Finite Group Databasebeta

← #30600363

Subgroups — #30600363

Order 15616.

600 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C2×#16840784C211yesmaximal, characteristic, complemented
2#21461546C211yesmaximal, characteristic, complemented
3C2×#16840617C212yesmaximal, complemented
4#21460695C212yesmaximal, complemented
5#21454119C211yesmaximal, characteristic
6#16840784—24no—
7C2×C4⋊#92420—21no—
8C2×#14519834—21no—
9C2×#14519834—21no—
10C2×#14519832C2212yes—
11C2×#14519832C2212yes—
12#16840628—21no—
13#16840617—24no—
14C2×#14519764C2211yescharacteristic
15C2×#14519757C411yescharacteristic
16C2×#14519748C411yescharacteristic
17C2×C42×D61C2211yescharacteristic
18#16839668C411yescharacteristic
19#16839655—21no—
20#16839655—21no—
21#16839638—21no—
22#16839638—21no—
23#16839627C2211yescharacteristic
24C61×#390C411yescharacteristic
25C22×#92420—22no—
26C22×#92420—22no—
27C4⋊#92420—24no—
28#14519834—24no—
29#14519834—24no—
30#14519832—22no—
31#14519832—22no—
32#14519832—22no—
33#14519832D414yescomplemented
34C22×Dic122C2×C411yescharacteristic
35C22×C22⋊D61—21no—
36C22×C4×D61C2×C411yescharacteristic
37C22×C4×D61D411yescharacteristic
38C22×C4×D61D411yescharacteristic
39#14519764—22no—
40#14519764—22no—
41#14519757—42no—
42#14519748—42no—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61C2×C411yescharacteristic
45C2×C22⋊Dic61—21no—
46C2×C22⋊Dic61—21no—
47C2×C22⋊Dic61—21no—
48C2×#11758198—21no—
49C2×C4×Dic61D411yescharacteristic
50C2×C4×Dic61D411yescharacteristic
51C2×C4×Dic61C2311yescharacteristic
52C42×D61—22no—
53C42×D61—22no—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57#14519684—21no—
58C22×D4×C61C2×C411yescharacteristic
59C2×C4⋊C4×C61C2×C411yescharacteristic
60C2×C22⋊C4×C61—21no—
61C2×C22⋊C4×C61—21no—
62C2×C42×C61C2×C411yescharacteristic
63C24×C61—21no—
64C23×D61C22⋊C411yescharacteristic
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—42no—
74C2×C22⋊D61—42no—
75C22×Dic61—21no—
76C22×Dic61—21no—
77C22×Dic61C22⋊C411yescharacteristic
78C22×Dic61C2×D411yescharacteristic
79C22×Dic61C2×D411yescharacteristic
80C2×C4×D61—22no—
81C2×C4×D61—22no—
82C2×C4×D61—22no—
83C2×C4×D61—21no—
84C2×C4×D61—22no—
85C2×C4×D61—21no—
86C2×C4×D61—41no—
87C2×C4×D61—41no—
88C2×C4×D61—41no—
89C2×C4×D61—41no—
90C2×Dic122—22no—
91C2×Dic122—41no—
92C2×Dic122—41no—
93C2×#92420—24no—
94C2×#92420—24no—
95C2×#92420—42no—
96C2×#92420—42no—
97C2×#92420—42no—
98C2×#92420—42no—
99C4⋊C4×C61—41no—
100C22⋊C4×C61—42no—
101C22⋊C4×C61—42no—
102C42×C61—22no—
103C22⋊Dic61—42no—
104C22⋊Dic61—41no—
105C22⋊Dic61—42no—
106C22⋊Dic61—41no—
107#11758198—42no—
108C4⋊Dic61—42no—
109C4⋊Dic61—41no—
110C4×Dic61—21no—
111C4×Dic61—22no—
112C4×Dic61—21no—
113C4×Dic61C2×D412yes—
114C4×Dic61#4112yes—
115C23×C61—41no—
116C23×C61—41no—
117C23×C61—41no—
118C23×C61—41no—
119C23×C61—41no—
120C23×C61—41no—
121C23×C61—41no—
122C23×C61C2×C22⋊C411yescharacteristic
123C22×D61—21no—
