Finite Group Databasebeta

← #30598404

Subgroups — #30598404

Order 15616.

782 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 114 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#21461683C211yesmaximal, characteristic
2#21461334C212yesmaximal, complemented
3#21461147C212yesmaximal, complemented
4#21461058C211yesmaximal, characteristic
5C2×#16840470C211yesmaximal, characteristic
6#16840779C2212yes—
7#16840765C2212yes—
8C2×#14519837—21no—
9#16840642—21no—
10#16840621—24no—
11Q8⋊#92420—24no—
12#16840606—21no—
13#16840605—21no—
14#16840470—24no—
15#16840470—24no—
16C2×#14519776C2211yescharacteristic
17C2×#14519767—21no—
18C2×#14519757—21no—
19C2×#14519754—21no—
20C2×#14519753C411yescharacteristic
21C2×Q8×Dic61C2211yescharacteristic
22C2×C4×C22⋊D61C2211yescharacteristic
23C2×#14519737—21no—
24C22×C4⋊Dic61C411yescharacteristic
25C2×C4⋊C4×D61C411yescharacteristic
26C2×C22⋊Q8×C61C411yescharacteristic
27C2×#11758217—22no—
28#14519844—22no—
29#14519844—22no—
30C4⋊#92420—22no—
31#14519837—42no—
32C22⋊#92420—22no—
33C4×#92420—22no—
34C22×Dic122—21no—
35C22×C22⋊D61D411yescharacteristic
36C23×Dic61—21no—
37C22×C4×D61—21no—
38C22×C4×D61C2×C411yescharacteristic
39#14519776—22no—
40#14519776D414yes—
41#14519767—28no—
42#14519757—28no—
43#14519754—28no—
44#14519753—22no—
45#14519753—22no—
46Q8×Dic61—22no—
47Q8×Dic61D414yes—
48C4×C22⋊D61—22no—
49C4×C22⋊D61—22no—
50#14519737—28no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61—22no—
54C2×C4⋊Dic61—22no—
55C2×C4⋊Dic61C2×C411yescharacteristic
56C2×C4⋊Dic61C2×C411yescharacteristic
57C2×#11758196—21no—
58C2×#11758196—21no—
59C2×#11758196—21no—
60C2×#11758196D411yescharacteristic
61C4⋊C4×D61—22no—
62C4⋊C4×D61—22no—
63C2×C22⋊Dic61D411yescharacteristic
64C2×#11758198—21no—
65C2×#11758198D411yescharacteristic
66C2×C4×Dic61—21no—
67C2×C4×Dic61C2311yescharacteristic
68C22×Q8×C61C2×C411yescharacteristic
69C23×C4×C61C2×C411yescharacteristic
70C22⋊Q8×C61—22no—
71C22⋊Q8×C61—22no—
72C2×C4⋊C4×C61—21no—
73C2×C4⋊C4×C61C2×C411yescharacteristic
74C2×C22⋊C4×C61—21no—
75C24×C61C22⋊C411yescharacteristic
76C23×D61C22⋊C411yescharacteristic
77C2×Q8×C61—21no—
78C2×Q8×C61—42no—
79C2×Q8×C61C22⋊C412yes—
80C22×C4×C61—21no—
81C22×C4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61—42no—
85C22×C4×C61C22⋊C411yescharacteristic
86C22×C4×C61C22×C411yescharacteristic
87C2×C22⋊D61—22no—
88C2×C22⋊D61—22no—
89C2×C22⋊D61SD1612yes—
90C2×C22⋊D61Q1612yes—
91C22×Dic61—21no—
92C22×Dic61—21no—
93C22×Dic61—24no—
94C22×Dic61—82no—
95C22×Dic61C22⋊C411yescharacteristic
96C22×Dic61C2×D411yescharacteristic
97C22×Dic61C2×D411yescharacteristic
98C2×C4×D61—21no—
99C2×C4×D61—21no—
100C2×C4×D61—24no—
101C2×C4×D61—42no—
102C2×C4×D61—82no—
103C2×Dic122—24no—
104C2×Dic122—82no—
105#11758217—42no—
106#11758217—42no—
107C2×#92420—42no—
108C2×#92420—42no—
109C4⋊C4×C61—24no—
110C4⋊C4×C61—21no—
111C4⋊C4×C61C22⋊C412yes—
112C22⋊C4×C61—24no—
113C22⋊Dic61—22no—
114#11758198—24no—
115#11758198—22no—
116C4⋊Dic61—24no—
117C4⋊Dic61—21no—
