Finite Group Databasebeta

← #30600983

Subgroups — #30600983

Order 15616.

570 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 94 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#21461681C211yesmaximal, characteristic
3#21461258C212yesmaximal, complemented
4#21460693C212yesmaximal, complemented
5#21450994C211yesmaximal, characteristic
6#16840789—24no—
7#16840779—21no—
8C2×#14519835—21no—
9D61⋊(C22⋊C8)C2212yes—
10C2×C22⋊#92420—21no—
11#16840704C2212yes—
12#16840640—21no—
13#16840615—24no—
14C2×#14519757C2211yescharacteristic
15C2×#14519737C2211yescharacteristic
16#16839700C411yescharacteristic, complemented
17#16839683C411yescharacteristic, complemented
18#16839674C411yescharacteristic, complemented
19#16839648—21no—
20#16839638—21no—
21#16839637—21no—
22#16839619—21no—
23#95×D61C2211yescharacteristic
24C61×#399C411yescharacteristic, complemented
25C2×D61⋊C8—22no—
26C22×#92420—22no—
27C2×C4⋊#6923—22no—
28C2×C22⋊#6923—22no—
29#14519844—22no—
30#14519835—42no—
31#14519834D414yes—
32C22⋊#92420—22no—
33C22⋊#92420—22no—
34C4×#92420—22no—
35(C2×#6923)⋊C4—22no—
36C22×Dic122C2×C411yescharacteristic
37C22×C22⋊D61D411yescharacteristic
38C23×Dic61D411yescharacteristic
39C22×C4×D61—21no—
40C22×C4×D61C2311yescharacteristic
41#14519757—22no—
42#14519757—22no—
43#14519737—22no—
44#14519737—22no—
45C2×C4⋊Dic61—21no—
46C2×C4⋊Dic61—21no—
47C2×#11758196C2×C411yescharacteristic
48C2×C22⋊Dic61—21no—
49C2×#11758198—21no—
50C2×#11758198D411yescharacteristic
51C2×C4×Dic61D411yescharacteristic
52#14519684—21no—
53#14519684—21no—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57#14519684C2×C411yescharacteristic
58C2×C4⋊C4×C61—21no—
59C2×C4⋊C4×C61C2×C411yescharacteristic
60C2×C22⋊C4×C61—21no—
61C2×C22⋊C4×C61C2×C411yescharacteristic
62#95×C61C2×C411yescharacteristic
63C24×C61C22⋊C411yescharacteristic
64C23×D61C2×D411yescharacteristic
65C22×#6923—42no—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61C22⋊C411yescharacteristic
69C22×C4×C61C22×C411yescharacteristic
70C2×C22⋊D61—22no—
71C2×C22⋊D61—22no—
72C2×C22⋊D61SD1612yes—
73C2×C22⋊D61Q1612yes—
74C22×Dic61—21no—
75C22×Dic61—21no—
76C22×Dic61—21no—
77C22×Dic61—21no—
78C22×Dic61—81no—
79C22×Dic61C22⋊C411yescharacteristic
80C22×Dic61C22⋊C411yescharacteristic
81C22×Dic61C2×D411yescharacteristic
82C2×C4×D61—42no—
83C2×C4×D61—42no—
84C2×C4×D61—81no—
85C2×C4×D61—81no—
86C2×C4×D61C2×D412yes—
87C2×C4×D61#4112yes—
88C2×Dic122—22no—
89C2×Dic122—81no—
90C22⋊#6923—42no—
91C22⋊#6923—42no—
92C2×#92420—24no—
93C2×#92420—82no—
94C4⋊#6923—82no—
95D61⋊C8—82no—
96C4⋊C4×C61—22no—
97C4⋊C4×C61—42no—
98C22⋊C4×C61—22no—
99C22⋊C4×C61—42no—
100C22⋊Dic61—81no—
101#11758198—22no—
102#11758198—42no—
103C4⋊Dic61—41no—
104C4⋊Dic61—41no—
105C4⋊Dic61—81no—
106#11758196—22no—
107C4×Dic61—21no—
108C4×Dic61—21no—
109C23×C61—21no—
110C23×C61—21no—
