Finite Group Databasebeta

← #30605200

Subgroups — #30605200

Order 15616.

464 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 66 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#21461679C211yesmaximal, characteristic, complemented
2#21461078C211yesmaximal, characteristic, complemented
3#21460816C212yesmaximal, complemented
4#21460797C212yesmaximal, complemented
5#21451704C211yesmaximal, characteristic
6#16840783—21no—
7C2×#14519835—21no—
8#16840704—21no—
9#16840640C2212yes—
10#16840606C2212yes—
11#16840605—21no—
12C2×C4×C22⋊D61C2211yescharacteristic
13#16839700C411yescharacteristic, complemented
14#16839697C2211yescharacteristic
15#16839676—21no—
16#16839676C411yescharacteristic, complemented
17#16839670—21no—
18#16839670C2211yescharacteristic
19#16839650—21no—
20#16839638—21no—
21#16839625C411yescharacteristic, complemented
22C2×#121×C61C411yescharacteristic, complemented
23C2×C4⋊#6923—41no—
24C2×C22⋊#6923—22no—
25C4⋊#92420—41no—
26#14519835—42no—
27C22⋊#92420—22no—
28#14519832—41no—
29(C2×#6923)⋊C4—41no—
30C22×C22⋊D61—21no—
31C22×C22⋊D61D411yescharacteristic
32C22×C4×D61D411yescharacteristic
33C4×C22⋊D61—41no—
34C4×C22⋊D61—41no—
35C2×#11758196—21no—
36C2×#11758196C2×C411yescharacteristic
37C2×C22⋊Dic61—21no—
38C2×C22⋊Dic61—21no—
39C2×C22⋊Dic61—21no—
40C2×C22⋊Dic61—21no—
41C2×C22⋊Dic61D411yescharacteristic
42C2×#11758198—21no—
43C2×#11758198C2311yescharacteristic
44C2×C4×Dic61D411yescharacteristic
45#14519684—21no—
46#14519684—21no—
47#14519684—21no—
48#14519684C2×C411yescharacteristic
49#14519684C2×C411yescharacteristic
50C22×D4×C61—21no—
51C22×D4×C61C2×C411yescharacteristic
52C23×C4×C61C2×C411yescharacteristic
53#121×C61—41no—
54#121×C61—41no—
55C2×C4⋊C4×C61C2×C411yescharacteristic
56C2×C22⋊C4×C61—21no—
57C24×C61—21no—
58C24×C61C22⋊C411yescharacteristic
59C23×D61C2×D411yescharacteristic
60C22×#6923—42no—
61C2×D4×C61—41no—
62C2×D4×C61—42no—
63C2×D4×C61—81no—
64C2×D4×C61—81no—
65C2×D4×C61—81no—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—81no—
70C22×C4×C61C22⋊C411yescharacteristic
71C22×C4×C61C22×C411yescharacteristic
72C2×C22⋊D61—21no—
73C2×C22⋊D61—21no—
74C2×C22⋊D61—41no—
75C2×C22⋊D61—41no—
76C2×C22⋊D61—41no—
77C2×C22⋊D61—41no—
78C2×C22⋊D61—81no—
79C2×C22⋊D61—81no—
80C22×Dic61—21no—
81C22×Dic61—21no—
82C22×Dic61C22⋊C411yescharacteristic
83C22×Dic61C22⋊C411yescharacteristic
84C22×Dic61C2×D411yescharacteristic
85C2×C4×D61—21no—
86C2×C4×D61—21no—
87C2×C4×D61—81no—
88C22⋊#6923—42no—
89C22⋊#6923—42no—
90C2×#92420—42no—
91C4⋊#6923—82no—
92C4⋊C4×C61—41no—
93C22⋊C4×C61—41no—
94C22⋊C4×C61—41no—
95C22⋊Dic61—41no—
96C22⋊Dic61—42no—
97C22⋊Dic61—81no—
98C22⋊Dic61—81no—
99C22⋊Dic61—81no—
100#11758198—41no—
101#11758198—81no—
102#11758196—41no—
103#11758196—81no—
104C4×Dic61—21no—
105C4×Dic61—21no—
106C23×C61—21no—
107C23×C61—21no—
108C23×C61—21no—
