Finite Group Databasebeta

← #30598709

Subgroups — #30598709

Order 15616.

976 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 976 conjugacy classes of subgroups — too many to draw. Only its 100 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#21461655C211yesmaximal, characteristic
2#21461648C211yesmaximal, characteristic
3#21461229C214yesmaximal, complemented
4C2×#16840336C211yesmaximal, characteristic
5C2×#14519838—21no—
6C2×#14519837—21no—
7C2×#14519836—21no—
8C2×#14519835—21no—
9#16840734C2212yes—
10#16840705C2212yes—
11#16840616—28no—
12C22×#11758222C2211yescharacteristic
13#16840336—28no—
14C2×#14519775—21no—
15C2×#14519766—21no—
16C2×#14519760—21no—
17C2×#14519760—21no—
18C2×#14519753C412yescomplemented
19C2×#14519753C2212yes—
20C2×#14519736C411yescharacteristic, complemented
21C23×Q8×C61C411yescharacteristic, complemented
22C2×#11758217—22no—
23C2×C22⋊#6923—22no—
24#14519838—42no—
25#14519837—42no—
26#14519836—42no—
27#14519835—42no—
28#14519834—24no—
29(C2×#6923)⋊C4—24no—
30C2×#11758222—24no—
31C2×#11758222—42no—
32C2×#11758222—42no—
33C22×Dic122D411yescharacteristic
34C22×D244D411yescharacteristic
35C22×C22⋊D61D411yescharacteristic
36C22×C22⋊D61D411yescharacteristic
37C22×C4×D61C2×C411yescharacteristic
38C22×C4×D61C2311yescharacteristic
39#14519775—28no—
40#14519766—28no—
41#14519760—28no—
42#14519760—28no—
43#14519753—24no—
44#14519753—24no—
45#14519753—24no—
46#14519753D418yes—
47#14519736—24no—
48C2×C4⋊Dic61C2×C412yes—
49C2×#11758196—22no—
50C2×#11758196—22no—
51C2×C22⋊Dic61—21no—
52C2×#11758198—22no—
53C2×#11758198—22no—
54C2×C4×Dic61—21no—
55C22×Q8×C61—24no—
56C22×Q8×C61—21no—
57C22×Q8×C61—42no—
58C22×Q8×C61—42no—
59C22×Q8×C61C2×C412yes—
60C23×C4×C61—21no—
61C23×C4×C61C2×C411yescharacteristic
62C24×C61C22⋊C411yescharacteristic
63C23×D61C22⋊C411yescharacteristic
64C23×D61C2×D411yescharacteristic
65C22×#6923—42no—
66C2×Q8×C61—24no—
67C2×Q8×C61—22no—
68C2×Q8×C61—416no—
69C2×Q8×C61—42no—
70C2×Q8×C61—42no—
71C2×Q8×C61—42no—
72C2×Q8×C61—42no—
73C2×Q8×C61C22⋊C414yes—
74C22×C4×C61—22no—
75C22×C4×C61—22no—
76C22×C4×C61—22no—
77C22×C4×C61—24no—
78C22×C4×C61—22no—
79C22×C4×C61—42no—
80C22×C4×C61—42no—
81C22×C4×C61C22⋊C411yescharacteristic
82C22×C4×C61C22×C411yescharacteristic
83C2×C22⋊D61—22no—
84C2×C22⋊D61—22no—
85C2×C22⋊D61—22no—
86C2×C22⋊D61—22no—
87C2×C22⋊D61—41no—
88C2×C22⋊D61—41no—
89C22×Dic61—22no—
90C22×Dic61C22⋊C411yescharacteristic
91C22×Dic61C2×D411yescharacteristic
92#11758222—416no—
93C2×D244—22no—
94C2×D244—22no—