124C22×D61—21no—
125C22×D61—21no—
126C22×D61—21no—
127C22×D61—21no—
128C22×D61—21no—
129C22×D61#10012yes—
130D4×C61—82no—
131C2×C4×C61—21no—
132C2×C4×C61—21no—
133C2×C4×C61—21no—
134C2×C4×C61—21no—
135C2×C4×C61—21no—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61#10412yes—
142C2×C4×C61C4×D412yes—
143C22⋊D61—84no—
144C2×Dic61—21no—
145C2×Dic61—21no—
146C2×Dic61—21no—
147C2×Dic61—21no—
148C2×Dic61—21no—
149C2×Dic61—21no—
150C2×Dic61—21no—
151C2×Dic61—41no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61#12011yescharacteristic
158C2×Dic61#12112yes—
159C2×Dic61#12311yescharacteristic
160C4×D61—44no—
161C4×D61—42no—
162C4×D61—42no—
163C4×D61—44no—
164Dic122—82no—
165#92420—44no—
166#92420—44no—
167#9124—611nomaximal, Sylow
168C22×C61—81no—
169C22×C61—81no—
170C22×C61—81no—
171C22×C61—81no—
172C22×C61—81no—
173C22×C61—81no—
174C22×C61—81no—
175C22×C61#38611yescharacteristic
176C22×C61C2×#10011yescharacteristic
177C22×C61#41311yescharacteristic
178C22×C61C2×#10411yescharacteristic
179C22×C61#42111yescharacteristic
180C22×C61#42111yescharacteristic
181C22×C61#42111yescharacteristic
182C2×D61—22no—
183C2×D61—22no—
184C2×D61—22no—
185C2×D61—41no—
186C2×D61—41no—
187C2×D61—41no—
188C2×D61—41no—
189C4×C61—22no—
190C4×C61—22no—
191C4×C61—41no—
192C4×C61—81no—
193Dic61—22no—
194Dic61—21no—
195Dic61—21no—
196Dic61—41no—
197Dic61—81no—
198Dic61—81no—
199C2×#432—611no—
200#2850—611no—
201#1749—611no—
202C2×#329—612no—
203#1295—612no—
204C2×C61—161no—
205C2×C61#152911yescharacteristic
206C2×C61#153011yescharacteristic
207C2×C61#153311yescharacteristic
208C2×C61#153511yescharacteristic
209C2×C61#187111yescharacteristic
210C2×C61#189711yescharacteristic
211C2×C61#190011yescharacteristic
212D61—42no—
213C2×#124—611no—
214C2×C22⋊Q8—611no—
215C2×#121—611no—
216C22×C42—611no—
217#432—1224no—
218C2×C4⋊C8—1221no—
219C2×C22⋊C8—1221no—
220C2×C22⋊C8—1221no—
221C2×#97—612no—
222C2×#97—612no—
223#391—611no—
224#390#692311yescharacteristic
225#388—1221no—
226#388—1221no—
227#386—1221no—
228#386—1221no—
229#382—611no—
230#336—1221no—
231#329—1224no—
232C61#912411yescharacteristic, complemented, Sylow
233C22×Q8—611no—
234C22×D4—1221no—
235C22×D4C2×#692311yescharacteristic
236C23×C4—611no—
237C23×C4—611no—
238C23×C4—611no—
239C22×C8—1222no—
240C22×C8—1222no—
241#124—1222no—
242#124—1222no—
243C22⋊Q8—2442no—
244#121—2442no—
245C2×C4⋊C4—611no—
246C2×C4⋊C4—1221no—
247C2×C4⋊C4C2×#692311yescharacteristic
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—1221no—
251C2×C22⋊C4—1221no—
252C2×C22⋊C4—1221no—
253C2×C22⋊C4—1221no—
254C2×C42—611no—
255C2×C42—611no—
256C2×C42—611no—
257C2×C42—1222no—
258C2×C42—1222no—
259C2×C42C2×#692311yescharacteristic
260C4⋊C8—1224no—
261C22⋊C8—1224no—
262C22⋊C8—1224no—
263#97—614no—
264#97—1222no—
265#97—1222no—
266#97—1222no—
267#95—1221no—
268#95—1221no—
269#95—1221no—
270#95—1221no—
271C24—21no—
272C24—611no—
