Finite Group Databasebeta

← #30601115

Subgroups — #30601115

Order 15616.

516 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 76 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#21462501C211yesmaximal, characteristic, complemented
2#21461717C211yesmaximal, characteristic, complemented
3#21461224C212yesmaximal
4#21460667C212yesmaximal, complemented
5#21450283C211yesmaximal, characteristic
6#16840787—24no—
7#16840765—21no—
8#16840734C2212yes—
9C4×#11758217—21no—
10#16840705C2212yes—
11#16840640—21no—
12#16840640—21no—
13#16840613—24no—
14C2×#14519748C2211yescharacteristic
15C2×#14519735C2211yescharacteristic
16#16839678—21no—
17#16839678—21no—
18#16839665C411yescharacteristic, complemented
19#16839655—21no—
20#16839655—21no—
21#16839632C411yescharacteristic, complemented
22#16839627C411yescharacteristic, complemented
23C4×#11758198C2211yescharacteristic
24C61×#390C411yescharacteristic, complemented
25C2×#11758217—22no—
26C2×C22⋊#6923—22no—
27#14519844—22no—
28C4⋊#92420—22no—
29#14519834—22no—
30#14519834D414yes—
31#14519832—22no—
32C4×#92420—22no—
33(C2×#6923)⋊C4—22no—
34(C2×#6923)⋊C4—22no—
35C22×C22⋊D61—21no—
36C22×C4×D61C2311yescharacteristic
37#14519748—22no—
38#14519748—22no—
39#14519735—22no—
40#14519735—22no—
41C2×C4⋊Dic61—21no—
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61C2×C411yescharacteristic
44C2×C4⋊Dic61C2×C411yescharacteristic
45C2×C22⋊Dic61—21no—
46C2×C22⋊Dic61—21no—
47C2×C22⋊Dic61—21no—
48C2×#11758198—21no—
49C2×#11758198D411yescharacteristic
50C2×#11758198D411yescharacteristic
51C2×C4×Dic61C2×C411yescharacteristic
52C2×C4×Dic61D411yescharacteristic
53C2×C4×Dic61D411yescharacteristic
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57C22×D4×C61C2×C411yescharacteristic
58C2×C4⋊C4×C61C2×C411yescharacteristic
59C2×C22⋊C4×C61—21no—
60C2×C22⋊C4×C61—21no—
61C2×C42×C61C2×C411yescharacteristic
62C24×C61—21no—
63C23×D61C2×D411yescharacteristic
64C22×#6923—42no—
65C2×D4×C61—22no—
66C2×D4×C61—41no—
67C2×D4×C61—41no—
68C22×C4×C61—21no—
69C22×C4×C61C22⋊C411yescharacteristic
70C22×C4×C61C22⋊C411yescharacteristic
71C22×C4×C61C22×C411yescharacteristic
72C2×C22⋊D61—42no—
73C2×C22⋊D61—81no—
74C2×C22⋊D61—81no—
75C22×Dic61—21no—
76C22×Dic61—21no—
77C22×Dic61C22⋊C411yescharacteristic
78C22×Dic61C22⋊C411yescharacteristic
79C22×Dic61C2×D411yescharacteristic
80C2×C4×D61—41no—
81C2×C4×D61—41no—
82C2×C4×D61C2×D412yes—
83C2×C4×D61#4112yes—
84C22⋊#6923—42no—
85C22⋊#6923—42no—
86#11758217—42no—
87#11758217—42no—
88C2×#92420—24no—
89C4⋊C4×C61—22no—
90C22⋊C4×C61—42no—
91C22⋊C4×C61—42no—
92C42×C61—41no—
93C22⋊Dic61—41no—
94C22⋊Dic61—41no—
95C22⋊Dic61—81no—
96C22⋊Dic61—81no—
97#11758198—22no—
98#11758198—22no—
99#11758198—42no—
100C4⋊Dic61—22no—
101C4⋊Dic61—22no—
102C4⋊Dic61—81no—
103C4⋊Dic61—81no—
104C4×Dic61—21no—
105C4×Dic61—21no—
106C4×Dic61—21no—
107C4×Dic61—21no—
108C4×Dic61—41no—
109C23×C61—41no—
110C23×C61—41no—
111C23×C61—41no—
112C23×C61—41no—
113C23×C61—41no—
114C23×C61—41no—
115C23×C61—41no—
116C23×C61C2×C22⋊C411yescharacteristic
117C22×D61—21no—
118C22×D61—21no—
119C22×D61—41no—
120C22×D61—41no—
121C22×D61#12112yes—
122C2×#6923—82no—
123C2×#6923—82no—
124C2×#6923—82no—
125D4×C61—82no—
126C2×C4×C61—21no—
127C2×C4×C61—21no—
128C2×C4×C61—22no—
129C2×C4×C61—21no—
130C2×C4×C61—21no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61C4×D412yes—
136C22⋊D61—84no—
137C2×Dic61—21no—
138C2×Dic61—21no—
139C2×Dic61—21no—
140C2×Dic61—21no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143C2×Dic61—41no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149C2×Dic61#12011yescharacteristic
150C2×Dic61#12311yescharacteristic
151C4×D61—42no—
152C4×D61—42no—
153#92420—44no—
154#9446—611nomaximal, Sylow
155C22×C61—21no—
156C22×C61—21no—
157C22×C61—81no—
158C22×C61—81no—
159C22×C61—81no—
160C22×C61—81no—
161C22×C61—81no—
162C22×C61—81no—
163C22×C61—81no—
164C22×C61#38611yescharacteristic
165C22×C61#41011yescharacteristic
166C22×C61#41011yescharacteristic
167C2×D61—22no—
168C2×D61—22no—
169C2×D61—22no—
170C2×D61—81no—
171C2×D61—81no—
172#6923—162no—
173C4×C61—22no—
