Finite Group Databasebeta

← #30601171

Subgroups — #30601171

Order 15616.

598 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 96 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#21462668C211yesmaximal, characteristic
2#21461722C212yesmaximal, complemented
3#21461681C211yesmaximal, characteristic
4#21460681C212yesmaximal, complemented
5#21450120C211yesmaximal, characteristic
6#16840789—24no—
7#16840779—21no—
8C2×#14519835—21no—
9D61⋊(C22⋊C8)C2212yes—
10C2×C22⋊#92420—21no—
11#16840704C2212yes—
12#16840640—21no—
13#16840635—24no—
14C2×#14519759—21no—
15C2×#14519737C2211yescharacteristic
16C2×C4⋊C4×D61C2211yescharacteristic
17#16839694C411yescharacteristic, complemented
18#16839693C411yescharacteristic, complemented
19#16839687C411yescharacteristic, complemented
20#16839659—21no—
21#16839624—21no—
22#16839621C2211yescharacteristic
23#16839620—21no—
24C61×#397C411yescharacteristic, complemented
25C2×D61⋊C8D414yes—
26C22×#92420—22no—
27C2×C4⋊#6923—22no—
28C2×C22⋊#6923—22no—
29#14519844—22no—
30#14519835—42no—
31#14519834—22no—
32C22⋊#92420—22no—
33C22⋊#92420—22no—
34C4×#92420—22no—
35(C2×#6923)⋊C4—22no—
36C22×C22⋊D61D411yescharacteristic
37C23×Dic61D411yescharacteristic
38C22×C4×D61—21no—
39C22×C4×D61C2311yescharacteristic
40#14519759—42no—
41#14519759—42no—
42#14519737—22no—
43#14519737—22no—
44C2×C4⋊Dic61—21no—
45C2×C4⋊Dic61C2×C411yescharacteristic
46C2×#11758196—21no—
47C2×#11758196—21no—
48C2×#11758196—21no—
49C2×#11758196C2×C411yescharacteristic
50C4⋊C4×D61—22no—
51C4⋊C4×D61—22no—
52C2×C22⋊Dic61—21no—
53C2×#11758198—21no—
54C2×#11758198D411yescharacteristic
55C2×C4×Dic61D411yescharacteristic
56#14519684—21no—
57#14519684—21no—
58#14519684—21no—
59#14519684C2×C411yescharacteristic
60C2×C4⋊C4×C61C2×C411yescharacteristic
61C2×C22⋊C4×C61—21no—
62C2×C22⋊C4×C61C2×C411yescharacteristic
63#95×C61—21no—
64#95×C61C2×C411yescharacteristic
65C24×C61C22⋊C411yescharacteristic
66C23×D61C2×D411yescharacteristic
67C22×#6923—42no—
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⋊D61D812yes—
75C2×C22⋊D61SD1612yes—
76C22×Dic61—21no—
77C22×Dic61—21no—
78C22×Dic61—21no—
79C22×Dic61—21no—
80C22×Dic61—42no—
81C22×Dic61C22⋊C411yescharacteristic
82C22×Dic61C22⋊C411yescharacteristic
83C22×Dic61C2×D411yescharacteristic
84C2×C4×D61—42no—
85C2×C4×D61—42no—
86C2×C4×D61—41no—
87C2×C4×D61—41no—
88C2×C4×D61—42no—
89C2×C4×D61C2×D412yes—
90C2×C4×D61#4112yes—
91C22⋊#6923—42no—
92C22⋊#6923—42no—
93C2×#92420—24no—
94C2×#92420—82no—
95C4⋊#6923—44no—
96D61⋊C8—44no—
97C4⋊C4×C61—22no—
98C22⋊C4×C61—22no—
99C22⋊C4×C61—42no—
100C22⋊Dic61—42no—
101#11758198—22no—
102#11758198—41no—
103#11758198—41no—
104C4⋊Dic61—22no—
105C4⋊Dic61—42no—
106#11758196—22no—
107#11758196—42no—
108#11758196—42no—
109#11758196—41no—
110#11758196—41no—
111C4×Dic61—21no—
112C4×Dic61—21no—
113C23×C61—21no—
