Dic122—21no—
91C2×Dic122—21no—
92C2×Dic122—21no—
93C2×Dic122—42no—
94C2×C8×C61—22no—
95C2×C8×C61—41no—
96C2×C8×C61—42no—
97C4⋊C4×C61—21no—
98C4⋊C4×C61—21no—
99C4⋊C4×C61—21no—
100C4⋊C4×C61—21no—
101C4⋊C4×C61—41no—
102C4⋊C4×C61—41no—
103C22⋊C4×C61—41no—
104C42×C61—41no—
105C42×C61C2×D411yescharacteristic
106C42×C61C2×D411yescharacteristic
107C42×C61C2×D411yescharacteristic
108C42×C61C2×D411yescharacteristic
109C22⋊Dic61—41no—
110#11758202—44no—
111C4⋊Dic61—21no—
112C4⋊Dic61—21no—
113C4⋊Dic61—21no—
114C4⋊Dic61—21no—
115C4⋊Dic61—21no—
116C4⋊Dic61—21no—
117#11758196—41no—
118#11758196—41no—
119#11758196—41no—
120#11758196—41no—
121C4×Dic61—41no—
122C4×Dic61—41no—
123C4×Dic61—41no—
124C2×#92418—22no—
125C2×#92418—22no—
126C2×#92418—22no—
127C2×#92418Q1614yescomplemented
128Dic244—44no—
129C23×C61#12111yescharacteristic
130Q8×C61—42no—
131Q8×C61—42no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61#12011yescharacteristic
137C2×C4×C61#12011yescharacteristic
138C2×C4×C61#12011yescharacteristic
139C2×C4×C61#12111yescharacteristic
140C2×C4×C61#12111yescharacteristic
141C2×C4×C61#12711yescharacteristic
142C2×C4×C61C2×Q1612yes—
143C2×C4×C61#13712yes—
144C2×Dic61—22no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—42no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151Dic122—22no—
152Dic122—22no—
153Dic122—41no—
154Dic122—41no—
155Dic122—42no—
156Dic122—42no—
157C8×C61—42no—
158C8×C61—81no—
159#92418—24no—
160#92418—42no—
161#10037—611nomaximal, Sylow
162C22×C61—21no—
163C22×C61—21no—
164C22×C61#39211yescharacteristic
165C22×C61#46212yes—
166C4×C61—21no—
167C4×C61—21no—
168C4×C61—21no—
169C4×C61—21no—
170C4×C61—81no—
171C4×C61—81no—
172C4×C61#45112yes—
173C4×C61#49411yescharacteristic
174C4×C61#49711yescharacteristic
175Dic61—42no—
176Dic61—81no—
177Dic61—81no—
178Dic61—81no—
179#3162—611no—
180#3104—611no—
181#3104—611no—
182#1985—611no—
183#1481—611no—
184#1481D6111yescharacteristic
185#1275—611no—
186C2×C61—22no—
187C2×C61#201011yescharacteristic
188C2×C61#201611yescharacteristic
189C2×C61#201711yescharacteristic
190#533—611no—
191#533C2×D6111yescharacteristic
192C2×C4×Q8—611no—
193#501—1221no—
194#501—1221no—
195#501—1221no—
196#501—1221no—
197#484—1221no—
198#484—1221no—
199#477—1221no—
200#477—1221no—
201#462—1221no—
202#462—1221no—
203#451—1221no—
204#451—1221no—
205C4×Q16—1222no—
206C4×Q16—1222no—
207#431—611no—
208#431C2×D6111yescharacteristic
209#427—1222no—
210C2×C4⋊C8C2×D6111yescharacteristic
211C2×#97—611no—
212#336—1222no—
213Q8⋊C8—22no—
214Q8⋊C8—1222no—
215C61#1003711yescharacteristic, complemented, Sylow
216C22×Q8—611no—
217C2×Q16—1222no—
218C2×Q16—1222no—
219C22×C8—21no—
220C22×C8—612no—
221C4⋊Q8—611no—
222C4⋊Q8—611no—
223C4⋊Q8D24411yescharacteristic
224C4⋊Q8D24411yescharacteristic
225#125—21no—
226#125—1221no—
227C22⋊Q8—21no—
228C22⋊Q8—21no—
229C22⋊Q8—1221no—
230C22⋊Q8—1221no—
231C4×Q8—21no—
232C4×Q8—21no—
233C4×Q8—1221no—
234C4×Q8—1221no—
235C4×Q8—1221no—
