Finite Group Databasebeta

← #30606831

Subgroups — #30606831

Order 15616.

990 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 990 conjugacy classes of subgroups — too many to draw. Only its 96 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×#16840669C212yesmaximal, complemented
2C2×#16840668C212yesmaximal, complemented
3#21461650C212yesmaximal
4C2×#16840337C211yesmaximal, characteristic
5C2×#14519836—22no—
6C2×#14519836C2214yes—
7#16840705—22no—
8#16840669—24no—
9#16840668—24no—
10C22×#11758226C2211yescharacteristic
11C22×Q8×D61C411yescharacteristic, complemented
12#16840337—24no—
13#16840337—24no—
14C2×#14519775C411yescharacteristic, complemented
15C2×#14519766C2211yescharacteristic
16C2×#14519765C2211yescharacteristic
17C2×#14519762—22no—
18C2×#14519755—22no—
19C2×#14519734C411yescharacteristic, complemented
20C2×C4⋊Q8×C61C411yescharacteristic, complemented
21C2×C22⋊#6923—24no—
22#14519836—24no—
23#14519836—24no—
24#14519836—44no—
25(C2×#6923)⋊C4—44no—
26C2×#11758226—22no—
27C2×#11758226—22no—
28C2×#11758226—42no—
29C2×#11758226—42no—
30C22×Dic122—21no—
31C22×Dic122C2×C411yescharacteristic
32C2×Q8×D61—22no—
33C2×Q8×D61—22no—
34C2×Q8×D61—42no—
35C2×Q8×D61—42no—
36C22×D244D412yes—
37C22×D244C2311yescharacteristic
38C22×C4×D61—21no—
39C22×C4×D61C2×C411yescharacteristic
40C22×C4×D61D412yes—
41#14519775—22no—
42#14519775—22no—
43#14519766—22no—
44#14519766D414yes—
45#14519765—22no—
46#14519765D414yes—
47#14519762—216no—
48#14519755—216no—
49#14519734—22no—
50#14519734—22no—
51C2×#11758196—22no—
52C2×#11758196—21no—
53C2×#11758198—21no—
54C2×#11758198—22no—
55C2×#11758198—22no—
56C2×C4×Dic61C2×C411yescharacteristic
57C22×Q8×C61C2×C411yescharacteristic
58C22×Q8×C61C2×C411yescharacteristic
59C4⋊Q8×C61—22no—
60C4⋊Q8×C61—22no—
61C2×C4⋊C4×C61—22no—
62C2×C42×C61C2×C411yescharacteristic
63C23×D61C22⋊C411yescharacteristic
64C23×D61C2×D412yes—
65C22×#6923—44no—
66C2×Q8×C61—21no—
67C2×Q8×C61—21no—
68C2×Q8×C61—21no—
69C2×Q8×C61—41no—
70C2×Q8×C61—41no—
71C2×Q8×C61—41no—
72C2×Q8×C61—41no—
73C2×Q8×C61C22⋊C412yes—
74C22×C4×C61—21no—
75C22×C4×C61—21no—
76C22×C4×C61C22⋊C412yes—
77C22×C4×C61C22×C411yescharacteristic
78C22×Dic61—21no—
79C22×Dic61—21no—
80C22×Dic61C22⋊C411yescharacteristic
81#11758226—48no—
82#11758226—48no—
83Q8×D61—48no—
84Q8×D61—48no—
85C2×D244—24no—
86C2×D244—24no—
87C2×D244—42no—
88C2×D244—41no—
89C2×D244—41no—
90C2×D244C2×D412yes—
91C2×D244C2×D412yes—
92C2×C4×D61—22no—
93C2×C4×D61—21no—
94C2×C4×D61—21no—
95C2×C4×D61—22no—
96C2×C4×D61—24no—
97C2×C4×D61—24no—
98C2×C4×D61—24no—
99C2×C4×D61—42no—
100C2×C4×D61—42no—
101C2×C4×D61—42no—
102C2×Dic122—24no—
103C2×Dic122—21no—
104C2×Dic122—42no—
105C2×Dic122—42no—
106C2×Dic122—41no—
107C2×Dic122—41no—
108C2×Dic122C22⋊C412yes—
109C22⋊#6923—44no—
110C22⋊#6923—44no—