114C23×C61—21no—
115C23×C61—41no—
116C23×C61—41no—
117C23×C61#10212yes—
118C23×C61C2×C22⋊C411yescharacteristic
119C22×D61—41no—
120C22×D61—41no—
121C22×D61#12112yes—
122C22×D61C2×D811yescharacteristic
123C22×D61C2×SD1611yescharacteristic
124C22×D61#13512yes—
125C2×#6923—82no—
126C2×#6923—82no—
127C2×#6923—82no—
128C2×C4×C61—21no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61C4×D412yes—
138C22⋊D61—24no—
139C22⋊D61—24no—
140C2×Dic61—21no—
141C2×Dic61—21no—
142C2×Dic61—22no—
143C2×Dic61—21no—
144C2×Dic61—42no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61—41no—
152C2×Dic61—81no—
153C2×Dic61#10212yes—
154C2×Dic61#12011yescharacteristic
155C2×Dic61#12311yescharacteristic
156C4×D61—24no—
157C4×D61—42no—
158C4×D61—82no—
159C4×D61—82no—
160#92420—44no—
161#9470—611nomaximal, Sylow
162C22×C61—21no—
163C22×C61—21no—
164C22×C61—22no—
165C22×C61—42no—
166C22×C61—41no—
167C22×C61—41no—
168C22×C61—81no—
169C22×C61#38611yescharacteristic
170C22×C61C2×#10211yescharacteristic
171C22×C61#41611yescharacteristic
172C2×D61—22no—
173C2×D61—22no—
174C2×D61—41no—
175C2×D61—41no—
176C2×D61—42no—
177C2×D61#46512yes—
178C2×D61#46612yes—
179#6923—162no—
180C4×C61—22no—
181C4×C61—41no—
182C4×C61—81no—
183C4×C61—81no—
184Dic61—21no—
185Dic61—22no—
186Dic61—41no—
187Dic61—81no—
188Dic61—81no—
189Dic61#45812yes—
190#2972—611no—
191#2486—611no—
192#1823—612no—
193#1799—611no—
194#1289—612no—
195C2×C61—21no—
196C2×C61—21no—
197C2×C61—42no—
198C2×C61#189011yescharacteristic
199C2×C61#189311yescharacteristic
200C2×C61#190911yescharacteristic
201D61—22no—
202D61—22no—
203C2×#123—1221no—
204C2×C4×D4—611no—
205C22×C4⋊C4—611no—
206C8×D4—1224no—
207#431—1221no—
208C2×#99—1221no—
209C2×C22⋊C8—612no—
210C2×C22⋊C8—1221no—
211#398—611no—
212#398—611no—
213#397—611no—
214#397#692311yescharacteristic, complemented
215#389—1221no—
216#381—1221no—
217#380—611no—
218#380—612no—
219#380—1221no—
220#342—1221no—
221#340—1224no—
222C61#947011yescharacteristic, complemented, Sylow
223C22×D4—611no—
224C23×C4—611no—
225C23×C4—611no—
226C23×C4—1221no—
227C22×C8—614no—
228C22×C8—1222no—
229#123—2442no—
230#123—2442no—
231C4×D4—1222no—
232C4×D4—1222no—
233C2×C4⋊C4—611no—
234C2×C4⋊C4—611no—
235C2×C4⋊C4—1222no—
236C2×C4⋊C4—1222no—
237C2×C4⋊C4—1221no—
238C2×C4⋊C4—1221no—
239C2×C4⋊C4—1221no—
240C2×C4⋊C4—1221no—
241C2×C4⋊C4—1222no—
242C2×C4⋊C4C2×#692311yescharacteristic
243C2×C22⋊C4—21no—
244C2×C22⋊C4—611no—
245C2×C22⋊C4—1221no—
246C2×C22⋊C4—1221no—
247C2×C22⋊C4—1222no—
248C2×C22⋊C4C2×#692311yescharacteristic
249C2×C42—611no—
250C4⋊C8—1222no—
251#99—2442no—
252C22⋊C8—1222no—
253C22⋊C8—1222no—
254C22⋊C8—1222no—
255C4×C8—1222no—
256#95—21no—
257#95—611no—
258#95—1221no—
259#95—1221no—
260#95—1222no—
261#95—1221no—
262#95C2×#692311yescharacteristic
263C24—611no—
264C24C22⋊#692311yescharacteristic