273C2×Q8—1222no—
274C2×Q8—2441no—
275C2×Q8—2441no—
276C2×D4—22no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—2442no—
280C2×D4—2442no—
281C2×D4—2442no—
282C22×C4—21no—
283C22×C4—611no—
284C22×C4—611no—
285C22×C4—611no—
286C22×C4—1221no—
287C22×C4—1222no—
288C22×C4—1221no—
289C22×C4—1222no—
290C22×C4—1222no—
291C22×C4—1222no—
292C22×C4—1221no—
293C22×C4—1221no—
294C22×C4—2441no—
295C22×C4—2441no—
296C22×C4—2441no—
297C22×C4—2441no—
298C22×C4C22⋊#692311yescharacteristic
299C22×C4C22⋊#692311yescharacteristic
300C22×C4C22×#692311yescharacteristic
301C2×C8—1224no—
302C2×C8—1224no—
303C2×C8—2442no—
304C2×C8—2442no—
305C2×C8—2442no—
306C2×C8—2442no—
307C4⋊C4—41no—
308C4⋊C4—2441no—
309C4⋊C4—2442no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—2442no—
313C22⋊C4—2441no—
314C22⋊C4—2442no—
315C22⋊C4—2441no—
316C22⋊C4—2442no—
317C42—22no—
318C42—612no—
319C42—612no—
320C42—1221no—
321C42—1221no—
322C42—1222no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—612no—
331C23—1221no—
332C23—1221no—
333C23—1221no—
334C23—1221no—
335C23—1221no—
336C23—1221no—
337C23C2×C22⋊#692311yescharacteristic
338Q8—4882no—
339D4—82no—
340D4—4884no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—611no—
352C2×C4—612no—
353C2×C4—611no—
354C2×C4—1221no—
355C2×C4—1221no—
356C2×C4—1221no—
357C2×C4—1221no—
358C2×C4—1221no—
359C2×C4—1221no—
360C2×C4—1221no—
361C2×C4—2444no—
362C2×C4—2444no—
363C2×C4—2442no—
364C2×C4—2442no—
365C2×C4—2441no—
366C2×C4—2441no—
367C2×C4—2441no—
368C2×C4—2441no—
369C2×C4—2441no—
370C2×C4—2441no—
371C2×C4#1451984112yes—
372C2×C4D4×#692312yes—
373C8—2444no—
374C8—2444no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—1222no—
383C22—1222no—
384C22—1222no—
385C22—2441no—
386C22—2441no—
387C22—2441no—
388C22—2441no—
389C22#1684071011yescharacteristic
390C22D61⋊#10011yescharacteristic
391C22#1684074811yescharacteristic
392C22C2×#1451984111yescharacteristic
393C22#1684075911yescharacteristic
394C22#1684075911yescharacteristic
395C22#1684076011yescharacteristic
396C4—22no—
397C4—22no—
398C4—41no—
399C4—81no—
400C4—1222no—
401C4—1221no—
402C4—1221no—
403C4—4881no—
404C4—4881no—
405C4—2441no—
406C2—161no—
407C2—2442no—
408C2#2146128411yescharacteristic
409C2#2146128911yescharacteristic
410C2#2146129911yescharacteristic
411C2#2146130311yescharacteristic
412C2#2146180211yescharacteristic
413C2#2146183911yescharacteristic
414C2#2146184511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519748C4
cosocleC2×C4×Dic61C23
hypercenter#390#6923
hypocenterC61#9124
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#16840784C2semidirect
2. #21461546C2semidirect
3. C2×#16840617C2semidirect
4. #21460695C2semidirect
33. #14519832D4semidirect
232. C61#9124semidirect

Maximal subgroups

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

subgroupindex
C2×#168407842
#214615462
C2×#168406172
#214606952
#214541192
#912461