95C2×D244—41no—
96C2×C4×D61—22no—
97C2×C4×D61—41no—
98C2×C4×D61—41no—
99C2×C4×D61—41no—
100C2×C4×D61—41no—
101C2×C4×D61C2×D414yes—
102C2×Dic122—22no—
103C2×Dic122—22no—
104C2×Dic122—41no—
105C22⋊#6923—42no—
106C22⋊#6923—42no—
107#11758217—42no—
108#11758217—42no—
109C2×#92420—42no—
110C22⋊Dic61—24no—
111#11758198—28no—
112#11758198—28no—
113C4⋊Dic61—22no—
114C4⋊Dic61C22⋊C414yes—
115#11758196—28no—
116#11758196—28no—
117C4×Dic61—24no—
118C23×C61—21no—
119C23×C61—21no—
120C23×C61—42no—
121C23×C61#9912yes—
122C23×C61C2×C22⋊C411yescharacteristic
123C22×D61—21no—
124C22×D61—21no—
125C22×D61—21no—
126C22×D61—21no—
127C22×D61—21no—
128C22×D61—42no—
129C22×D61—42no—
130C22×D61#12012yes—
131C2×#6923—82no—
132C2×#6923—84no—
133Q8×C61—48no—
134Q8×C61—44no—
135Q8×C61—44no—
136C2×C4×C61—28no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61—21no—
140C2×C4×C61—21no—
141C2×C4×C61—24no—
142C2×C4×C61—48no—
143C2×C4×C61—42no—
144C2×C4×C61—42no—
145C2×C4×C61#9912yes—
146C2×C4×C61C2×C22⋊C412yes—
147C22⋊D61—44no—
148C22⋊D61—44no—
149C2×Dic61—21no—
150C2×Dic61—21no—
151C2×Dic61—24no—
152C2×Dic61—21no—
153C2×Dic61—21no—
154C2×Dic61—21no—
155C2×Dic61—44no—
156C2×Dic61#12012yes—
157D244—44no—
158C4×D61—44no—
159C4×D61—44no—
160Dic122—44no—
161#92420—82no—
162#8520—611nomaximal, Sylow
163C22×C61—22no—
164C22×C61—22no—
165C22×C61—42no—
166C22×C61—42no—
167C22×C61—42no—
168C22×C61#37911yescharacteristic
169C22×C61C2×#9911yescharacteristic
170C22×C61C2×#9911yescharacteristic
171C2×D61—24no—
172C2×D61—22no—
173C2×D61—42no—
174C2×D61—41no—
175C2×D61—81no—
176C2×D61—81no—
177C2×D61—81no—
178C2×D61—81no—
179#6923—162no—
180C4×C61—22no—
181C4×C61—22no—
182C4×C61—44no—
183Dic61—22no—
184Dic61—44no—
185Dic61—41no—
186C2×#536—611no—
187#1785—611no—
188#1782—611no—
189#1511—614no—
190C2×C61—21no—
191C2×C61—21no—
192C2×C61—42no—
193C2×C61#178111yescharacteristic
194C2×C61#179612yes—
195D61—42no—
196D61—81no—
197C22×#41—611no—
198C23×Q8#692311yescharacteristic, complemented
199#536—1228no—
200C2×C4⋊Q8—1221no—
201C2×#124—1221no—
202C2×#123—1221no—
203C2×#123—1221no—
204C2×C22⋊Q8—612no—
205C2×C22⋊Q8—612no—
206C2×#117—611no—
207C2×#101—1221no—
208C2×#100—1221no—
209C2×#99—1221no—
210C2×#99—1221no—
211#407—612no—
212#381—612no—
213#328—1228no—
214C61#852011yescharacteristic, complemented, Sylow
215C2×#41—1224no—
216C2×#41—2442no—
217C2×#41—2442no—
218C22×Q8—24no—
219C22×Q8—21no—
220C22×Q8—42no—
221C22×Q8—42no—
222C22×Q8—611no—
223C22×Q8C2×#692312yes—
224C22×D4—611no—
225C22×D4—611no—