111C4⋊C4×C61—28no—
112C42×C61—21no—
113C42×C61C22⋊C412yes—
114#11758198—24no—
115#11758198—28no—
116#11758198—28no—
117#11758196—24no—
118#11758196—28no—
119C4×Dic61—21no—
120C4×Dic61C22⋊C412yes—
121C23×C61C2×C22⋊C411yescharacteristic
122C22×D61—21no—
123C22×D61—21no—
124C22×D61—22no—
125C22×D61—22no—
126C22×D61—42no—
127C22×D61—44no—
128C22×D61#9912yes—
129C22×D61#12014yes—
130C2×#6923—84no—
131C2×#6923—88no—
132Q8×C61—42no—
133Q8×C61—42no—
134Q8×C61—42no—
135Q8×C61—42no—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—22no—
139C2×C4×C61—22no—
140C2×C4×C61—22no—
141C2×C4×C61—22no—
142C2×C4×C61—22no—
143C2×C4×C61—42no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61C2×C22⋊C411yescharacteristic
147C2×C4×C61C2×C22⋊C411yescharacteristic
148C2×Dic61—21no—
149C2×Dic61—21no—
150C2×Dic61—22no—
151C2×Dic61—22no—
152C2×Dic61—42no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61#9912yes—
156D244—42no—
157D244—42no—
158D244—48no—
159C4×D61—24no—
160C4×D61—24no—
161C4×D61—48no—
162C4×D61—48no—
163Dic122—48no—
164Dic122—42no—
165Dic122—42no—
166#12698—611nomaximal, Sylow
167C22×C61—22no—
168C22×C61#37911yescharacteristic
169C22×C61C2×#9911yescharacteristic
170C22×C61C2×#9911yescharacteristic
171C2×D61—22no—
172C2×D61—24no—
173C2×D61—42no—
174C2×D61—44no—
175C2×D61—42no—
176C2×D61—42no—
177C2×D61—44no—
178C2×D61—82no—
179#6923—164no—
180C4×C61—21no—
181C4×C61—21no—
182C4×C61—42no—
183C4×C61—42no—
184C4×C61—42no—
185Dic61—22no—
186Dic61—42no—
187Dic61—42no—
188C2×#536—611no—
189C2×#354—612no—
190C2×#354—612no—
191#1782—612no—
192C2×C61—21no—
193C2×C61—21no—
194C2×C61#178111yescharacteristic
195C2×C61#212912yes—
196D61—44no—
197D61—42no—
198C22×#41—611no—
199C23×Q8—611no—
200#536—1224no—
201#536—1224no—
202C2×C4⋊Q8—611no—
203C2×C4⋊Q8#692311yescharacteristic, complemented
204C2×#124—611no—
205C2×#124—611no—
206C2×#123—1222no—
207C2×C22⋊Q8—1222no—
208C2×#117—611no—
209C2×#99—614no—
210C2×#99—1222no—
211#381—1222no—
212#354—1224no—
213#354—1224no—
214C61#1269811yescharacteristic, complemented, Sylow
215C2×#41—1222no—
216C2×#41—1222no—
217C2×#41—2442no—
218C2×#41—2442no—
219C22×Q8—611no—
220C22×Q8—1221no—
221C22×Q8—1222no—
222C22×Q8—1222no—
223C22×Q8—2442no—
224C22×Q8—2442no—
225C22×Q8C2×#692311yescharacteristic
226C22×Q8C2×#692311yescharacteristic
227C22×D4—612no—
228C22×D4—611no—
229C23×C4—611no—
230C23×C4—612no—
231C23×C4—1221no—
232C4⋊Q8—22no—
233C4⋊Q8—22no—
234C4⋊Q8—1222no—
235C4⋊Q8—1222no—
236#124—614no—
237#124—614no—
238#124—1222no—
239#124—1222no—
240#123—12216no—
241C22⋊Q8—12216no—
242#117—1222no—
243#117—1222no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—1221no—
246C2×C4⋊C4—1222no—
247C2×C22⋊C4—1221no—
248C2×C22⋊C4—1222no—
249C2×C22⋊C4—1222no—
250C2×C22⋊C4—1224no—
251C2×C42—611no—
