Finite Group Databasebeta

← #30598708

Subgroups — #30598708

Order 15616.

754 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 92 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#21461720C211yesmaximal, characteristic, complemented
2#21461716C211yesmaximal, characteristic, complemented
3#21461229C212yesmaximal, complemented
4#21461224C212yesmaximal
5C2×#16840381C211yesmaximal, characteristic
6#16840734C2212yes—
7#16840705C2212yes—
8#16840642—21no—
9#16840642—21no—
10#16840640—21no—
11#16840640—21no—
12#16840616—24no—
13#16840613—24no—
14#16840381—24no—
15#16840381—24no—
16C2×#14519777C411yescharacteristic, complemented
17C2×#14519770—21no—
18C2×#14519763—21no—
19C2×#14519761—21no—
20C2×#14519754—21no—
21C2×#14519753C2211yescharacteristic
22C2×#14519748C2211yescharacteristic
23C2×Q8×Dic61C411yescharacteristic, complemented
24C2×D4×Dic61C411yescharacteristic, complemented
25C2×#14519734C2211yescharacteristic
26C2×#124×C61C411yescharacteristic, complemented
27C2×#11758217—22no—
28C2×C22⋊#6923—22no—
29#14519834—22no—
30#14519834—22no—
31(C2×#6923)⋊C4—22no—
32(C2×#6923)⋊C4—22no—
33C22×Dic122—21no—
34C22×C22⋊D61—21no—
35C23×Dic61—21no—
36C22×C4×D61C2311yescharacteristic
37#14519777—22no—
38#14519777—22no—
39#14519770—28no—
40#14519763—28no—
41#14519761—28no—
42#14519754—28no—
43#14519753—22no—
44#14519753D414yes—
45#14519748—22no—
46#14519748D414yes—
47Q8×Dic61—22no—
48Q8×Dic61—22no—
49D4×Dic61—22no—
50D4×Dic61—22no—
51#14519734—22no—
52#14519734—22no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61C2×C411yescharacteristic
56C2×C4⋊Dic61C2×C411yescharacteristic
57C2×#11758196—21no—
58C2×#11758196D411yescharacteristic
59C2×#11758196D411yescharacteristic
60C2×C22⋊Dic61—21no—
61C2×C22⋊Dic61—21no—
62C2×#11758198—21no—
63C2×#11758198D411yescharacteristic
64C2×#11758198D411yescharacteristic
65C2×C4×Dic61—21no—
66C2×C4×Dic61C2×C411yescharacteristic
67C22×Q8×C61C2×C411yescharacteristic
68C22×D4×C61C2×C411yescharacteristic
69#124×C61—22no—
70#124×C61—22no—
71C2×C22⋊C4×C61—21no—
72C2×C22⋊C4×C61—21no—
73C2×C42×C61C2×C411yescharacteristic
74C24×C61—21no—
75C23×D61C2×D411yescharacteristic
76C22×#6923—42no—
77C2×Q8×C61—21no—
78C2×Q8×C61—42no—
79C2×Q8×C61C22⋊C412yes—
80C2×D4×C61—21no—
81C2×D4×C61—42no—
82C2×D4×C61C22⋊C412yes—
83C22×C4×C61—21no—
84C22×C4×C61C22⋊C411yescharacteristic
85C22×C4×C61C22⋊C411yescharacteristic
86C22×C4×C61C22×C411yescharacteristic
87C2×C22⋊D61—24no—
88C2×C22⋊D61—82no—
89C22×Dic61—21no—
90C22×Dic61—21no—
91C22×Dic61—24no—
92C22×Dic61—82no—
93C22×Dic61C22⋊C411yescharacteristic
94C22×Dic61C22⋊C411yescharacteristic
95C22×Dic61C2×D411yescharacteristic
96C2×C4×D61—42no—
97C2×C4×D61C2×D412yes—
98C2×C4×D61C2×D412yes—
99C2×Dic122—24no—
100C2×Dic122—82no—
101C22⋊#6923—44no—
102#11758217—44no—
103C2×#92420—42no—
104C22⋊C4×C61—24no—
105C22⋊C4×C61—24no—
106C42×C61—21no—
107C42×C61—21no—
108C22⋊Dic61—24no—