111C23×C61—41no—
112C23×C61—41no—
113C23×C61#10312yes—
114C23×C61C2×C22⋊C411yescharacteristic
115C22×D61—41no—
116C22×D61—41no—
117C22×D61#12112yes—
118C22×D61C2×SD1611yescharacteristic
119C22×D61C2×Q1611yescharacteristic
120C22×D61#13512yes—
121C2×#6923—82no—
122C2×#6923—82no—
123C2×#6923—82no—
124C2×C4×C61—22no—
125C2×C4×C61—21no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61C4×D412yes—
134C22⋊D61—24no—
135C22⋊D61—24no—
136C2×Dic61—21no—
137C2×Dic61—21no—
138C2×Dic61—21no—
139C2×Dic61—41no—
140C2×Dic61—42no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143C2×Dic61—41no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—41no—
148C2×Dic61—81no—
149C2×Dic61#10312yes—
150C2×Dic61#12011yescharacteristic
151C2×Dic61#12311yescharacteristic
152C4×D61—42no—
153C4×D61—42no—
154C4×D61—84no—
155Dic122—82no—
156#92420—44no—
157#9383—611nomaximal, Sylow
158C22×C61—21no—
159C22×C61—21no—
160C22×C61—22no—
161C22×C61—42no—
162C22×C61—41no—
163C22×C61—41no—
164C22×C61—81no—
165C22×C61#38611yescharacteristic
166C22×C61C2×#10311yescharacteristic
167C22×C61#41611yescharacteristic
168C2×D61—22no—
169C2×D61—22no—
170C2×D61—42no—
171C2×D61—41no—
172C2×D61—41no—
173C2×D61#46512yes—
174C2×D61#46712yes—
175#6923—162no—
176C4×C61—22no—
177C4×C61—42no—
178C4×C61—41no—
179C4×C61—81no—
180Dic61—22no—
181Dic61—21no—
182Dic61—21no—
183Dic61—41no—
184Dic61—81no—
185Dic61—81no—
186#2972—611no—
187#2564—611no—
188#1799—611no—
189#1520—612no—
190#1294—612no—
191C2×C61—21no—
192C2×C61—21no—
193C2×C61—42no—
194C2×C61#188411yescharacteristic
195C2×C61#189211yescharacteristic
196C2×C61#191311yescharacteristic
197D61—22no—
198D61—22no—
199C2×C22⋊Q8—611no—
200C2×C4×D4—611no—
201C8×D4—1224no—
202#431—1221no—
203C2×#99—1221no—
204C2×C22⋊C8—612no—
205C2×C22⋊C8—1221no—
206#400—611no—
207#399#692311yescharacteristic, complemented
208#395—611no—
209#393—611no—
210#387—1221no—
211#386—1221no—
212#385—1221no—
213#380—612no—
214#380—1221no—
215C2×#95—611no—
216#342—1221no—
217#328—1224no—
218C61#938311yescharacteristic, complemented, Sylow
219C22×Q8—611no—
220C22×D4—611no—
221C23×C4—611no—
222C23×C4—611no—
223C23×C4—1221no—
224C22×C8—1222no—
225C22×C8—1222no—
226C22⋊Q8—1222no—
227C22⋊Q8—1222no—
228C4×D4—1222no—
229C4×D4—1222no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—611no—
232C2×C4⋊C4—1221no—
233C2×C4⋊C4—1222no—
234C2×C4⋊C4—1221no—
235C2×C4⋊C4C2×#692311yescharacteristic
236C2×C22⋊C4—21no—
237C2×C22⋊C4—611no—
238C2×C22⋊C4—1221no—
239C2×C22⋊C4—1221no—
240C2×C22⋊C4—1222no—
241C2×C22⋊C4C2×#692311yescharacteristic
242C2×C42—611no—
243C4⋊C8—1222no—
244#99—2442no—
245C22⋊C8—614no—
246C22⋊C8—1222no—
247C22⋊C8—1222no—
248C4×C8—1222no—
249#95—611no—
250#95—1221no—
251#95—1221no—
252#95—1221no—
253#95—1221no—
254#95—1222no—
255#95—1221no—
256#95C2×#692311yescharacteristic