252C2×C42C2×#692311yescharacteristic
253#99—1224no—
254#99—1224no—
255#99—2444no—
256#95—2444no—
257C24—612no—
258C24—611no—
259#41—2448no—
260#41—2448no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—612no—
269C2×Q8—1224no—
270C2×Q8—1221no—
271C2×Q8—2441no—
272C2×Q8—2441no—
273C2×Q8—2442no—
274C2×Q8—2442no—
275C2×Q8—2448no—
276C2×Q8—2448no—
277C2×Q8C22⋊#692312yes—
278C2×D4—612no—
279C2×D4—612no—
280C2×D4—1224no—
281C2×D4—1224no—
282C2×D4—2441no—
283C2×D4—2441no—
284C2×D4—2442no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—611no—
288C22×C4—1222no—
289C22×C4—1224no—
290C22×C4—1222no—
291C22×C4—1224no—
292C22×C4—1224no—
293C22×C4—1221no—
294C22×C4—1221no—
295C22×C4—1221no—
296C22×C4—1221no—
297C22×C4—2442no—
298C22×C4—2442no—
299C22×C4—2442no—
300C22×C4—2444no—
301C22×C4C22⋊#692312yes—
302C22×C4C22×#692311yescharacteristic
303C4⋊C4—28no—
304C4⋊C4—1224no—
305C4⋊C4—1228no—
306C22⋊C4—1228no—
307C22⋊C4—1228no—
308C22⋊C4—1224no—
309C22⋊C4—2444no—
310C22⋊C4—2444no—
311C42—21no—
312C42—612no—
313C42—1221no—
314C42C22⋊#692312yes—
315C23—612no—
316C23—614no—
317C23—1222no—
318C23—1221no—
319C23—1222no—
320C23—1221no—
321C23—2444no—
322C23—2442no—
323C23C2×C22⋊#692311yescharacteristic
324Q8—42no—
325Q8—42no—
326Q8—42no—
327Q8—42no—
328Q8—2442no—
329Q8—2442no—
330Q8—2448no—
331D4—2442no—
332D4—2442no—
333D4—2448no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—612no—
345C2×C4—1224no—
346C2×C4—1224no—
347C2×C4—1221no—
348C2×C4—1221no—
349C2×C4—1222no—
350C2×C4—1222no—
351C2×C4—4884no—
352C2×C4—4888no—
353C2×C4—2448no—
354C2×C4—2448no—
355C2×C4—2442no—
356C2×C4—2441no—
357C2×C4—2441no—
358C2×C4C2×C22⋊#692311yescharacteristic
359C2×C4C2×C22⋊#692311yescharacteristic
360C22—22no—
361C22—1222no—
362C22—1224no—
363C22—4882no—
364C22—2444no—
365C22—2442no—
366C22—2442no—
367C22—2444no—
368C22—2442no—
369C22#1684070311yescharacteristic
370C22D61⋊#9911yescharacteristic
371C22D61⋊#9911yescharacteristic
372C4—21no—
373C4—21no—
374C4—42no—
375C4—42no—
376C4—42no—
377C4—1222no—
378C4—9764no—
379C4—2442no—
380C4—2442no—
381C2—21no—
382C2—21no—
383C2—2444no—
384C2—2442no—
385C2#2146164611yescharacteristic
386C2#2146210212yes—
subgroupG/H
centerC22D61⋊#99
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4⋊Q8×C61C4
socleC22×C61C2×#99
prime radicalC22×#11758226C22
cosocleC22×D244C23
hypercenterC2×C4⋊Q8#6923
hypocenterC61#12698
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16840669C2semidirect
2. C2×#16840668C2semidirect
11. C22×Q8×D61C4semidirect
14. C2×#14519775C4semidirect
19. C2×#14519734C4semidirect
20. C2×C4⋊Q8×C61C4semidirect
203. C2×C4⋊Q8#6923semidirect
214. C61#12698semidirect

Maximal subgroups

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

subgroupindex
C2×#168406692
C2×#168406682
#214616502
C2×#168403372
#1269861