Finite Group Databasebeta

← #30605465

Subgroups — #30605465

Order 15616.

452 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 76 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#21461680C211yesmaximal, characteristic
2#21461056C211yesmaximal, characteristic
3#21460816C212yesmaximal, complemented
4#21460814C212yesmaximal, complemented
5#21451177C211yesmaximal, characteristic
6C2×#14519836—21no—
7D61⋊(C22⋊C8)—21no—
8#16840704—21no—
9#16840640C2212yes—
10#16840607—21no—
11#16840606C2212yes—
12C2×C4×D244C2211yescharacteristic
13#16839697C411yescharacteristic, complemented
14#16839697C2211yescharacteristic
15#16839673C411yescharacteristic, complemented
16#16839673C2211yescharacteristic
17#16839664—21no—
18#16839660—21no—
19#16839633—21no—
20#16839633—21no—
21#16839628C411yescharacteristic, complemented
22C2×C4×Q8×C61C411yescharacteristic, complemented
23C2×D61⋊C8—41no—
24C2×C4⋊#6923—41no—
25C2×C22⋊#6923—22no—
26#14519836—42no—
27#14519834—41no—
28C22⋊#92420—22no—
29(C2×#6923)⋊C4—41no—
30C22×D244D411yescharacteristic
31C22×C4×D61D411yescharacteristic
32C22×C4×D61D411yescharacteristic
33C4×D244—41no—
34C4×D244—41no—
35C2×C4⋊Dic61—21no—
36C2×C4⋊Dic61D411yescharacteristic
37C2×#11758196—21no—
38C2×#11758196—21no—
39C2×#11758198—21no—
40C2×#11758198—21no—
41C2×#11758198—21no—
42C2×#11758198—21no—
43C2×#11758198C2×C411yescharacteristic
44C2×#11758198C2311yescharacteristic
45#14519684—21no—
46#14519684—21no—
47#14519684—21no—
48#14519684C2×C411yescharacteristic
49#14519684C2×C411yescharacteristic
50C22×Q8×C61C2×C411yescharacteristic
51C4×Q8×C61—41no—
52C4×Q8×C61—41no—
53C2×C4⋊C4×C61—21no—
54C2×C4⋊C4×C61C2×C411yescharacteristic
55C2×C42×C61—21no—
56C2×C42×C61C2×C411yescharacteristic
57C23×D61C22⋊C411yescharacteristic
58C23×D61C2×D411yescharacteristic
59C22×#6923—42no—
60C2×Q8×C61—42no—
61C2×Q8×C61—41no—
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22⋊C411yescharacteristic
66C22×C4×C61C22×C411yescharacteristic
67C22×Dic61—21no—
68C22×Dic61—21no—
69C22×Dic61C22⋊C411yescharacteristic
70C22×Dic61C2×D411yescharacteristic
71C2×D244—41no—
72C2×D244—41no—
73C2×D244SD1612yes—
74C2×D244Q1612yes—
75C2×C4×D61—21no—
76C2×C4×D61—21no—
77C2×C4×D61—21no—
78C2×C4×D61—21no—
79C2×C4×D61—81no—
80C2×C4×D61—81no—
81C22⋊#6923—42no—
82C22⋊#6923—42no—
83C2×#92420—42no—
84C4⋊#6923—82no—
85D61⋊C8—82no—
86C4⋊C4×C61—41no—
87C4⋊C4×C61—41no—
88C4⋊C4×C61—41no—
89C42×C61—21no—
90C42×C61—21no—
91C42×C61—42no—
92#11758198—41no—
93#11758198—41no—
94#11758198—42no—
95#11758198—42no—
96#11758198—81no—
97#11758198—81no—
98C4⋊Dic61—41no—
99C4⋊Dic61—42no—
100#11758196—81no—
101#11758196—81no—
102C23×C61C2×C22⋊C411yescharacteristic
103C22×D61—21no—
104C22×D61—21no—
105C22×D61—21no—
106C22×D61—21no—
107C22×D61—41no—
108C22×D61—41no—
109C22×D61—41no—
110C22×D61—41no—
111C22×D61#10312yes—
112C22×D61C2×SD1611yescharacteristic
113C22×D61C2×Q1611yescharacteristic
114C2×#6923—82no—
115C2×#6923—82no—
116C2×#6923—82no—
117Q8×C61—42no—
118Q8×C61—42no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—22no—
122C2×C4×C61—21no—
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×C61#9911yescharacteristic
131C2×C4×C61#9911yescharacteristic
132C2×C4×C61#10312yes—
133C2×Dic61—21no—
134C2×Dic61—21no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61—41no—
139C2×Dic61—41no—
140C2×Dic61—41no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143D244—42no—
144D244—42no—
145C4×D61—42no—
146C4×D61—42no—
147C4×D61—81no—
148C4×D61—81no—
149#92420—82no—
150#11897—611nomaximal, Sylow
151C22×C61—21no—
152C22×C61—21no—
153C22×C61C2×#9911yescharacteristic
154C22×C61C2×#10311yescharacteristic
155C22×C61#42111yescharacteristic
156C2×D61—21no—
157C2×D61—22no—
158C2×D61—21no—
159C2×D61—41no—
160C2×D61—41no—
161C2×D61—41no—
162C2×D61—41no—
163C2×D61—42no—
164C2×D61—41no—
165C2×D61—41no—
166C2×D61—81no—
167C2×D61—81no—
168#6923—162no—
169C4×C61—21no—
170C4×C61—21no—
171C4×C61—22no—
172C4×C61—42no—
173C4×C61—41no—
174C4×C61—81no—
175Dic61—41no—
176Dic61—41no—
177Dic61—81no—
178Dic61—81no—
