Finite Group Databasebeta

← #30598248

Subgroups — #30598248

Order 15616.

1030 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 1030 conjugacy classes of subgroups — too many to draw. Only its 152 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#21461676C211yesmaximal, characteristic
2#21461676C211yesmaximal, characteristic
3C2×#16840621C214yesmaximal, complemented
4C2×C4⋊Q8×D61C211yesmaximal, characteristic
5C2×#14519844C2212yes—
6C2×#14519844C2212yes—
7C2×#14519834—21no—
8C2×#14519834—21no—
9#16840628—21no—
10#16840628—21no—
11#16840621—28no—
12C22×Q8×D61—21no—
13C2×#14519777C411yescharacteristic
14C2×#14519776—21no—
15C2×#14519776C2212yes—
16C2×#14519775—21no—
17C4⋊Q8×D61—28no—
18C2×C4⋊C4×D61—21no—
19C2×C4⋊C4×D61C412yes—
20C2×C42×D61C2211yescharacteristic
21C2×C4⋊Q8×C61C411yescharacteristic
22C22×#92420—22no—
23C22×#92420—22no—
24#14519844—24no—
25#14519844—24no—
26#14519834—42no—
27#14519834—42no—
28C22×Dic122—21no—
29C22×Dic122—22no—
30C2×Q8×D61—28no—
31C2×Q8×D61—44no—
32C2×Q8×D61—42no—
33C2×Q8×D61—42no—
34C22×C4×D61—22no—
35C22×C4×D61C2×C411yescharacteristic
36C22×C4×D61D411yescharacteristic
37C22×C4×D61D411yescharacteristic
38#14519777—24no—
39#14519776—28no—
40#14519776—24no—
41#14519776D418yes—
42#14519775—28no—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61C2×C412yes—
45C2×#11758196—22no—
46C2×#11758196—22no—
47C4⋊C4×D61—28no—
48C4⋊C4×D61—24no—
49C4⋊C4×D61—24no—
50C2×C4×Dic61D411yescharacteristic
51C2×C4×Dic61D411yescharacteristic
52C2×C4×Dic61C2311yescharacteristic
53C42×D61—24no—
54C22×Q8×C61—21no—
55C4⋊Q8×C61—24no—
56C2×C4⋊C4×C61—21no—
57C2×C4⋊C4×C61C2×C412yes—
58C2×C42×C61C2×C411yescharacteristic
59C23×D61C22⋊C411yescharacteristic
60C2×Q8×C61—24no—
61C2×Q8×C61—42no—
62C2×Q8×C61—42no—
63C22×C4×C61—22no—
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22⋊C411yescharacteristic
66C22×C4×C61C22×C411yescharacteristic
67C22×Dic61—22no—
68C22×Dic61C22⋊C411yescharacteristic
69C22×Dic61C2×D411yescharacteristic
70C22×Dic61C2×D411yescharacteristic
71Q8×D61—416no—
72Q8×D61—416no—
73C2×C4×D61—22no—
74C2×C4×D61—22no—
75C2×C4×D61—22no—
76C2×C4×D61—22no—
77C2×C4×D61—22no—
78C2×C4×D61—22no—
79C2×C4×D61—22no—
80C2×C4×D61—28no—
81C2×C4×D61—44no—
82C2×C4×D61—44no—
83C2×C4×D61D812yes—
84C2×C4×D61SD1612yes—
85C2×C4×D61SD1612yes—
86C2×C4×D61Q1612yes—
87C2×Dic122—24no—
88C2×Dic122—28no—
89C2×Dic122—42no—
90C2×Dic122—42no—
91C2×Dic122—42no—
92C2×Dic122—42no—
93C2×Dic122—44no—
94C2×#92420—42no—
95C2×#92420—42no—
96C2×#92420—42no—
97C2×#92420—42no—
98C2×#92420—42no—
99C2×#92420—42no—
100C4⋊C4×C61—24no—
101C4⋊C4×C61—22no—
102C4⋊C4×C61C22⋊C414yes—
103C42×C61—22no—
104C4⋊Dic61—24no—
105C4⋊Dic61—22no—
106C4⋊Dic61C22⋊C414yes—
107#11758196—28no—
108#11758196—28no—
109C4×Dic61—22no—
110C4×Dic61—22no—
111C4×Dic61C2×D414yes—
112C23×C61C2×C22⋊C411yescharacteristic
113C22×D61—21no—