109C22⋊Dic61—24no—
110#11758198—24no—
111#11758198—22no—
112#11758198—22no—
113C4⋊Dic61—24no—
114C4⋊Dic61—24no—
115C4⋊Dic61—21no—
116C4⋊Dic61—21no—
117C4⋊Dic61C22⋊C412yes—
118C4⋊Dic61C22⋊C412yes—
119#11758196—24no—
120#11758196—22no—
121#11758196—22no—
122C4×Dic61—24no—
123C4×Dic61—21no—
124C4×Dic61—21no—
125C23×C61—22no—
126C23×C61—42no—
127C23×C61—42no—
128C23×C61—42no—
129C23×C61C2×C22⋊C411yescharacteristic
130C22×D61—22no—
131C22×D61—41no—
132C22×D61—41no—
133C22×D61#12012yes—
134C2×#6923—82no—
135C2×#6923—82no—
136C2×#6923—82no—
137Q8×C61—42no—
138Q8×C61—42no—
139D4×C61—42no—
140D4×C61—42no—
141C2×C4×C61—22no—
142C2×C4×C61—22no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—42no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61C2×C22⋊C411yescharacteristic
149C2×C4×C61C2×C22⋊C411yescharacteristic
150C22⋊D61—84no—
151C2×Dic61—22no—
152C2×Dic61—21no—
153C2×Dic61—21no—
154C2×Dic61—22no—
155C2×Dic61—22no—
156C2×Dic61—42no—
157C2×Dic61—42no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—84no—
161C2×Dic61#12012yes—
162C4×D61—42no—
163C4×D61—42no—
164Dic122—84no—
165#92420—82no—
166#8519—611nomaximal, Sylow
167C22×C61—21no—
168C22×C61—21no—
169C22×C61—42no—
170C22×C61—42no—
171C22×C61—42no—
172C22×C61—84no—
173C22×C61#37911yescharacteristic
174C22×C61#41012yes—
175C2×D61—22no—
176C2×D61—22no—
177C2×D61—22no—
178C2×D61—82no—
179#6923—162no—
180C4×C61—21no—
181C4×C61—21no—
182C4×C61—41no—
183C4×C61—41no—
184C4×C61—42no—
185Dic61—22no—
186Dic61—42no—
187Dic61—42no—
188Dic61—41no—
189Dic61—41no—
190C2×#542—611no—
191#1822—611no—
192#1820—611no—
193#1511—612no—
194#1510—612no—
195C2×C61—22no—
196C2×C61—82no—
197C2×C61#178211yescharacteristic
198C2×C61#179511yescharacteristic
199C2×C61#179611yescharacteristic
200D61—42no—
201#542—1224no—
202#542—1224no—
203C2×C4⋊Q8—611no—
204C2×#126—1221no—
205C2×#124—1221no—
206C2×#124#692311yescharacteristic, complemented
207C2×#123—1221no—
208C2×C22⋊Q8—611no—
209C2×C22⋊Q8—1221no—
210C2×#121—611no—
211C2×C4×Q8—611no—
212C2×C4×D4—611no—
213C2×#117—611no—
214#407—612no—
215#381—612no—
216#343—1221no—
217#343—1221no—
218#342—1221no—
219#342—1221no—
220#328—1224no—
221#327—1224no—
222C61#851911yescharacteristic, complemented, Sylow
223C22×Q8—1221no—
224C22×Q8C2×#692311yescharacteristic
225C22×D4—1221no—
226C22×D4C2×#692311yescharacteristic
227C23×C4—611no—
228C23×C4—1221no—
229C2×M16—1222no—
230C4⋊Q8—1222no—
231C4⋊Q8—1222no—
232#126—1228no—
233#124—22no—
234#124—22no—
235#124—1228no—
236#123—1228no—
237C22⋊Q8—614no—
238C22⋊Q8—1228no—
239C22⋊Q8—1222no—
240#121—614no—
241#121—1222no—
242C4×Q8—1222no—
243C4×Q8—1222no—
244C4×D4—1222no—
245C4×D4—1222no—
246#117—1222no—
247#117—1222no—
248C2×C4⋊C4—611no—
249C2×C4⋊C4—611no—
250C2×C4⋊C4—611no—
251C2×C4⋊C4—611no—
252C2×C4⋊C4—1221no—
253C2×C4⋊C4—1221no—
254C2×C4⋊C4—1221no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—611no—
258C2×C22⋊C4—611no—