179#2580—611no—
180#1799—611no—
181#1459—611no—
182#1352—612no—
183#1351—612no—
184C2×C61—21no—
185C2×C61—21no—
186C2×C61#152011yescharacteristic
187C2×C61#212011yescharacteristic
188C2×C61#212711yescharacteristic
189D61—41no—
190D61—42no—
191D61—41no—
192C2×C4×Q8#692311yescharacteristic, complemented
193C2×C4×D4—611no—
194C2×#99—1221no—
195C2×C22⋊C8—1221no—
196#399—611no—
197#399—611no—
198#393—611no—
199#393—611no—
200#390—1221no—
201#389—1221no—
202#385—1221no—
203#385—1221no—
204#382—611no—
205#380—1221no—
206#342—612no—
207#324—1221no—
208#323—612no—
209C61#1189711yescharacteristic, complemented, Sylow
210C22×Q8C2×#692311yescharacteristic
211C22×D4—611no—
212C23×C4—611no—
213C23×C4—611no—
214C22×C8—2441no—
215C4×Q8—41no—
216C4×Q8—41no—
217C4×D4—2441no—
218C4×D4—2441no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—611no—
221C2×C4⋊C4—1221no—
222C2×C4⋊C4—1221no—
223C2×C4⋊C4—1221no—
224C2×C4⋊C4—2441no—
225C2×C4⋊C4C2×#692311yescharacteristic
226C2×C22⋊C4—611no—
227C2×C22⋊C4—611no—
228C2×C22⋊C4—1221no—
229C2×C22⋊C4—1221no—
230C2×C22⋊C4—1221no—
231C2×C22⋊C4—1221no—
232C2×C22⋊C4—1222no—
233C2×C42—21no—
234C2×C42C2×#692311yescharacteristic
235#99—2442no—
236C22⋊C8—1222no—
237C22⋊C8—2441no—
238#95—611no—
239#95—611no—
240#95—1221no—
241#95—1221no—
242#95—1221no—
243#95—2441no—
244C24—611no—
245C24—611no—
246C2×Q8—42no—
247C2×Q8—41no—
248C2×D4—612no—
249C2×D4—612no—
250C2×D4—2441no—
251C2×D4—2441no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—611no—
255C22×C4—611no—
256C22×C4—1221no—
257C22×C4—1221no—
258C22×C4—1221no—
259C22×C4—1221no—
260C22×C4—1221no—
261C22×C4—1221no—
262C22×C4—4881no—
263C22×C4—4881no—
264C22×C4—2442no—
265C22×C4C22⋊#692311yescharacteristic
266C22×C4C22⋊#692311yescharacteristic
267C22×C4C22×#692311yescharacteristic
268C2×C8—4882no—
269C2×C8—2442no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—4881no—
274C4⋊C4—4881no—
275C4⋊C4—4882no—
276C4⋊C4—2441no—
277C4⋊C4—2442no—
278C22⋊C4—4881no—
279C22⋊C4—4881no—
280C22⋊C4—2441no—
281C22⋊C4—2441no—
282C22⋊C4—2442no—
283C22⋊C4—2442no—
284C22⋊C4—2442no—
285C22⋊C4—2442no—
286C42—21no—
287C42—21no—
288C42—42no—
289C23—612no—
290C23—611no—
291C23—611no—
292C23—1221no—
293C23—1221no—
294C23—1221no—
295C23—1221no—
296C23—2441no—
297C23—2441no—
298C23—2441no—
299C23—2441no—
300C23C2×C22⋊#692311yescharacteristic
301Q8—42no—
302Q8—42no—
303D4—2442no—
304D4—2442no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—1221no—
317C2×C4—1221no—
318C2×C4—4881no—
319C2×C4—4881no—
320C2×C4—4882no—
321C2×C4—4882no—
322C2×C4—4882no—
323C2×C4—2442no—
324C2×C4—2442no—
325C2×C4—2441no—
326C2×C4—2441no—
327C2×C4—2441no—
328C2×C4—2441no—
329C2×C4—2441no—
330C2×C4—2441no—
331C2×C4—2441no—
332C2×C4—2441no—
333C2×C4#1451983511yescharacteristic
334C2×C4#1451983511yescharacteristic
335C2×C4#1451984012yes—
336C8—4882no—
337C22—21no—
338C22—21no—
339C22—1221no—
340C22—1222no—
341C22—1221no—
342C22—4881no—
343C22—4881no—
344C22—2441no—
345C22—2442no—
346C22—2441no—
347C22—2441no—
348C22—2441no—
349C22—2441no—
350C22—2441no—
351C22C2×#1451983511yescharacteristic
352C22C2×#1451984011yescharacteristic
353C22#1684076111yescharacteristic
354C4—22no—
355C4—21no—
356C4—21no—
357C4—42no—
358C4—41no—
359C4—81no—
360C4—9762no—
361C4—4881no—
362C4—4881no—
363C4—2441no—
364C4—2441no—
365C2—21no—
366C2—21no—
367C2—2441no—
368C2—2441no—
369C2—2442no—
370C2#2146125911yescharacteristic
371C2#2146207011yescharacteristic
372C2#2146209411yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4×Q8×C61C4
socleC22×C61C2×#99
prime radicalC22×D244D4
cosocleC2×#11758198C23
hypercenterC2×C4×Q8#6923
hypocenterC61#11897
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
3. #21460816C2semidirect
4. #21460814C2semidirect
13. #16839697C4semidirect
15. #16839673C4semidirect
21. #16839628C4semidirect
22. C2×C4×Q8×C61C4semidirect
192. C2×C4×Q8#6923semidirect
209. C61#11897semidirect

Maximal subgroups

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

subgroupindex
#214616802
#214610562
#214608162
#214608142
#214511772
#1189761