Finite Group Databasebeta

← #30601141

Subgroups — #30601141

Order 15616.

554 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 82 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#21462670C211yesmaximal, characteristic
2#21461348C212yesmaximal, complemented
3#21461079C211yesmaximal, characteristic
4#21460670C212yesmaximal, complemented
5#21450770C211yesmaximal, characteristic
6#16840793—24no—
7#16840782C2212yes—
8#16840782C2212yes—
9#16840732—21no—
10C2×C22⋊#92420—21no—
11#16840621—24no—
12#16840605—21no—
13#16840605—21no—
14C2×#14519776C2211yescharacteristic
15C2×#14519754—21no—
16C2×#14519754—21no—
17C2×D4×Dic61C2211yescharacteristic
18#16839698—21no—
19#16839698—21no—
20#16839665C411yescharacteristic
21#16839632C411yescharacteristic
22#16839631C411yescharacteristic
23C4×#11758196C2211yescharacteristic
24C61×#390C411yescharacteristic
25C2×#11758217—22no—
26C22×#92420—22no—
27#14519844—22no—
28#14519844D414yes—
29C22⋊#92420—22no—
30C22⋊#92420—22no—
31C22⋊#92420—22no—
32C22⋊#92420—22no—
33#14519832—22no—
34#14519832—22no—
35C22×Dic122—21no—
36C23×Dic61D411yescharacteristic
37C23×Dic61D411yescharacteristic
38#14519776—22no—
39#14519776—22no—
40#14519754—44no—
41#14519754—44no—
42D4×Dic61—22no—
43D4×Dic61—22no—
44C2×C4⋊Dic61—21no—
45C2×C4⋊Dic61—21no—
46C2×C4⋊Dic61C2×C411yescharacteristic
47C2×C4⋊Dic61C2×C411yescharacteristic
48C2×#11758196—21no—
49C2×#11758196—21no—
50C2×#11758196—21no—
51C2×#11758196—21no—
52C2×C22⋊Dic61—21no—
53C2×C22⋊Dic61—21no—
54C2×C22⋊Dic61D411yescharacteristic
55C2×C22⋊Dic61D411yescharacteristic
56C2×C4×Dic61C2×C411yescharacteristic
57C2×C4×Dic61C2311yescharacteristic
58#14519684—21no—
59C22×D4×C61C2×C411yescharacteristic
60C2×C4⋊C4×C61C2×C411yescharacteristic
61C2×C22⋊C4×C61—21no—
62C2×C22⋊C4×C61—21no—
63C2×C42×C61C2×C411yescharacteristic
64C24×C61C22⋊C411yescharacteristic
65C24×C61C22⋊C411yescharacteristic
66C2×D4×C61—22no—
67C2×D4×C61—22no—
68C2×D4×C61—41no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61C22×C411yescharacteristic
72C22×Dic61—21no—
73C22×Dic61—21no—
74C22×Dic61—21no—
75C22×Dic61—21no—
76C22×Dic61—21no—
77C22×Dic61—21no—
78C22×Dic61—42no—
79C22×Dic61—42no—
80C22×Dic61C22⋊C411yescharacteristic
81C22×Dic61C22⋊C411yescharacteristic
82C22×Dic61C2×D411yescharacteristic
83C22×Dic61C2×D411yescharacteristic
84C2×Dic122—42no—
85C2×Dic122—42no—
86C2×Dic122—41no—
87C2×Dic122—41no—
88#11758217—82no—
89C2×#92420—24no—
90C2×#92420—42no—
91C2×#92420—82no—
92C4⋊C4×C61—22no—
93C22⋊C4×C61—42no—
94C22⋊C4×C61—42no—
95C42×C61—41no—
96C22⋊Dic61—22no—
97C22⋊Dic61—22no—
98C22⋊Dic61—42no—
99C22⋊Dic61—42no—
100C4⋊Dic61—22no—
101C4⋊Dic61—22no—
102C4⋊Dic61—42no—
103C4⋊Dic61—42no—
104#11758196—24no—
105#11758196—42no—
106#11758196—42no—
107#11758196—42no—
108C4×Dic61—41no—
109C4×Dic61C2×D412yes—
110C4×Dic61#4112yes—
111C23×C61—21no—
112C23×C61—21no—
113C23×C61—21no—
114C23×C61—21no—
115C23×C61—21no—
116C23×C61—41no—
117C23×C61—41no—
118C23×C61—41no—
119C23×C61—41no—
120C23×C61#10112yes—
121C23×C61C2×C22⋊C411yescharacteristic
122D4×C61—44no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—22no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61C4×D412yes—
132C2×Dic61—22no—
133C2×Dic61—22no—
134C2×Dic61—21no—
135C2×Dic61—21no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61—41no—
139C2×Dic61—42no—
140C2×Dic61—42no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143C2×Dic61—41no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146C2×Dic61—81no—
147C2×Dic61—81no—
148C2×Dic61#12011yescharacteristic
149C2×Dic61#12112yes—
150C2×Dic61#12311yescharacteristic
151Dic122—82no—
152Dic122—82no—
153#92420—44no—
154#92420—82no—
155#9455—611nomaximal, Sylow
156C22×C61—21no—
157C22×C61—21no—
158C22×C61—22no—
159C22×C61—42no—
160C22×C61—42no—
161C22×C61—41no—
162C22×C61—41no—
163C22×C61—41no—
164C22×C61—41no—
165C22×C61—41no—
166C22×C61—81no—
167C22×C61—81no—
168C22×C61#38611yescharacteristic
169C22×C61C2×#10111yescharacteristic
170C22×C61#41311yescharacteristic