109C23×C61—41no—
110C23×C61—41no—
111C23×C61—41no—
112C23×C61—41no—
113C23×C61—41no—
114C23×C61—41no—
115C23×C61—41no—
116C23×C61—41no—
117C23×C61—41no—
118C23×C61C2×C22⋊C411yescharacteristic
119C22×D61—21no—
120C22×D61—21no—
121C22×D61—41no—
122C22×D61—41no—
123C22×D61AGL(1,ℤ/8)11yescharacteristic
124C22×D61#13711yescharacteristic
125C2×#6923—82no—
126C2×#6923—82no—
127C2×#6923—82no—
128D4×C61—82no—
129D4×C61—84no—
130C2×C4×C61—21no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—81no—
137C2×C4×C61—81no—
138C2×C4×C61#9911yescharacteristic
139C2×C4×C61#9911yescharacteristic
140C22⋊D61—42no—
141C22⋊D61—42no—
142C22⋊D61—82no—
143C22⋊D61—82no—
144C2×Dic61—21no—
145C2×Dic61—21no—
146C2×Dic61—21no—
147C2×Dic61—21no—
148C2×Dic61—21no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61—41no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C4×D61—42no—
158C4×D61—81no—
159#92420—82no—
160#11824—611nomaximal, Sylow
161C22×C61—21no—
162C22×C61—21no—
163C22×C61—41no—
164C22×C61—41no—
165C22×C61—41no—
166C22×C61—81no—
167C22×C61—81no—
168C22×C61—81no—
169C22×C61—81no—
170C22×C61—81no—
171C22×C61—81no—
172C22×C61—81no—
173C22×C61—81no—
174C22×C61—81no—
175C22×C61C2×#9911yescharacteristic
176C22×C61#41711yescharacteristic
177C22×C61#42111yescharacteristic
178C2×D61—21no—
179C2×D61—21no—
180C2×D61—41no—
181C2×D61—41no—
182C2×D61—41no—
183C2×D61—41no—
184C2×D61—81no—
185#6923—162no—
186C4×C61—21no—
187C4×C61—21no—
188C4×C61—41no—
189C4×C61—81no—
190Dic61—41no—
191Dic61—41no—
192Dic61—41no—
193Dic61—81no—
194Dic61—81no—
195#2635—611no—
196#1799—611no—
197#1466—611no—
198#1352—612no—
199#1343—612no—
200C2×C61—21no—
201C2×C61—21no—
202C2×C61—81no—
203C2×C61—161no—
204C2×C61#151311yescharacteristic
205C2×C61#212111yescharacteristic
206C2×C61#212811yescharacteristic
207D61—41no—
208D61—41no—
209C2×#121#692311yescharacteristic, complemented
210C2×C4×D4—611no—
211#432—1221no—
212C2×#99—1221no—
213#400—611no—
214#399—611no—
215#394—611no—
216#394—1221no—
217#392—611no—
218#392—1221no—
219#388—1221no—
220#386—1221no—
221#381—611no—
222#380—1221no—
223#342—612no—
224#323—612no—
225#323—1221no—
226C61#1182411yescharacteristic, complemented, Sylow
227C22×D4—21no—
228C22×D4—611no—
229C22×D4—1221no—
230C22×D4C2×#692311yescharacteristic
231C23×C4—611no—
232C23×C4C2×#692311yescharacteristic
233#121—41no—
234#121—41no—
235C4×D4—2441no—
236C4×D4—2441no—
237C2×C4⋊C4—611no—
238C2×C4⋊C4—1221no—
239C2×C4⋊C4—2441no—
240C2×C4⋊C4C2×#692311yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—611no—
243C2×C22⋊C4—611no—
244C2×C22⋊C4—1221no—
245C2×C22⋊C4—1221no—
246C2×C22⋊C4—1221no—
247C2×C22⋊C4—1221no—
248C2×C22⋊C4—1221no—
249C2×C22⋊C4—1222no—
250C2×C42—611no—
251C4⋊C8—2441no—
252#99—2442no—
253C22⋊C8—1222no—
254#97—2441no—
255#95—611no—
256#95—611no—
257#95—1221no—
258#95—1221no—
259#95—1221no—
