Finite Group Databasebeta

← #30600501

Subgroups — #30600501

Order 15616.

532 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 80 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#21462672C211yesmaximal, characteristic
2#21461566C211yesmaximal, characteristic
3#21461219C212yesmaximal, complemented
4#21460666C212yesmaximal, complemented
5#21451742C211yesmaximal, characteristic
6#16840791—24no—
7#16840783—21no—
8C2×#14519835—21no—
9#16840733C2212yes—
10C2×C22⋊#92420—21no—
11C4×C22⋊#6923C2212yes—
12#16840640—21no—
13#16840616—24no—
14C2×#14519753C2211yescharacteristic
15C2×#14519737C2211yescharacteristic
16#16839694C411yescharacteristic, complemented
17#16839691—21no—
18#16839687C411yescharacteristic, complemented
19#16839681C411yescharacteristic, complemented
20#16839678—21no—
21#16839645—21no—
22#16839634—21no—
23#16839624C2211yescharacteristic
24C61×#393C411yescharacteristic, complemented
25C2×#11758213—22no—
26C22×#92420—22no—
27C2×C22⋊#6923—22no—
28C2×C4×#6923—22no—
29C4⋊#92420—22no—
30#14519835—24no—
31#14519834D414yes—
32C22⋊#92420—22no—
33C22⋊#92420—22no—
34#14519832—22no—
35(C2×#6923)⋊C4—22no—
36C22×C22⋊D61D411yescharacteristic
37C23×Dic61D411yescharacteristic
38C22×C4×D61C2311yescharacteristic
39#14519753—22no—
40#14519753—22no—
41#14519737—22no—
42#14519737—22no—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61C2×C411yescharacteristic
45C2×#11758196—21no—
46C2×#11758196—21no—
47C2×#11758196C2×C411yescharacteristic
48C2×C22⋊Dic61—21no—
49C2×#11758198—21no—
50C2×#11758198—21no—
51C2×#11758198D411yescharacteristic
52C2×C4×Dic61D411yescharacteristic
53#14519684—21no—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57#14519684C2×C411yescharacteristic
58C22×Q8×C61C2×C411yescharacteristic
59C2×C22⋊C4×C61—21no—
60C2×C22⋊C4×C61C2×C411yescharacteristic
61#95×C61—21no—
62#95×C61C2×C411yescharacteristic
63C24×C61C22⋊C411yescharacteristic
64C23×D61C2×D411yescharacteristic
65C22×#6923—42no—
66C2×Q8×C61—22no—
67C2×Q8×C61—81no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61C22⋊C411yescharacteristic
71C22×C4×C61C22×C411yescharacteristic
72C2×C22⋊D61—22no—
73C2×C22⋊D61—22no—
74C2×C22⋊D61—41no—
75C22×Dic61—21no—
76C22×Dic61—21no—
77C22×Dic61—21no—
78C22×Dic61—21no—
79C22×Dic61—81no—
80C22×Dic61C22⋊C411yescharacteristic
81C22×Dic61C22⋊C411yescharacteristic
82C22×Dic61C2×D411yescharacteristic
83C2×C4×D61—81no—
84C2×C4×D61C2×D412yes—
85C2×C4×D61#4112yes—
86C22⋊#6923—24no—
87C22⋊#6923—42no—
88C2×#92420—24no—
89C2×#92420—82no—
90C4×#6923—82no—
91#11758213—82no—
92C22⋊C4×C61—22no—
93C22⋊C4×C61—81no—
94C22⋊Dic61—42no—
95#11758198—22no—
96#11758198—42no—
97#11758198—81no—
98C4⋊Dic61—22no—
99C4⋊Dic61—42no—
100#11758196—22no—
101#11758196—41no—
102#11758196—41no—
103#11758196—81no—
104C4×Dic61—21no—
105C4×Dic61—21no—
106C23×C61—21no—
107C23×C61—21no—
108C23×C61—41no—
109C23×C61—41no—
110C23×C61#10412yes—
111C23×C61C2×C22⋊C411yescharacteristic
112C22×D61—21no—
113C22×D61—21no—
114C22×D61—41no—
115C22×D61—41no—
116C22×D61#12112yes—
117C2×#6923—44no—
118C2×#6923—82no—
119C2×#6923—82no—
120Q8×C61—82no—
121C2×C4×C61—21no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61C4×D412yes—
131C22⋊D61—44no—
132C2×Dic61—21no—
133C2×Dic61—21no—
134C2×Dic61—21no—
135C2×Dic61—21no—
136C2×Dic61—41no—
137C2×Dic61—42no—
138C2×Dic61—41no—
139C2×Dic61—41no—
140C2×Dic61—41no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143C2×Dic61—41no—
144C2×Dic61—41no—
145C2×Dic61—81no—
146C2×Dic61#12011yescharacteristic
147C2×Dic61#12311yescharacteristic
148C4×D61—42no—
149C4×D61—42no—
150#92420—44no—
151#9183—611nomaximal, Sylow
152C22×C61—21no—
153C22×C61—21no—
154C22×C61—22no—
155C22×C61—42no—
156C22×C61—41no—
157C22×C61—41no—
158C22×C61—81no—
159C22×C61#38611yescharacteristic
160C22×C61C2×#10411yescharacteristic
161C22×C61#42111yescharacteristic
162C2×D61—22no—
163C2×D61—22no—
164C2×D61—22no—
165C2×D61—81no—
166C2×D61—81no—
167#6923—84no—
168C4×C61—22no—
169C4×C61—41no—
170C4×C61—81no—
171C4×C61—81no—
172Dic61—21no—
173Dic61—21no—
174Dic61—41no—
175Dic61—41no—
176Dic61—81no—
177Dic61—81no—
