Finite Group Databasebeta

← #30598392

Subgroups — #30598392

Order 15616.

738 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 104 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#21461328C212yesmaximal, complemented
3#21461147C212yesmaximal, complemented
4#21461058C211yesmaximal, characteristic
5C2×#16840483C211yesmaximal, characteristic
6#16840779C2212yes—
7#16840765C2212yes—
8C2×#14519837—21no—
9#16840642—21no—
10#16840621—24no—
11Q8⋊#92420—24no—
12#16840606—21no—
13#16840605—21no—
14#16840483—24no—
15#16840483—24no—
16C2×#14519776C2211yescharacteristic
17C2×#14519772—21no—
18C2×#14519770—21no—
19C2×#14519763—21no—
20C2×#14519753C411yescharacteristic
21C2×Q8×Dic61C2211yescharacteristic
22C2×#14519742—21no—
23C2×C4×C22⋊D61C2211yescharacteristic
24C2×#14519736C411yescharacteristic
25C2×#14519735C411yescharacteristic
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
36C22×C4×D61C2×C411yescharacteristic
37#14519776—22no—
38#14519776D414yes—
39#14519772—28no—
40#14519770—28no—
41#14519763—28no—
42#14519753—22no—
43#14519753—22no—
44Q8×Dic61—22no—
45Q8×Dic61D414yes—
46#14519742—28no—
47C4×C22⋊D61—22no—
48C4×C22⋊D61—22no—
49#14519736—22no—
50#14519736—22no—
51#14519735—22no—
52#14519735—22no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61C2×C411yescharacteristic
55C2×C4⋊Dic61C2×C411yescharacteristic
56C2×#11758196—21no—
57C2×#11758196—21no—
58C2×#11758196D411yescharacteristic
59C2×C22⋊Dic61—21no—
60C2×C22⋊Dic61D411yescharacteristic
61C2×#11758198—21no—
62C2×#11758198—21no—
63C2×#11758198D411yescharacteristic
64C2×C4×Dic61—21no—
65C2×C4×Dic61—21no—
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⋊D61—21no—
90C2×C22⋊D61—21no—
91C22×Dic61—21no—
92C22×Dic61—21no—
93C22×Dic61C22⋊C411yescharacteristic
94C22×Dic61C2×D411yescharacteristic
95C22×Dic61C2×D411yescharacteristic
96C2×C4×D61—21no—
97C2×C4×D61—21no—
98C2×C4×D61—42no—
99C2×Dic122—24no—
100C2×Dic122—42no—
101C2×Dic122—41no—
102C2×Dic122—41no—
103#11758217—42no—
104#11758217—42no—
105C2×#92420—42no—
106C2×#92420—42no—
107C4⋊C4×C61—24no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61C22⋊C412yes—
110C22⋊C4×C61—24no—
111C22⋊Dic61—24no—
112C22⋊Dic61—22no—
113#11758198—24no—
114#11758198—24no—
115#11758198—22no—
116C4⋊Dic61—21no—
117C4⋊Dic61—21no—
118C4⋊Dic61—24no—
119C4⋊Dic61C22⋊C412yes—
120C4⋊Dic61C22⋊C412yes—
121#11758196—24no—
122#11758196—24no—
123#11758196—22no—
124C4×Dic61—24no—
125C4×Dic61—24no—
126C4×Dic61—24no—
127C4×Dic61C2×D412yes—
128C4×Dic61C2×D412yes—
129C23×C61—21no—
130C23×C61—21no—
131C23×C61—21no—
132C23×C61—41no—
133C23×C61—41no—
134C23×C61C2×C22⋊C411yescharacteristic
135C22×D61—21no—
136C22×D61—21no—
137C22×D61—21no—
138C22×D61—41no—
139C22×D61—41no—