265C2×D4—612no—
266C2×D4—612no—
267C2×D4—1222no—
268C2×D4—1222no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—611no—
272C22×C4—611no—
273C22×C4—612no—
274C22×C4—612no—
275C22×C4—611no—
276C22×C4—1221no—
277C22×C4—1221no—
278C22×C4—1221no—
279C22×C4—1221no—
280C22×C4—2442no—
281C22×C4—2442no—
282C22×C4—2441no—
283C22×C4—2441no—
284C22×C4—2442no—
285C22×C4—2442no—
286C22×C4—2442no—
287C22×C4C22⋊#692311yescharacteristic
288C22×C4C22×#692311yescharacteristic
289C2×C8—1224no—
290C2×C8—4882no—
291C2×C8—2444no—
292C4⋊C4—22no—
293C4⋊C4—1222no—
294C4⋊C4—1222no—
295C4⋊C4—2441no—
296C4⋊C4—2441no—
297C4⋊C4—2442no—
298C4⋊C4—2442no—
299C4⋊C4—2442no—
300C4⋊C4—2444no—
301C22⋊C4—22no—
302C22⋊C4—42no—
303C22⋊C4—1222no—
304C22⋊C4—2441no—
305C22⋊C4—2441no—
306C22⋊C4—2442no—
307C22⋊C4—2442no—
308C22⋊C4—2442no—
309C42—1221no—
310C42—1221no—
311C23—21no—
312C23—21no—
313C23—41no—
314C23—41no—
315C23—612no—
316C23—611no—
317C23—611no—
318C23—612no—
319C23—2441no—
320C23—2441no—
321C23#1451983912yes—
322C23C2×C22⋊#692311yescharacteristic
323D4—1224no—
324D4—1224no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—611no—
335C2×C4—611no—
336C2×C4—612no—
337C2×C4—1224no—
338C2×C4—1222no—
339C2×C4—1221no—
340C2×C4—1221no—
341C2×C4—1221no—
342C2×C4—4882no—
343C2×C4—4882no—
344C2×C4—4881no—
345C2×C4—4882no—
346C2×C4—4882no—
347C2×C4—4882no—
348C2×C4—2442no—
349C2×C4—2442no—
350C2×C4—2441no—
351C2×C4—2441no—
352C2×C4—2441no—
353C2×C4—2441no—
354C2×C4—2441no—
355C2×C4—2441no—
356C2×C4—2441no—
357C2×C4D4×#692312yes—
358C8—2444no—
359C22—21no—
360C22—21no—
361C22—22no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22—81no—
366C22—612no—
367C22—612no—
368C22—1222no—
369C22—1222no—
370C22—2441no—
371C22—2441no—
372C22—2442no—
373C22#1684071011yescharacteristic
374C22C2×#1451983911yescharacteristic
375C22#1684075211yescharacteristic
376C4—22no—
377C4—41no—
378C4—81no—
379C4—81no—
380C4—612no—
381C4—1221no—
382C4—1222no—
383C4—9762no—
384C4—4881no—
385C4—4881no—
386C4—2441no—
387C2—21no—
388C2—21no—
389C2—42no—
390C2—1222no—
391C2—1222no—
392C2#2146182911yescharacteristic
393C2#2146183311yescharacteristic
394C2#2146186011yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#397C4
socleC22×C61#386
prime radicalC22×C22⋊D61D4
cosocleC22×C4×D61C23
hypercenter#397#6923
hypocenterC61#9470
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #397

Complements

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

NH ≅ G/Ntype
2. #21461722C2semidirect
4. #21460681C2semidirect
17. #16839694C4semidirect
18. #16839693C4semidirect
19. #16839687C4semidirect
24. C61×#397C4semidirect
214. #397#6923semidirect
222. C61#9470semidirect

Maximal subgroups

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

subgroupindex
#214626682
#214617222
#214616812
#214606812
#214501202
#947061