257C24—611no—
258C24C22⋊#692311yescharacteristic
259C2×Q8—1222no—
260C2×Q8—4881no—
261C2×D4—612no—
262C2×D4—612no—
263C2×D4—1222no—
264C2×D4—1222no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—611no—
268C22×C4—611no—
269C22×C4—612no—
270C22×C4—612no—
271C22×C4—611no—
272C22×C4—1221no—
273C22×C4—1221no—
274C22×C4—1221no—
275C22×C4—1221no—
276C22×C4—4881no—
277C22×C4—4881no—
278C22×C4—4881no—
279C22×C4—2442no—
280C22×C4—2442no—
281C22×C4—2442no—
282C22×C4C22⋊#692311yescharacteristic
283C22×C4C22×#692311yescharacteristic
284C2×C8—1224no—
285C2×C8—4882no—
286C2×C8—4882no—
287C4⋊C4—22no—
288C4⋊C4—42no—
289C4⋊C4—1222no—
290C4⋊C4—4881no—
291C4⋊C4—4882no—
292C4⋊C4—2441no—
293C4⋊C4—2441no—
294C22⋊C4—22no—
295C22⋊C4—42no—
296C22⋊C4—1222no—
297C22⋊C4—4881no—
298C22⋊C4—2442no—
299C22⋊C4—2442no—
300C22⋊C4—2442no—
301C42—1221no—
302C42—1221no—
303C23—21no—
304C23—21no—
305C23—41no—
306C23—41no—
307C23—612no—
308C23—611no—
309C23—611no—
310C23—612no—
311C23—2441no—
312C23—2441no—
313C23#1451984012yes—
314C23C2×C22⋊#692311yescharacteristic
315Q8—4882no—
316D4—1224no—
317D4—1224no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—611no—
328C2×C4—611no—
329C2×C4—612no—
330C2×C4—1221no—
331C2×C4—1221no—
332C2×C4—1221no—
333C2×C4—4884no—
334C2×C4—4881no—
335C2×C4—4882no—
336C2×C4—4882no—
337C2×C4—4882no—
338C2×C4—2442no—
339C2×C4—2442no—
340C2×C4—2441no—
341C2×C4—2442no—
342C2×C4—2441no—
343C2×C4—2441no—
344C2×C4—2441no—
345C2×C4—2441no—
346C2×C4—2441no—
347C2×C4—2441no—
348C2×C4—2441no—
349C2×C4D4×#692312yes—
350C8—2444no—
351C22—21no—
352C22—21no—
353C22—22no—
354C22—42no—
355C22—41no—
356C22—41no—
357C22—81no—
358C22—612no—
359C22—612no—
360C22—1222no—
361C22—1222no—
362C22—2442no—
363C22—2441no—
364C22—2441no—
365C22#1684071011yescharacteristic
366C22C2×#1451984011yescharacteristic
367C22#1684075211yescharacteristic
368C4—22no—
369C4—42no—
370C4—41no—
371C4—81no—
372C4—1222no—
373C4—1221no—
374C4—1221no—
375C4—9762no—
376C4—4881no—
377C4—4881no—
378C4—2441no—
379C2—21no—
380C2—21no—
381C2—42no—
382C2—1222no—
383C2—1222no—
384C2#2146182211yescharacteristic
385C2#2146183211yescharacteristic
386C2#2146186911yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#399C4
socleC22×C61#386
prime radicalC22×C22⋊D61D4
cosocleC22×C4×D61C23
hypercenter#399#6923
hypocenterC61#9383
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #399

Complements

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

NH ≅ G/Ntype
3. #21461258C2semidirect
4. #21460693C2semidirect
16. #16839700C4semidirect
17. #16839683C4semidirect
18. #16839674C4semidirect
24. C61×#399C4semidirect
207. #399#6923semidirect
218. C61#9383semidirect

Maximal subgroups

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

subgroupindex
#214626682
#214616812
#214612582
#214606932
#214509942
#938361