140Q8×C61—24no—
141Q8×C61—42no—
142C2×C4×C61—22no—
143C2×C4×C61—21no—
144C2×C4×C61—22no—
145C2×C4×C61—22no—
146C2×C4×C61—42no—
147C2×C4×C61—42no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—42no—
151C2×C4×C61—41no—
152C2×C4×C61C2×C22⋊C411yescharacteristic
153C2×C4×C61C2×C22⋊C411yescharacteristic
154C22⋊D61—42no—
155C22⋊D61—42no—
156C2×Dic61—22no—
157C2×Dic61—21no—
158C2×Dic61—22no—
159C2×Dic61—41no—
160C2×Dic61—42no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61#12012yes—
165C2×Dic61#12012yes—
166C2×Dic61#13512yes—
167C2×Dic61AGL(1,ℤ/8)11yescharacteristic
168C2×Dic61#13711yescharacteristic
169C4×D61—42no—
170C4×D61—42no—
171Dic122—44no—
172Dic122—44no—
173#92420—82no—
174#92420—82no—
175#8389—611nomaximal, Sylow
176C22×C61—21no—
177C22×C61—21no—
178C22×C61—41no—
179C22×C61—41no—
180C22×C61—41no—
181C22×C61—81no—
182C22×C61—81no—
183C22×C61#37911yescharacteristic
184C22×C61#41611yescharacteristic
185C22×C61#41711yescharacteristic
186C2×D61—41no—
187C2×D61—41no—
188C2×D61—41no—
189C2×D61—81no—
190C2×D61—81no—
191C4×C61—21no—
192C4×C61—41no—
193C4×C61—42no—
194C4×C61—42no—
195C4×C61#41312yes—
196Dic61—22no—
197Dic61—42no—
198Dic61—41no—
199Dic61—42no—
200Dic61#44912yes—
201Dic61#45212yes—
202C2×#556—611no—
203#1800—611no—
204#1546—612no—
205#1487—612no—
206#1460—611no—
207C2×C61—21no—
208C2×C61—21no—
209C2×C61—81no—
210C2×C61#178511yescharacteristic
211C2×C61#178811yescharacteristic
212C2×C61#179611yescharacteristic
213D61—81no—
214#556—1224no—
215#556—1224no—
216C2×C4⋊Q8—611no—
217C2×#126—1221no—
218C2×#126—1221no—
219C2×#124—1221no—
220C2×C22⋊Q8—611no—
221C2×C22⋊Q8#692311yescharacteristic
222C2×C4×Q8—611no—
223C2×C4×Q8—1221no—
224C2×C4×D4—611no—
225C2×#117—611no—
226C2×#117—611no—
227#431—612no—
228#423—612no—
229C2×#100—1221no—
230#343—1221no—
231#332—1224no—
232Q8⋊C8—1224no—
233#323—1221no—
234#323—1221no—
235C61#838911yescharacteristic, complemented, Sylow
236C22×Q8—1221no—
237C22×Q8C2×#692311yescharacteristic
238C22×D4—611no—
239C23×C4—611no—
240C23×C4C2×#692311yescharacteristic
241C2×M16—1222no—
242C4⋊Q8—614no—
243C4⋊Q8—1222no—
244#126—1228no—
245#126—1228no—
246#124—1228no—
247C22⋊Q8—22no—
248C22⋊Q8—22no—
249C22⋊Q8—1222no—
250C22⋊Q8—1222no—
251C4×Q8—614no—
252C4×Q8—1222no—
253C4×Q8—1228no—
254C4×D4—1222no—
255C4×D4—1222no—
256#117—1222no—
257#117—1222no—
258#117—1222no—
259#117—1222no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—611no—
262C2×C4⋊C4—611no—
263C2×C4⋊C4—611no—
264C2×C4⋊C4—1221no—
265C2×C4⋊C4—1221no—
266C2×C4⋊C4—1221no—
267C2×C4⋊C4C2×#692311yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—611no—
270C2×C22⋊C4—611no—
271C2×C22⋊C4—1221no—
272C2×C22⋊C4—1221no—
273C2×C22⋊C4—1221no—
274C2×C42—611no—