118C4⋊Dic61—21no—
119C4⋊Dic61—24no—
120C4⋊Dic61C22⋊C412yes—
121C4⋊Dic61C22⋊C412yes—
122#11758196—24no—
123#11758196—24no—
124#11758196—24no—
125#11758196—22no—
126C4×Dic61—24no—
127C4×Dic61C2×D412yes—
128C4×Dic61C2×D412yes—
129C23×C61—21no—
130C23×C61—21no—
131C23×C61—41no—
132C23×C61—41no—
133C23×C61#10312yes—
134C23×C61C2×C22⋊C411yescharacteristic
135C22×D61—21no—
136C22×D61—21no—
137C22×D61—41no—
138C22×D61—41no—
139C22×D61#10312yes—
140Q8×C61—24no—
141Q8×C61—42no—
142C2×C4×C61—22no—
143C2×C4×C61—22no—
144C2×C4×C61—22no—
145C2×C4×C61—21no—
146C2×C4×C61—42no—
147C2×C4×C61—42no—
148C2×C4×C61—42no—
149C2×C4×C61—42no—
150C2×C4×C61—41no—
151C2×C4×C61C2×C22⋊C411yescharacteristic
152C2×C4×C61C2×C22⋊C411yescharacteristic
153C22⋊D61—24no—
154C22⋊D61—24no—
155C2×Dic61—22no—
156C2×Dic61—21no—
157C2×Dic61—22no—
158C2×Dic61—41no—
159C2×Dic61—42no—
160C2×Dic61—42no—
161C2×Dic61—41no—
162C2×Dic61—84no—
163C2×Dic61#12012yes—
164C2×Dic61#12012yes—
165C2×Dic61C2×SD1611yescharacteristic
166C2×Dic61C2×Q1611yescharacteristic
167C2×Dic61#13712yes—
168C4×D61—42no—
169C4×D61—42no—
170C4×D61—84no—
171Dic122—84no—
172#92420—82no—
173#92420—82no—
174#8395—611nomaximal, Sylow
175C22×C61—21no—
176C22×C61—21no—
177C22×C61—22no—
178C22×C61—42no—
179C22×C61—42no—
180C22×C61—41no—
181C22×C61—41no—
182C22×C61#37911yescharacteristic
183C22×C61C2×#10311yescharacteristic
184C22×C61#41911yescharacteristic
185C2×D61—22no—
186C2×D61—42no—
187C2×D61—41no—
188C2×D61—41no—
189C2×D61—42no—
190C4×C61—21no—
191C4×C61—42no—
192C4×C61—41no—
193C4×C61—42no—
194C4×C61#41312yes—
195Dic61—22no—
196Dic61—42no—
197Dic61—42no—
198Dic61—41no—
199Dic61#44812yes—
200Dic61#45112yes—
201C2×#554—611no—
202#1800—611no—
203#1549—612no—
204#1487—612no—
205#1460—611no—
206C2×C61—21no—
207C2×C61—21no—
208C2×C61—42no—
209C2×C61#178511yescharacteristic
210C2×C61#178711yescharacteristic
211C2×C61#179511yescharacteristic
212D61—42no—
213#554—1224no—
214#554—1224no—
215C2×C4⋊Q8—611no—
216C2×#125—1221no—
217C2×C22⋊Q8—611no—
218C2×C22⋊Q8—1221no—
219C2×C22⋊Q8—1221no—
220C2×C22⋊Q8#692311yescharacteristic
221C2×C4×Q8—611no—
222C2×C4×D4—611no—
223C2×C4×D4—1221no—
224C22×C4⋊C4—611no—
225C22×C4⋊C4—611no—
226#431—612no—
227#423—612no—
228C2×#100—1221no—
229#343—1221no—
230#332—1224no—
231Q8⋊C8—1224no—
232#323—1221no—
233#323—1221no—
234C61#839511yescharacteristic, complemented, Sylow
235C22×Q8—1221no—
236C22×Q8C2×#692311yescharacteristic
237C22×D4—611no—
238C23×C4—611no—
239C23×C4—1221no—
240C23×C4—1221no—
241C23×C4C2×#692311yescharacteristic
242C2×M16—1222no—
243C4⋊Q8—614no—
244C4⋊Q8—1222no—
245#125—1228no—
246C22⋊Q8—22no—
247C22⋊Q8—22no—
248C22⋊Q8—1228no—
249C22⋊Q8—1228no—
250C22⋊Q8—1222no—
251C22⋊Q8—1222no—
252C4×Q8—614no—
253C4×Q8—1222no—
254C4×D4—1222no—
255C4×D4—1222no—
256C4×D4—1228no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—611no—
259C2×C4⋊C4—611no—
260C2×C4⋊C4—611no—
261C2×C4⋊C4—1222no—
262C2×C4⋊C4—1222no—
263C2×C4⋊C4—1222no—
264C2×C4⋊C4—1221no—
265C2×C4⋊C4—1221no—