174C4×C61—42no—
175C4×C61—41no—
176C4×C61—41no—
177Dic61—21no—
178Dic61—21no—
179Dic61—41no—
180Dic61—41no—
181Dic61—81no—
182Dic61—81no—
183#2921—611no—
184#2500—611no—
185#1820—611no—
186#1510—612no—
187#1284—612no—
188C2×C61—21no—
189C2×C61—21no—
190C2×C61—161no—
191C2×C61#188411yescharacteristic
192C2×C61#189911yescharacteristic
193C2×C61#191011yescharacteristic
194D61—42no—
195C2×#121—611no—
196C2×#117—611no—
197#433—1224no—
198#423—1221no—
199#407—612no—
200C4×M16—1221no—
201#394—1221no—
202#394—1221no—
203#390—611no—
204#390#692311yescharacteristic, complemented
205#388—1221no—
206#388—1221no—
207#384—611no—
208#382—611no—
209#381—612no—
210C4×C22⋊C4—611no—
211#342—1221no—
212#342—1221no—
213#327—1224no—
214C61#944611yescharacteristic, complemented, Sylow
215C22×D4—1221no—
216C22×D4C2×#692311yescharacteristic
217C23×C4—611no—
218C2×M16—1222no—
219#121—1222no—
220#121—1222no—
221#117—1222no—
222#117—1222no—
223C2×C4⋊C4—611no—
224C2×C4⋊C4—611no—
225C2×C4⋊C4—1221no—
226C2×C4⋊C4—1221no—
227C2×C4⋊C4C2×#692311yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—611no—
231C2×C22⋊C4—611no—
232C2×C22⋊C4—1221no—
233C2×C22⋊C4—1221no—
234C2×C22⋊C4—1221no—
235C2×C22⋊C4—1221no—
236C2×C22⋊C4—1222no—
237C2×C42—611no—
238C2×C42—611no—
239C2×C42—611no—
240C2×C42C2×#692311yescharacteristic
241C4⋊C8—1222no—
242C4⋊C8—1222no—
243C22⋊C8—614no—
244C22⋊C8—1222no—
245#97—1222no—
246C4×C8—1222no—
247#95—1221no—
248#95—1222no—
249#95—1222no—
250#95—1221no—
251#95—1221no—
252C24—21no—
253C24—611no—
254C2×D4—22no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—4881no—
258C2×D4—4881no—
259C2×D4—2442no—
260C22×C4—21no—
261C22×C4—611no—
262C22×C4—612no—
263C22×C4—612no—
264C22×C4—611no—
265C22×C4—611no—
266C22×C4—1221no—
267C22×C4—1221no—
268C22×C4—2441no—
269C22×C4—2441no—
270C22×C4—2442no—
271C22×C4C22⋊#692311yescharacteristic
272C22×C4C22⋊#692311yescharacteristic
273C22×C4C22×#692311yescharacteristic
274M16—2442no—
275M16—2442no—
276C2×C8—1224no—
277C4⋊C4—22no—
278C4⋊C4—1222no—
279C4⋊C4—1222no—
280C4⋊C4—4881no—
281C4⋊C4—4881no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—1222no—
285C22⋊C4—1222no—
286C22⋊C4—4881no—
287C22⋊C4—4881no—
288C22⋊C4—2442no—
289C22⋊C4—2441no—
290C22⋊C4—2441no—
291C22⋊C4—2442no—
292C22⋊C4—2442no—
293C42—41no—
294C42—1221no—
295C42—1221no—
296C42—1221no—
297C42—1221no—
298C42—2441no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—612no—
307C23—1221no—
308C23—1221no—
309C23—2441no—
310C23—2441no—
311C23C2×C22⋊#692311yescharacteristic
312D4—82no—
313D4—4884no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—611no—
324C2×C4—611no—
325C2×C4—1221no—
326C2×C4—1221no—
327C2×C4—1221no—
328C2×C4—1221no—
329C2×C4—4882no—
330C2×C4—4882no—
331C2×C4—4882no—
332C2×C4—2442no—
333C2×C4—2442no—
334C2×C4—2441no—
335C2×C4—2441no—
336C2×C4—2441no—
337C2×C4—2441no—
338C2×C4—2441no—
339C2×C4—2441no—
340C2×C4—2441no—
341C2×C4—2441no—
342C2×C4D4×#692312yes—
343C8—2444no—
344C22—21no—
345C22—21no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—1222no—
354C22—1222no—
355C22—1222no—
356C22—4881no—
357C22—4881no—
358C22#1684071011yescharacteristic
359C22#1684074111yescharacteristic
360C22#1684074211yescharacteristic
361C4—22no—
362C4—42no—
363C4—41no—
364C4—41no—
365C4—1221no—
366C4—1221no—
367C4—9762no—
368C4—4881no—
369C4—4881no—
370C4—2441no—
371C4—2441no—
372C2—21no—
373C2—21no—
374C2—161no—
375C2—2442no—
376C2#2146182211yescharacteristic
377C2#2146184211yescharacteristic
378C2#2146186311yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC22×C61#386
prime radicalC2×#14519748C22
cosocleC22×C4×D61C23
hypercenter#390#6923
hypocenterC61#9446
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #21462501C2semidirect
2. #21461717C2semidirect
4. #21460667C2semidirect
18. #16839665C4semidirect
21. #16839632C4semidirect
22. #16839627C4semidirect
24. C61×#390C4semidirect
204. #390#6923semidirect
214. C61#9446semidirect

Maximal subgroups

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

subgroupindex
#214625012
#214617172
#214612242
#214606672
#214502832
#944661