Finite Group Databasebeta

← #30598201

Subgroups — #30598201

Order 15616.

748 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 748 conjugacy classes of subgroups — too many to draw. Only its 108 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#21461123C212yesmaximal, complemented
2#21461118C212yesmaximal, complemented
3#21461050C211yesmaximal, characteristic, complemented
4#21461050C211yesmaximal, characteristic, complemented
5C2×#16840379C211yesmaximal, characteristic
6#16840778C2212yes—
7#16840778C2212yes—
8#16840612—24no—
9D4⋊#92420—24no—
10#16840607—21no—
11#16840607—21no—
12#16840605—21no—
13#16840605—21no—
14#16840379—24no—
15#16840379—24no—
16C2×#14519776C411yescharacteristic
17C2×#14519770—21no—
18C2×#14519764—21no—
19C2×#14519764C411yescharacteristic
20C2×#14519758—21no—
21C2×#14519756C411yescharacteristic
22C2×#14519754—21no—
23C2×#14519742C2211yescharacteristic
24C2×D4×Dic61C2211yescharacteristic
25C2×#14519736C2211yescharacteristic
26C2×#121×C61C411yescharacteristic
27C4⋊#92420—22no—
28C4⋊#92420—22no—
29C22⋊#92420—22no—
30C22⋊#92420—22no—
31C4×#92420—22no—
32C4×#92420—22no—
33C22×Dic122—21no—
34C22×Dic122C2×C411yescharacteristic
35C23×Dic61—21no—
36#14519776—22no—
37#14519776—22no—
38#14519770—28no—
39#14519764—28no—
40#14519764—22no—
41#14519764—22no—
42#14519758—28no—
43#14519756—22no—
44#14519756—22no—
45#14519754—28no—
46#14519742—22no—
47#14519742D414yes—
48D4×Dic61—22no—
49D4×Dic61D414yes—
50#14519736—22no—
51#14519736—22no—
52C2×C4⋊Dic61C2×C411yescharacteristic
53C2×C4⋊Dic61D411yescharacteristic
54C2×C4⋊Dic61D411yescharacteristic
55C2×#11758196—21no—
56C2×#11758196—21no—
57C2×#11758196—21no—
58C2×C22⋊Dic61—21no—
59C2×C22⋊Dic61—21no—
60C2×C22⋊Dic61—21no—
61C2×C22⋊Dic61—21no—
62C2×C22⋊Dic61D411yescharacteristic
63C2×C22⋊Dic61D411yescharacteristic
64C2×C4×Dic61—21no—
65C2×C4×Dic61C2×C411yescharacteristic
66C2×C4×Dic61C2311yescharacteristic
67C22×D4×C61—21no—
68C22×D4×C61C2×C411yescharacteristic
69C23×C4×C61C2×C411yescharacteristic
70#121×C61—22no—
71#121×C61—22no—
72C2×C4⋊C4×C61C2×C411yescharacteristic
73C2×C22⋊C4×C61—21no—
74C24×C61—21no—
75C24×C61C22⋊C411yescharacteristic
76C2×D4×C61—24no—
77C2×D4×C61—21no—
78C2×D4×C61—24no—
79C2×D4×C61—82no—
80C2×D4×C61C22⋊C412yes—
81C22×C4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61—24no—
85C22×C4×C61C22⋊C411yescharacteristic
86C22×C4×C61C22×C411yescharacteristic
87C22×Dic61—21no—
88C22×Dic61—21no—
89C22×Dic61—24no—
90C22×Dic61—82no—
91C22×Dic61C22⋊C411yescharacteristic
92C22×Dic61C22⋊C411yescharacteristic
93C22×Dic61C2×D411yescharacteristic
94C22×Dic61C2×D411yescharacteristic
95C2×Dic122—24no—
96C2×Dic122—21no—
97C2×Dic122—24no—
98C2×Dic122—82no—
99C2×Dic122C22⋊C412yes—
100C2×#92420—42no—
101C2×#92420—42no—
102C4⋊C4×C61—21no—
103C4⋊C4×C61C22⋊C412yes—