260#95—2441no—
261C24—21no—
262C24—611no—
263C24C22⋊#692311yescharacteristic
264C2×D4—41no—
265C2×D4—42no—
266C2×D4—81no—
267C2×D4—81no—
268C2×D4—81no—
269C2×D4—1221no—
270C2×D4—1221no—
271C2×D4—4881no—
272C2×D4—4881no—
273C2×D4—2441no—
274C2×D4—2441no—
275C2×D4—2441no—
276C2×D4—2441no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—81no—
281C22×C4—611no—
282C22×C4—611no—
283C22×C4—611no—
284C22×C4—1221no—
285C22×C4—1221no—
286C22×C4—1221no—
287C22×C4—1221no—
288C22×C4—4881no—
289C22×C4—2442no—
290C22×C4C22⋊#692311yescharacteristic
291C22×C4C22×#692311yescharacteristic
292C2×C8—2442no—
293C4⋊C4—41no—
294C4⋊C4—4881no—
295C4⋊C4—4882no—
296C4⋊C4—2441no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—4881no—
300C22⋊C4—4881no—
301C22⋊C4—4881no—
302C22⋊C4—4881no—
303C22⋊C4—2441no—
304C22⋊C4—2441no—
305C22⋊C4—2442no—
306C22⋊C4—2442no—
307C22⋊C4—2442no—
308C42—1221no—
309C42—1221no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—611no—
323C23—611no—
324C23—1221no—
325C23—1221no—
326C23—2441no—
327C23—2441no—
328C23C2×C22⋊#692311yescharacteristic
329D4—82no—
330D4—84no—
331D4—4882no—
332D4—4882no—
333D4—2442no—
334D4—2442no—
335C2×C4—21no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—1221no—
344C2×C4—1221no—
345C2×C4—1221no—
346C2×C4—1221no—
347C2×C4—1221no—
348C2×C4—4881no—
349C2×C4—4882no—
350C2×C4—4882no—
351C2×C4—4882no—
352C2×C4—2442no—
353C2×C4—2441no—
354C2×C4—2441no—
355C2×C4—2441no—
356C2×C4—2441no—
357C2×C4—2441no—
358C2×C4—2441no—
359C2×C4—2441no—
360C2×C4—2441no—
361C2×C4#1451983511yescharacteristic
362C2×C4#1451983511yescharacteristic
363C8—4882no—
364C22—21no—
365C22—21no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—1221no—
379C22—1221no—
380C22—4881no—
381C22—2441no—
382C22—2441no—
383C22—2441no—
384C22—2441no—
385C22C2×#1451983511yescharacteristic
386C22#1684075311yescharacteristic
387C22#1684075911yescharacteristic
388C4—21no—
389C4—21no—
390C4—41no—
391C4—81no—
392C4—9762no—
393C4—4881no—
394C4—4881no—
395C4—2441no—
396C4—2441no—
397C4—2441no—
398C2—21no—
399C2—21no—
400C2—81no—
401C2—161no—
402C2—2441no—
403C2—2441no—
404C2#2146123411yescharacteristic
405C2#2146207411yescharacteristic
406C2#2146209811yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#121×C61C4
socleC22×C61C2×#99
prime radical#21451704C2
cosocleC2×#11758198C23
hypercenterC2×#121#6923
hypocenterC61#11824
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. #21461679C2semidirect
2. #21461078C2semidirect
3. #21460816C2semidirect
4. #21460797C2semidirect
13. #16839700C4semidirect
16. #16839676C4semidirect
21. #16839625C4semidirect
22. C2×#121×C61C4semidirect
209. C2×#121#6923semidirect
226. C61#11824semidirect

Maximal subgroups

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

subgroupindex
#214616792
#214610782
#214608162
#214607972
#214517042
#1182461