171C4×C61—22no—
172C4×C61—42no—
173C4×C61—81no—
174Dic61—21no—
175Dic61—22no—
176Dic61—42no—
177Dic61—41no—
178Dic61—41no—
179Dic61—81no—
180Dic61#45312yes—
181#2973—611no—
182#2542—611no—
183#1556—612no—
184#1466—611no—
185#1285—612no—
186C2×C61—21no—
187C2×C61—21no—
188C2×C61—42no—
189C2×C61—81no—
190C2×C61#188711yescharacteristic
191C2×C61#190111yescharacteristic
192C2×C61#190911yescharacteristic
193C2×C4⋊Q8—611no—
194C2×C22⋊Q8—1221no—
195C2×C22⋊Q8—1221no—
196C2×C4×D4—611no—
197#435—1224no—
198#432—612no—
199#432—612no—
200#407—1221no—
201C2×C22⋊C8—1221no—
202#399—1221no—
203#399—1221no—
204#390—611no—
205#390#692311yescharacteristic
206#384—611no—
207#384—611no—
208C4×C4⋊C4—611no—
209#332—1224no—
210#323—1221no—
211#323—1221no—
212C61#945511yescharacteristic, complemented, Sylow
213C22×Q8—1221no—
214C22×D4C2×#692311yescharacteristic
215C23×C4—611no—
216C23×C4—611no—
217C2×M16—1222no—
218C22×C8—1222no—
219C4⋊Q8—1222no—
220C4⋊Q8—1222no—
221C22⋊Q8—2444no—
222C22⋊Q8—2444no—
223C4×D4—1222no—
224C4×D4—1222no—
225C2×C4⋊C4—611no—
226C2×C4⋊C4—611no—
227C2×C4⋊C4—1221no—
228C2×C4⋊C4—1221no—
229C2×C4⋊C4—1221no—
230C2×C4⋊C4—1221no—
231C2×C4⋊C4—1221no—
232C2×C4⋊C4—1221no—
233C2×C4⋊C4C2×#692311yescharacteristic
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—611no—
237C2×C22⋊C4—611no—
238C2×C22⋊C4—1221no—
239C2×C22⋊C4—1221no—
240C2×C42—611no—
241C2×C42—611no—
242C2×C42C2×#692311yescharacteristic
243C4⋊C8—614no—
244C4⋊C8—1222no—
245C22⋊C8—1222no—
246C22⋊C8—1222no—
247C22⋊C8—1222no—
248C22⋊C8—1222no—
249#97—1222no—
250#97—1222no—
251#95—1221no—
252C24C22⋊#692311yescharacteristic
253C24C22⋊#692311yescharacteristic
254C2×Q8—2441no—
255C2×Q8—2441no—
256C2×Q8—2442no—
257C2×Q8—2442no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—41no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—611no—
264C22×C4—611no—
265C22×C4—611no—
266C22×C4—611no—
267C22×C4—1221no—
268C22×C4—1221no—
269C22×C4—1221no—
270C22×C4—1221no—
271C22×C4—1221no—
272C22×C4—1221no—
273C22×C4—2442no—
274C22×C4—2442no—
275C22×C4C22×#692311yescharacteristic
276M16—4882no—
277C2×C8—1224no—
278C2×C8—4882no—
279C2×C8—2442no—
280C4⋊C4—22no—
281C4⋊C4—1224no—
282C4⋊C4—1222no—
283C4⋊C4—1222no—
284C4⋊C4—2442no—
285C4⋊C4—2442no—
286C4⋊C4—2442no—
287C4⋊C4—2442no—
288C4⋊C4—2442no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—1222no—
292C22⋊C4—1222no—
293C22⋊C4—2442no—
294C22⋊C4—2442no—
295C42—41no—
296C42—612no—
297C42—612no—
298C42—2441no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23#1451983812yes—
309C23C2×C22⋊#692311yescharacteristic
310Q8—4882no—
311Q8—4882no—
312D4—44no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—611no—
322C2×C4—612no—
323C2×C4—611no—
324C2×C4—1222no—
325C2×C4—1222no—
326C2×C4—1221no—
327C2×C4—1221no—
328C2×C4—4881no—
329C2×C4—4881no—
330C2×C4—2442no—
331C2×C4—2442no—
332C2×C4—2441no—
333C2×C4—2441no—
334C2×C4—2441no—
335C2×C4—2441no—
336C2×C4—2441no—
337C2×C4—2441no—
338C2×C4—2441no—
339C2×C4—2441no—
340C2×C4D4×#692312yes—
341C8—4882no—
342C8—2444no—
343C22—21no—
344C22—21no—
345C22—22no—
346C22—42no—
347C22—42no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—81no—
354C22—81no—
355C22#1684071011yescharacteristic
356C22C2×#1451983811yescharacteristic
357C22#1684074911yescharacteristic
358C4—22no—
359C4—42no—
360C4—81no—
361C4—612no—
362C4—1221no—
363C4—1222no—
364C4—4881no—
365C4—2442no—
366C4—2441no—
367C4—2441no—
368C2—21no—
369C2—21no—
370C2—42no—
371C2—81no—
372C2#2146182611yescharacteristic
373C2#2146184611yescharacteristic
374C2#2146186011yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC22×C61#386
prime radicalC22×D4×C61C2×C4
cosocleC2×C4×Dic61C23
hypercenter#390#6923
hypocenterC61#9455
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #21461348C2semidirect
4. #21460670C2semidirect
212. C61#9455semidirect

Maximal subgroups

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

subgroupindex
#214626702
#214613482
#214610792
#214606702
#214507702
#945561