104C22⋊C4×C61—24no—
105C22⋊Dic61—24no—
106C22⋊Dic61—24no—
107C22⋊Dic61—24no—
108C22⋊Dic61—24no—
109C22⋊Dic61—22no—
110C22⋊Dic61—22no—
111C4⋊Dic61—21no—
112C4⋊Dic61—22no—
113C4⋊Dic61—22no—
114C4⋊Dic61C22⋊C412yes—
115#11758196—24no—
116#11758196—24no—
117#11758196—24no—
118C4×Dic61—21no—
119C4×Dic61—21no—
120C4×Dic61—24no—
121C4×Dic61C2×D412yes—
122C4×Dic61C2×D412yes—
123C23×C61—21no—
124C23×C61—22no—
125C23×C61—22no—
126C23×C61—42no—
127C23×C61—42no—
128C23×C61—41no—
129C23×C61—41no—
130C23×C61—42no—
131C23×C61C2×C22⋊C411yescharacteristic
132D4×C61—24no—
133D4×C61—42no—
134D4×C61—84no—
135C2×C4×C61—22no—
136C2×C4×C61—21no—
137C2×C4×C61—22no—
138C2×C4×C61—42no—
139C2×C4×C61—42no—
140C2×C4×C61—42no—
141C2×C4×C61C2×C22⋊C411yescharacteristic
142C2×C4×C61C2×C22⋊C411yescharacteristic
143C2×C4×C61#13114yes—
144C2×Dic61—21no—
145C2×Dic61—22no—
146C2×Dic61—21no—
147C2×Dic61—22no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—42no—
151C2×Dic61—42no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—84no—
155C2×Dic61#12012yes—
156C2×Dic61#12012yes—
157Dic122—24no—
158Dic122—42no—
159Dic122—84no—
160#92420—82no—
161#92420—82no—
162#8325—611nomaximal, Sylow
163C22×C61—21no—
164C22×C61—21no—
165C22×C61—42no—
166C22×C61—41no—
167C22×C61—41no—
168C22×C61—42no—
169C22×C61—41no—
170C22×C61—42no—
171C22×C61—82no—
172C22×C61—84no—
173C22×C61#37911yescharacteristic
174C22×C61#40812yes—
175C4×C61—42no—
176C4×C61—41no—
177C4×C61#40812yes—
178C4×C61#40812yes—
179Dic61—22no—
180Dic61—22no—
181Dic61—41no—
182Dic61—42no—
183Dic61—41no—
184Dic61—42no—
185C2×#542—611no—
186#1479—612no—
187#1478—612no—
188#1458—611no—
189#1458—611no—
190C2×C61—22no—
191C2×C61—81no—
192C2×C61—82no—
193C2×C61#178011yescharacteristic
194C2×C61#179511yescharacteristic
195C2×C61#179611yescharacteristic
196#542—1224no—
197#542—1224no—
198C2×C4⋊Q8—611no—
199C2×#126—1221no—
200C2×#124—611no—
201C2×#124—1221no—
202C2×#123—1221no—
203C2×C22⋊Q8—611no—
204C2×C22⋊Q8—1221no—
205C2×#121#692311yescharacteristic
206C2×C4×Q8—611no—
207C2×C4×D4—611no—
208C2×#117—611no—
209#431—612no—
210#431—612no—
211Q8⋊C8—1224no—
212D4⋊C8—1224no—
213#324—1221no—
214#324—1221no—
215#323—1221no—
216#323—1221no—
217C61#832511yescharacteristic, complemented, Sylow
218C22×Q8—611no—
219C22×Q8—1221no—
220C22×D4—21no—
221C22×D4C2×#692311yescharacteristic
222C23×C4—1221no—
223C23×C4C2×#692311yescharacteristic
224C4⋊Q8—1222no—
225C4⋊Q8—1222no—
226#126—1228no—
227#124—1222no—
228#124—1228no—
229#124—1222no—
230#123—1228no—
231C22⋊Q8—1228no—
232C22⋊Q8—1222no—
233C22⋊Q8—1222no—
234#121—22no—
235#121—22no—
236C4×Q8—614no—
237C4×Q8—1222no—
238C4×D4—614no—
239C4×D4—1222no—
240#117—1222no—
241#117—1222no—
242C2×C4⋊C4—611no—
243C2×C4⋊C4—611no—
244C2×C4⋊C4—611no—
245C2×C4⋊C4—1221no—
246C2×C4⋊C4—1221no—
247C2×C4⋊C4—1221no—