178#2973—611no—
179#2638—611no—
180C4×#99—611no—
181#1508—612no—
182#1284—612no—
183C2×C61—21no—
184C2×C61—21no—
185C2×C61—42no—
186C2×C61#187111yescharacteristic
187C2×C61#190911yescharacteristic
188C2×C61#191311yescharacteristic
189D61—42no—
190C2×C22⋊Q8—611no—
191C2×C4×D4—611no—
192#435—1224no—
193#432—1221no—
194C2×#99—1221no—
195#407—612no—
196C2×C22⋊C8—1221no—
197#398—611no—
198#397—611no—
199#397—1221no—
200#395—611no—
201#394—1221no—
202#393#692311yescharacteristic, complemented
203#387—1221no—
204#385—1221no—
205#381—611no—
206C4×C22⋊C4—612no—
207#342—1221no—
208#328—1224no—
209C61#918311yescharacteristic, complemented, Sylow
210C22×Q8C2×#692311yescharacteristic
211C22×D4—611no—
212C23×C4—611no—
213C23×C4—611no—
214C2×M16—1222no—
215C22×C8—1222no—
216C22⋊Q8—1222no—
217C22⋊Q8—1222no—
218C4×D4—1222no—
219C4×D4—1222no—
220C2×C4⋊C4—611no—
221C2×C4⋊C4—611no—
222C2×C4⋊C4—1221no—
223C2×C4⋊C4—1221no—
224C2×C4⋊C4—1221no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—611no—
227C2×C22⋊C4—1222no—
228C2×C22⋊C4—1221no—
229C2×C22⋊C4—1221no—
230C2×C22⋊C4—1221no—
231C2×C22⋊C4C2×#692311yescharacteristic
232C2×C42—611no—
233C2×C42—1222no—
234C4⋊C8—1222no—
235#99—1224no—
236C22⋊C8—614no—
237C22⋊C8—1222no—
238C22⋊C8—1222no—
239#97—1222no—
240#95—21no—
241#95—611no—
242#95—1221no—
243#95—1221no—
244#95—1221no—
245#95—1222no—
246#95—1221no—
247#95C2×#692311yescharacteristic
248C24—611no—
249C24C22⋊#692311yescharacteristic
250C2×Q8—22no—
251C2×Q8—81no—
252C2×D4—1222no—
253C2×D4—1222no—
254C2×D4—2441no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—611no—
258C22×C4—612no—
259C22×C4—612no—
260C22×C4—611no—
261C22×C4—611no—
262C22×C4—1221no—
263C22×C4—1221no—
264C22×C4—1221no—
265C22×C4—1221no—
266C22×C4—4881no—
267C22×C4—4881no—
268C22×C4—2442no—
269C22×C4C22⋊#692311yescharacteristic
270C22×C4C22×#692311yescharacteristic
271M16—4882no—
272C2×C8—1224no—
273C2×C8—4882no—
274C4⋊C4—1222no—
275C4⋊C4—1222no—
276C4⋊C4—4881no—
277C4⋊C4—2441no—
278C4⋊C4—2441no—
279C4⋊C4—2442no—
280C22⋊C4—22no—
281C22⋊C4—81no—
282C22⋊C4—1222no—
283C22⋊C4—1224no—
284C22⋊C4—4881no—
285C22⋊C4—2442no—
286C22⋊C4—2442no—
287C22⋊C4—2442no—
288C42—1221no—
289C42—1221no—
290C42—4882no—
291C23—21no—
292C23—21no—
293C23—41no—
294C23—41no—
295C23—612no—
296C23—1221no—
297C23—1221no—
298C23—2441no—
299C23—2441no—
300C23#1451984212yes—
301C23C2×C22⋊#692311yescharacteristic
302Q8—82no—
303D4—2444no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—611no—
314C2×C4—611no—
315C2×C4—1221no—
316C2×C4—1221no—
317C2×C4—1221no—
318C2×C4—1221no—
319C2×C4—4881no—
320C2×C4—4882no—
321C2×C4—4882no—
322C2×C4—2442no—
323C2×C4—2442no—
324C2×C4—2441no—
325C2×C4—2442no—
326C2×C4—2441no—
327C2×C4—2441no—
328C2×C4—2444no—
329C2×C4—2441no—
330C2×C4—2441no—
331C2×C4—2441no—
332C2×C4—2441no—
333C2×C4—2441no—
334C2×C4D4×#692312yes—
335C8—2444no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—42no—
340C22—41no—
341C22—41no—
342C22—81no—
343C22—1222no—
344C22—1222no—
345C22—1222no—
346C22—4881no—
347C22—4881no—
348C22#1684071011yescharacteristic
349C22C2×#1451984211yescharacteristic
350C22#1684075911yescharacteristic
351C4—22no—
352C4—41no—
353C4—81no—
354C4—81no—
355C4—1221no—
356C4—1221no—
357C4—4881no—
358C4—4884no—
359C4—4881no—
360C4—2441no—
361C4—2441no—
362C2—21no—
363C2—21no—
364C2—42no—
365C2—2442no—
366C2#2146180011yescharacteristic
367C2#2146186011yescharacteristic
368C2#2146186811yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#393C4
socleC22×C61#386
prime radicalC22×C22⋊D61D4
cosocleC22×C4×D61C23
hypercenter#393#6923
hypocenterC61#9183
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #393

Complements

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

NH ≅ G/Ntype
3. #21461219C2semidirect
4. #21460666C2semidirect
16. #16839694C4semidirect
18. #16839687C4semidirect
19. #16839681C4semidirect
24. C61×#393C4semidirect
202. #393#6923semidirect
209. C61#9183semidirect

Maximal subgroups

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

subgroupindex
#214626722
#214615662
#214612192
#214606662
#214517422
#918361