266C2×C4⋊C4—1221no—
267C2×C4⋊C4—1221no—
268C2×C4⋊C4—1221no—
269C2×C4⋊C4—1222no—
270C2×C4⋊C4C2×#692311yescharacteristic
271C2×C22⋊C4—21no—
272C2×C22⋊C4—611no—
273C2×C22⋊C4—611no—
274C2×C22⋊C4—1221no—
275C2×C42—611no—
276C2×C42—1221no—
277C4⋊C8—1222no—
278C4⋊C8—1222no—
279C4⋊C8—1222no—
280#100—2442no—
281C22⋊C8—1222no—
282C4×C8—1222no—
283C24—611no—
284C24C22⋊#692311yescharacteristic
285C2×Q8—21no—
286C2×Q8—42no—
287C2×Q8—1224no—
288C2×Q8—4882no—
289C2×Q8C22⋊#692312yes—
290C2×D4—612no—
291C2×D4—612no—
292C2×D4—1222no—
293C2×D4—1222no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—42no—
299C22×C4—611no—
300C22×C4—611no—
301C22×C4—611no—
302C22×C4—1224no—
303C22×C4—1224no—
304C22×C4—1221no—
305C22×C4—1221no—
306C22×C4—1221no—
307C22×C4—1221no—
308C22×C4—4882no—
309C22×C4—4882no—
310C22×C4—2442no—
311C22×C4C22⋊#692311yescharacteristic
312C22×C4C22×#692311yescharacteristic
313M16—2442no—
314M16—2442no—
315C2×C8—2442no—
316C2×C8—2442no—
317C4⋊C4—24no—
318C4⋊C4—21no—
319C4⋊C4—612no—
320C4⋊C4—612no—
321C4⋊C4—1222no—
322C4⋊C4—1224no—
323C4⋊C4—1224no—
324C4⋊C4—1224no—
325C4⋊C4—1224no—
326C4⋊C4—1224no—
327C4⋊C4—1221no—
328C4⋊C4—1221no—
329C4⋊C4C22⋊#692312yes—
330C22⋊C4—24no—
331C22⋊C4—1222no—
332C22⋊C4—1224no—
333C22⋊C4—1222no—
334C42—612no—
335C42—612no—
336C42—1224no—
337C23—21no—
338C23—21no—
339C23—41no—
340C23—41no—
341C23—612no—
342C23—1221no—
343C23—1221no—
344C23—2441no—
345C23—2441no—
346C23#1451984012yes—
347C23C2×C22⋊#692311yescharacteristic
348Q8—24no—
349Q8—42no—
350Q8—4884no—
351D4—1224no—
352D4—1224no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—611no—
363C2×C4—611no—
364C2×C4—612no—
365C2×C4—612no—
366C2×C4—612no—
367C2×C4—1222no—
368C2×C4—1222no—
369C2×C4—1221no—
370C2×C4—4884no—
371C2×C4—4884no—
372C2×C4—2442no—
373C2×C4—2441no—
374C2×C4—2442no—
375C2×C4—2442no—
376C2×C4—2442no—
377C2×C4—2441no—
378C2×C4C2×C22⋊#692311yescharacteristic
379C2×C4C2×C22⋊#692311yescharacteristic
380C8—4882no—
381C8—4882no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—42no—
386C22—42no—
387C22—41no—
388C22—41no—
389C22—1222no—
390C22—2442no—
391C22—2442no—
392C22—2441no—
393C22—2441no—
394C22#1684070311yescharacteristic
395C22C2×#1451984011yescharacteristic
396C22#1684075511yescharacteristic
397C4—21no—
398C4—41no—
399C4—42no—
400C4—42no—
401C4—612no—
402C4—612no—
403C4—1222no—
404C4—2441no—
405C4—2442no—
406C4—2442no—
407C4#1684074912yes—
408C2—21no—
409C2—21no—
410C2—42no—
411C2—2442no—
412C2#2146165611yescharacteristic
413C2#2146165811yescharacteristic
414C2#2146167011yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C22⋊Q8×C61C4
socleC22×C61#379
prime radicalC22×C22⋊D61D4
cosocleC2×C4×Dic61C23
hypercenterC2×C22⋊Q8#6923
hypocenterC61#8395
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #21461334C2semidirect
3. #21461147C2semidirect
234. C61#8395semidirect

Maximal subgroups

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

subgroupindex
#214616832
#214613342
#214611472
#214610582
C2×#168404702
#839561