248C2×C4⋊C4C2×#692311yescharacteristic
249C2×C22⋊C4—21no—
250C2×C22⋊C4—611no—
251C2×C22⋊C4—611no—
252C2×C22⋊C4—1221no—
253C2×C22⋊C4—1221no—
254C2×C22⋊C4—1221no—
255C2×C22⋊C4—1221no—
256C2×C42—611no—
257C2×C42—611no—
258C2×C42—1221no—
259C4⋊C8—1222no—
260C4⋊C8—1222no—
261C22⋊C8—1222no—
262C22⋊C8—1222no—
263C4×C8—1222no—
264C4×C8—1222no—
265C24—21no—
266C24C22⋊#692311yescharacteristic
267C2×Q8—612no—
268C2×Q8—1224no—
269C2×Q8—1221no—
270C2×Q8—1224no—
271C2×Q8—4882no—
272C2×D4—24no—
273C2×D4—21no—
274C2×D4—24no—
275C2×D4—82no—
276C2×D4C22⋊#692312yes—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—24no—
281C22×C4—611no—
282C22×C4—611no—
283C22×C4—611no—
284C22×C4—611no—
285C22×C4—1224no—
286C22×C4—1221no—
287C22×C4—1221no—
288C22×C4—4882no—
289C22×C4C22⋊#692311yescharacteristic
290C22×C4C22×#692311yescharacteristic
291C2×C8—2442no—
292C2×C8—2442no—
293C4⋊C4—21no—
294C4⋊C4—612no—
295C4⋊C4—1222no—
296C4⋊C4—1222no—
297C4⋊C4—1224no—
298C4⋊C4—1224no—
299C4⋊C4—1224no—
300C4⋊C4—1221no—
301C4⋊C4C22⋊#692312yes—
302C22⋊C4—24no—
303C22⋊C4—1222no—
304C22⋊C4—1222no—
305C22⋊C4—1224no—
306C22⋊C4—1224no—
307C22⋊C4—1224no—
308C22⋊C4—1224no—
309C42—612no—
310C42—612no—
311C42—1224no—
312C42—1221no—
313C42—1221no—
314C23—22no—
315C23—21no—
316C23—22no—
317C23—42no—
318C23—42no—
319C23—41no—
320C23—41no—
321C23—42no—
322C23C2×C22⋊#692311yescharacteristic
323Q8—1224no—
324Q8—4884no—
325Q8—2442no—
326D4—24no—
327D4—42no—
328D4—84no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—612no—
336C2×C4—612no—
337C2×C4—1221no—
338C2×C4—1222no—
339C2×C4—1221no—
340C2×C4—1222no—
341C2×C4—4884no—
342C2×C4—2441no—
343C2×C4—2441no—
344C2×C4—2442no—
345C2×C4—2442no—
346C2×C4—2441no—
347C2×C4—2441no—
348C2×C4C2×C22⋊#692311yescharacteristic
349C2×C4C2×C22⋊#692311yescharacteristic
350C2×C4#1451986414yes—
351C8—4882no—
352C8—4882no—
353C22—21no—
354C22—21no—
355C22—41no—
356C22—42no—
357C22—42no—
358C22—41no—
359C22—41no—
360C22—42no—
361C22—82no—
362C22—84no—
363C22#1684070311yescharacteristic
364C22#1684073512yes—
365C4—41no—
366C4—42no—
367C4—1222no—
368C4—1222no—
369C4—2442no—
370C4—2442no—
371C4—2441no—
372C4—2441no—
373C4#1684073512yes—
374C4#1684073612yes—
375C2—22no—
376C2—82no—
377C2—81no—
378C2#2146164311yescharacteristic
379C2#2146166911yescharacteristic
380C2#2146167211yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
Fitting, prime radicalC2×#121×C61C4
socleC22×C61#379
cosocleC2×C4×Dic61C23
hypercenterC2×#121#6923
hypocenterC61#8325
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. #21461123C2semidirect
2. #21461118C2semidirect
3. #21461050C2semidirect
4. #21461050C2semidirect
217. C61#8325semidirect

Maximal subgroups

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

subgroupindex
#214611232
#214611182
#214610502
C2×#168403792
#832561