114C22×D61—21no—
115C22×D61—21no—
116C22×D61—21no—
117C22×D61#10112yes—
118C22×D61#10212yes—
119C22×D61#10312yes—
120Q8×C61—48no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—24no—
128C2×C4×C61—42no—
129C2×C4×C61—42no—
130C2×C4×C61#10212yes—
131C2×C4×C61#10312yes—
132C2×C4×C61C2×C22⋊C412yes—
133C2×Dic61—21no—
134C2×Dic61—21no—
135C2×Dic61—24no—
136C2×Dic61—42no—
137C2×Dic61—42no—
138C2×Dic61#10112yes—
139C2×Dic61#12012yes—
140C2×Dic61#12012yes—
141C2×Dic61C2×D811yescharacteristic
142C2×Dic61C2×SD1611yescharacteristic
143C2×Dic61C2×SD1611yescharacteristic
144C2×Dic61C2×Q1611yescharacteristic
145C2×Dic61AGL(1,ℤ/8)12yes—
146C2×Dic61#13712yes—
147C4×D61—28no—
148C4×D61—24no—
149C4×D61—24no—
150C4×D61—24no—
151C4×D61—24no—
152C4×D61—416no—
153Dic122—48no—
154Dic122—48no—
155Dic122—48no—
156#92420—82no—
157#92420—82no—
158#8340—611nomaximal, Sylow
159C22×C61#37911yescharacteristic
160C22×C61C2×#10111yescharacteristic
161C22×C61C2×#10111yescharacteristic
162C22×C61C2×#10211yescharacteristic
163C22×C61C2×#10311yescharacteristic
164C22×C61#41811yescharacteristic
165C22×C61#41911yescharacteristic
166C2×D61—22no—
167C2×D61—22no—
168C2×D61—22no—
169C2×D61—22no—
170C2×D61—22no—
171C2×D61—22no—
172C2×D61#33214yescomplemented
173C4×C61—22no—
174C4×C61—22no—
175C4×C61—22no—
176C4×C61—44no—
177Dic61—22no—
178Dic61—44no—
179Dic61#44712yes—
180Dic61#44812yes—
181Dic61#45012yes—
182Dic61#45112yes—
183C22×C4⋊Q8—611no—
184#1797—611no—
185#1797—611no—
186C2×#332—614no—
187C2×C61C2×#33211yescharacteristic
188C2×C61#154611yescharacteristic
189C2×C61#154911yescharacteristic
190C2×C61#155611yescharacteristic
191C2×C61#178411yescharacteristic
192C2×C61#178611yescharacteristic
193C2×C61#178711yescharacteristic
194D61—24no—
195C23×Q8—1221no—
196C2×C4⋊Q8—612no—
197C2×C4⋊Q8—611no—
198C2×C4⋊Q8—1221no—
199C2×C4⋊Q8—1221no—
200C2×C4⋊Q8—1228no—
201C2×C4⋊Q8#692311yescharacteristic
202C22×C4⋊C4—612no—
203C22×C4⋊C4—1221no—
204C22×C42—611no—
205C2×C4⋊C8—612no—
206C2×C4⋊C8—612no—
207C2×C22⋊C8—1221no—
208C2×C22⋊C8—1221no—
209#336—1221no—
210#336—1221no—
211#332—1228no—
212C61#834011yescharacteristic, complemented, Sylow
213C22×Q8—21no—
214C22×Q8—1222no—
215C22×Q8—1228no—
216C22×Q8—1221no—
217C22×Q8—2442no—
218C22×Q8—2442no—
219C22×Q8—2444no—
220C23×C4—611no—
221C23×C4—611no—
222C23×C4—611no—
223C23×C4—1222no—
224C22×C8—1222no—
225C22×C8—1222no—
226C4⋊Q8—24no—
227C4⋊Q8—618no—
228C4⋊Q8—1224no—
229C4⋊Q8—1224no—
230C4⋊Q8—1228no—
231C4⋊Q8—1228no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—612no—
234C2×C4⋊C4—1224no—
235C2×C4⋊C4—1228no—
236C2×C4⋊C4—1222no—
237C2×C4⋊C4—1222no—
238C2×C4⋊C4—1221no—
239C2×C4⋊C4—1224no—
240C2×C4⋊C4C2×#692312yes—
241C2×C42—611no—
242C2×C42—611no—
243C2×C42—611no—
244C2×C42—1224no—
245C2×C42C2×#692311yescharacteristic
246C4⋊C8—1224no—
247C4⋊C8—1224no—
248C22⋊C8—2442no—
249C22⋊C8—2442no—