236C4×Q8—1221no—
237C4×Q8—1221no—
238C4×Q8—1221no—
239#117—21no—
240#117—1221no—
241C2×C4⋊C4—611no—
242C2×C4⋊C4—1221no—
243C2×C42—1221no—
244C2×C42C22×D6111yescharacteristic
245#107—1222no—
246#107—1222no—
247#107—1222no—
248#107—1222no—
249#106—1222no—
250#106—1222no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—1221no—
255#103—1222no—
256#103—1222no—
257#103—1222no—
258#103—1222no—
259C22⋊C8—21no—
260C22⋊C8—1221no—
261#97—1221no—
262#97—1221no—
263C4×C8—612no—
264C4×C8C22⋊D6112yes—
265C2×Q8—21no—
266C2×Q8—21no—
267C2×Q8—1221no—
268C2×Q8—1221no—
269C2×Q8—1221no—
270C2×Q8—1221no—
271C2×Q8—2442no—
272C22×C4—1221no—
273C22×C4—1221no—
274C22×C4#1175822211yescharacteristic
275C22×C4D4×D6111yescharacteristic
276C22×C4D4×D6111yescharacteristic
277Q16—2444no—
278Q16—2444no—
279C2×C8—22no—
280C2×C8—41no—
281C2×C8—42no—
282C2×C8—614no—
283C2×C8—1222no—
284C2×C8—1222no—
285C2×C8—1222no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—1221no—
293C4⋊C4—1221no—
294C4⋊C4—1221no—
295C4⋊C4—1221no—
296C4⋊C4—1221no—
297C4⋊C4—1221no—
298C4⋊C4—2441no—
299C4⋊C4—2441no—
300C4⋊C4—2441no—
301C4⋊C4—2441no—
302C22⋊C4—41no—
303C22⋊C4—2441no—
304C42—41no—
305C42—2441no—
306C42—2441no—
307C42—2441no—
308C42C2×D24411yescharacteristic
309C42D4×D6111yescharacteristic
310C42D4×D6111yescharacteristic
311C42C2×C22⋊D6111yescharacteristic
312C23#1451975011yescharacteristic
313Q8—42no—
314Q8—42no—
315Q8—1222no—
316Q8—1222no—
317Q8—2442no—
318Q8—2442no—
319Q8—2441no—
320Q8—2441no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—1222no—
326C2×C4—2442no—
327C2×C4—2441no—
328C2×C4—2441no—
329C2×C4—2441no—
330C2×C4—2441no—
331C2×C4—2441no—
332C2×C4#1451974511yescharacteristic
333C2×C4#1451974711yescharacteristic
334C2×C4#1451974711yescharacteristic
335C2×C4#1451975011yescharacteristic
336C2×C4#1451975111yescharacteristic
337C2×C4#1451977311yescharacteristic
338C2×C4Q16×D6112yes—
339C2×C4#1451980612yes—
340C8—42no—
341C8—81no—
342C8—1224no—
343C8—2442no—
344C22—21no—
345C22—21no—
346C22#1683967111yescharacteristic
347C22#1683997112yes—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—81no—
353C4—81no—
354C4—4881no—
355C4—4881no—
356C4—4881no—
357C4—2442no—
358C4#1683991912yes—
359C4#1684014011yescharacteristic
360C4#1684015311yescharacteristic
361C2—22no—
362C2#2144688511yescharacteristic
363C2#2144692011yescharacteristic
364C2#2144692311yescharacteristic
subgroupG/H
centerC2×C4#14519773
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1481C2
socleC22×C61#392
prime radicalC23×C61#121
hypercenter#1481D61
hypocenterC61#10037
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1481

Complements

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

NH ≅ G/Ntype
127. C2×#92418Q16semidirect
215. C61#10037semidirect

Maximal subgroups

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

subgroupindex
#214563412
#214558592
#214558542
#214467582
#214442212
#214436492
C61×#14812
#1003761