259C2×C22⋊C4—1221no—
260C2×C22⋊C4—1221no—
261C2×C22⋊C4—1221no—
262C2×C22⋊C4—1222no—
263C2×C42—611no—
264C2×C42—1221no—
265C2×C42C2×#692311yescharacteristic
266C22⋊C8—1222no—
267C22⋊C8—1222no—
268#95—1222no—
269#95—1222no—
270C24—21no—
271C24—611no—
272C2×Q8—21no—
273C2×Q8—42no—
274C2×Q8—1224no—
275C2×Q8—4882no—
276C2×Q8C22⋊#692312yes—
277C2×D4—21no—
278C2×D4—42no—
279C2×D4—1224no—
280C2×D4—4882no—
281C2×D4C22⋊#692312yes—
282C22×C4—21no—
283C22×C4—611no—
284C22×C4—612no—
285C22×C4—612no—
286C22×C4—611no—
287C22×C4—611no—
288C22×C4—1224no—
289C22×C4—1221no—
290C22×C4—1221no—
291C22×C4—4882no—
292C22×C4—2442no—
293C22×C4—2442no—
294C22×C4C22⋊#692311yescharacteristic
295C22×C4C22⋊#692311yescharacteristic
296C22×C4C22×#692311yescharacteristic
297M16—2444no—
298C2×C8—2442no—
299C4⋊C4—612no—
300C4⋊C4—612no—
301C4⋊C4—1222no—
302C4⋊C4—1222no—
303C4⋊C4—1224no—
304C4⋊C4—1224no—
305C4⋊C4—1224no—
306C4⋊C4—1221no—
307C4⋊C4—1221no—
308C22⋊C4—24no—
309C22⋊C4—24no—
310C22⋊C4—1222no—
311C22⋊C4—1222no—
312C22⋊C4—1224no—
313C22⋊C4—1224no—
314C22⋊C4—1224no—
315C22⋊C4—2444no—
316C42—21no—
317C42—21no—
318C42—1224no—
319C42—1221no—
320C42—1221no—
321C23—22no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—612no—
326C23—1222no—
327C23—2441no—
328C23—2441no—
329C23C2×C22⋊#692311yescharacteristic
330Q8—42no—
331Q8—42no—
332Q8—4884no—
333D4—42no—
334D4—42no—
335D4—4884no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—612no—
344C2×C4—1222no—
345C2×C4—1221no—
346C2×C4—1222no—
347C2×C4—1221no—
348C2×C4—1222no—
349C2×C4—4882no—
350C2×C4—4882no—
351C2×C4—4882no—
352C2×C4—4884no—
353C2×C4—2442no—
354C2×C4—2442no—
355C2×C4—2442no—
356C2×C4—2442no—
357C2×C4—2441no—
358C2×C4—2441no—
359C2×C4C2×C22⋊#692311yescharacteristic
360C2×C4C2×C22⋊#692311yescharacteristic
361C8—4882no—
362C22—21no—
363C22—21no—
364C22—42no—
365C22—42no—
366C22—42no—
367C22—84no—
368C22—1222no—
369C22—1222no—
370C22—1222no—
371C22—4882no—
372C22#1684070311yescharacteristic
373C22#1684074212yes—
374C4—21no—
375C4—21no—
376C4—42no—
377C4—41no—
378C4—41no—
379C4—1222no—
380C4—9762no—
381C4—2441no—
382C4—2442no—
383C4—2442no—
384C4—2441no—
385C2—22no—
386C2—82no—
387C2—2442no—
388C2#2146164911yescharacteristic
389C2#2146167011yescharacteristic
390C2#2146167211yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#124×C61C4
socleC22×C61#379
prime radicalC2×#14519748C22
cosocleC22×C4×D61C23
hypercenterC2×#124#6923
hypocenterC61#8519
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #21461720C2semidirect
2. #21461716C2semidirect
3. #21461229C2semidirect
16. C2×#14519777C4semidirect
23. C2×Q8×Dic61C4semidirect
24. C2×D4×Dic61C4semidirect
26. C2×#124×C61C4semidirect
206. C2×#124#6923semidirect
222. C61#8519semidirect

Maximal subgroups

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

subgroupindex
#214617202
#214617162
#214612292
#214612242
C2×#168403812
#851961