275C2×C42—1221no—
276C2×C42—1221no—
277C2×C42—1221no—
278C4⋊C8—1222no—
279C4⋊C8—1222no—
280C4⋊C8—1222no—
281#100—2442no—
282C22⋊C8—1222no—
283C4×C8—1222no—
284C24—611no—
285C24C22⋊#692311yescharacteristic
286C2×Q8—21no—
287C2×Q8—42no—
288C2×Q8—1224no—
289C2×Q8—2442no—
290C2×Q8—2441no—
291C2×Q8—2441no—
292C2×Q8C22⋊#692312yes—
293C2×D4—1221no—
294C2×D4—1222no—
295C2×D4—1222no—
296C2×D4—1221no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—42no—
302C22×C4—611no—
303C22×C4—611no—
304C22×C4—611no—
305C22×C4—1221no—
306C22×C4—1221no—
307C22×C4—1221no—
308C22×C4—1221no—
309C22×C4—2442no—
310C22×C4C22⋊#692311yescharacteristic
311C22×C4C22×#692311yescharacteristic
312M16—2442no—
313M16—2442no—
314C2×C8—2442no—
315C2×C8—2442no—
316C4⋊C4—24no—
317C4⋊C4—21no—
318C4⋊C4—612no—
319C4⋊C4—612no—
320C4⋊C4—1222no—
321C4⋊C4—1224no—
322C4⋊C4—1224no—
323C4⋊C4—1224no—
324C4⋊C4—1221no—
325C4⋊C4—1221no—
326C4⋊C4C22⋊#692312yes—
327C22⋊C4—24no—
328C22⋊C4—1222no—
329C22⋊C4—1224no—
330C22⋊C4—1224no—
331C22⋊C4—1222no—
332C22⋊C4—1224no—
333C42—612no—
334C42—612no—
335C42—1224no—
336C42—1224no—
337C42—1224no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—41no—
342C23—41no—
343C23—1221no—
344C23—1221no—
345C23—1221no—
346C23—2441no—
347C23—2441no—
348C23C2×C22⋊#692311yescharacteristic
349Q8—24no—
350Q8—42no—
351Q8—2444no—
352Q8—2444no—
353D4—2442no—
354D4—2442no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—612no—
366C2×C4—611no—
367C2×C4—611no—
368C2×C4—612no—
369C2×C4—612no—
370C2×C4—1222no—
371C2×C4—1221no—
372C2×C4—1222no—
373C2×C4—2442no—
374C2×C4—2441no—
375C2×C4—2442no—
376C2×C4—2441no—
377C2×C4—2441no—
378C2×C4—2442no—
379C2×C4—2441no—
380C2×C4C2×C22⋊#692311yescharacteristic
381C2×C4C2×C22⋊#692311yescharacteristic
382C8—4882no—
383C8—4882no—
384C22—21no—
385C22—21no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—81no—
390C22—81no—
391C22—4881no—
392C22—4881no—
393C22—2441no—
394C22—2441no—
395C22—2441no—
396C22#1684070311yescharacteristic
397C22#1684075211yescharacteristic
398C22#1684075311yescharacteristic
399C4—21no—
400C4—41no—
401C4—42no—
402C4—42no—
403C4—612no—
404C4—612no—
405C4—1222no—
406C4—2441no—
407C4—2442no—
408C4—2442no—
409C4#1684074912yes—
410C2—21no—
411C2—21no—
412C2—81no—
413C2—4881no—
414C2#2146165611yescharacteristic
415C2#2146165911yescharacteristic
416C2#2146167111yescharacteristic
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#8389
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. #21461328C2semidirect
3. #21461147C2semidirect
235. C61#8389semidirect

Maximal subgroups

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

subgroupindex
#214616832
#214613282
#214611472
#214610582
C2×#168404832
#838961