250C24—611no—
251C2×Q8—24no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—1228no—
255C2×Q8—1224no—
256C2×Q8—2442no—
257C2×Q8—2442no—
258C2×Q8—2442no—
259C2×Q8—2444no—
260C2×Q8—24416no—
261C2×Q8—24416no—
262C2×Q8—2442no—
263C22×C4—22no—
264C22×C4—611no—
265C22×C4—611no—
266C22×C4—611no—
267C22×C4—612no—
268C22×C4—612no—
269C22×C4—612no—
270C22×C4—612no—
271C22×C4—1222no—
272C22×C4—1222no—
273C22×C4—1222no—
274C22×C4—1222no—
275C22×C4—1222no—
276C22×C4—1222no—
277C22×C4—1228no—
278C22×C4—1222no—
279C22×C4—1222no—
280C22×C4—2444no—
281C22×C4—2444no—
282C22×C4C22⋊#692311yescharacteristic
283C22×C4C22⋊#692311yescharacteristic
284C22×C4C22×#692311yescharacteristic
285C2×C8—2442no—
286C2×C8—2442no—
287C2×C8—2442no—
288C2×C8—2442no—
289C2×C8—2442no—
290C2×C8—2442no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4—614no—
294C4⋊C4—1228no—
295C4⋊C4—1228no—
296C4⋊C4—1224no—
297C4⋊C4—1222no—
298C4⋊C4C22⋊#692314yes—
299C42—22no—
300C42—614no—
301C42—1222no—
302C42—1222no—
303C23—612no—
304C23—612no—
305C23—612no—
306C23—1221no—
307C23—1221no—
308C23—1221no—
309C23—1221no—
310C23C2×C22⋊#692311yescharacteristic
311Q8—48no—
312Q8—2448no—
313Q8—2448no—
314Q8—2448no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—24no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—611no—
325C2×C4—611no—
326C2×C4—611no—
327C2×C4—611no—
328C2×C4—612no—
329C2×C4—612no—
330C2×C4—612no—
331C2×C4—612no—
332C2×C4—612no—
333C2×C4—1224no—
334C2×C4—1224no—
335C2×C4—1224no—
336C2×C4—1224no—
337C2×C4—1228no—
338C2×C4—1221no—
339C2×C4—1221no—
340C2×C4—1224no—
341C2×C4—24416no—
342C2×C4—2442no—
343C2×C4—2442no—
344C2×C4#1451983912yes—
345C2×C4#1451984012yes—
346C2×C4C2×C22⋊#692312yes—
347C8—4882no—
348C8—4882no—
349C22—614no—
350C22—1222no—
351C22—1222no—
352C22—1222no—
353C22—1222no—
354C22—1222no—
355C22—1222no—
356C22#1684070311yescharacteristic
357C22D61⋊#10111yescharacteristic
358C22D61⋊#10111yescharacteristic
359C22C2×#1451983911yescharacteristic
360C22C2×#1451984011yescharacteristic
361C22#1684075411yescharacteristic
362C22#1684075511yescharacteristic
363C4—22no—
364C4—22no—
365C4—22no—
366C4—44no—
367C4—612no—
368C4—612no—
369C4—612no—
370C4—612no—
371C4—1222no—
372C4—2444no—
373C2—1224no—
374C2D61⋊#33211yescharacteristic
375C2#2146132911yescharacteristic
376C2#2146133511yescharacteristic
377C2#2146134911yescharacteristic
378C2#2146165411yescharacteristic
379C2#2146165711yescharacteristic
380C2#2146165811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4⋊Q8×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenterC2×C4⋊Q8#6923
hypocenterC61#8340
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×#16840621C2semidirect
172. C2×D61#332semidirect
212. C61#8340semidirect

Maximal subgroups

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

subgroupindex
#214616762
C2×#168406212
C2×C4⋊Q8×D612
#834061