226C22×D4—611no—
227C23×C4—21no—
228C23×C4—611no—
229C23×C4—611no—
230C23×C4C2×#692311yescharacteristic
231C2×M16—1222no—
232C4⋊Q8—1228no—
233#124—1228no—
234#123—1228no—
235#123—1228no—
236C22⋊Q8—618no—
237C22⋊Q8—1224no—
238C22⋊Q8—1224no—
239C22⋊Q8—1224no—
240#117—1224no—
241C2×C4⋊C4—612no—
242C2×C4⋊C4—1222no—
243C2×C4⋊C4—1222no—
244C2×C22⋊C4—1222no—
245C2×C22⋊C4—1222no—
246C2×C22⋊C4—1221no—
247C2×C22⋊C4—1222no—
248C2×C42—1221no—
249#101—2442no—
250#100—2442no—
251#99—2442no—
252#99—2442no—
253C22⋊C8—1224no—
254#95—1224no—
255C24—611no—
256C24—611no—
257C24C22⋊#692311yescharacteristic
258#41—24416no—
259C2×Q8—24no—
260C2×Q8—22no—
261C2×Q8—416no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—1222no—
267C2×Q8—1222no—
268C2×Q8—2441no—
269C2×Q8C22⋊#692314yes—
270C2×D4—1222no—
271C2×D4—1222no—
272C2×D4—1222no—
273C2×D4—1222no—
274C2×D4—1222no—
275C2×D4—1222no—
276C2×D4—2441no—
277C2×D4—2441no—
278C2×D4—2441no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—611no—
287C22×C4—614no—
288C22×C4—611no—
289C22×C4—1222no—
290C22×C4—1222no—
291C22×C4—2441no—
292C22×C4—2441no—
293C22×C4—2441no—
294C22×C4—2441no—
295C22×C4—2442no—
296C22×C4C22⋊#692311yescharacteristic
297C22×C4C22×#692311yescharacteristic
298M16—2442no—
299M16—2442no—
300C2×C8—2442no—
301C4⋊C4—614no—
302C4⋊C4—1228no—
303C4⋊C4—1228no—
304C4⋊C4—1222no—
305C22⋊C4—1228no—
306C22⋊C4—1228no—
307C22⋊C4—1224no—
308C22⋊C4—2442no—
309C22⋊C4—2442no—
310C42—1224no—
311C23—21no—
312C23—21no—
313C23—42no—
314C23—612no—
315C23—1221no—
316C23—1221no—
317C23—1221no—
318C23—1221no—
319C23—1221no—
320C23—2442no—
321C23—2442no—
322C23#1451983512yes—
323C23C2×C22⋊#692311yescharacteristic
324Q8—44no—
325Q8—44no—
326Q8—48no—
327Q8—2444no—
328D4—2444no—
329D4—2444no—
330D4—2444no—
331C2×C4—28no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—24no—
337C2×C4—48no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—612no—
341C2×C4—1221no—
342C2×C4—1221no—
343C2×C4—1221no—
344C2×C4—1221no—
345C2×C4—1224no—
346C2×C4—1221no—
347C2×C4—4882no—
348C2×C4—4884no—
349C2×C4—2444no—
350C2×C4—2444no—
351C2×C4—2444no—
352C2×C4#1451983512yes—
353C2×C4C2×C22⋊#692312yes—
354C8—4882no—
355C22—22no—
356C22—22no—
357C22—42no—
358C22—42no—
359C22—42no—
360C22—1222no—
361C22—1224no—
362C22—4881no—
363C22—4881no—
364C22—4881no—
365C22—4881no—
366C22—2441no—
367C22—2442no—
368C22#1684070311yescharacteristic
369C22C2×#1451983511yescharacteristic
370C22C2×#1451983511yescharacteristic
371C4—22no—
372C4—22no—
373C4—44no—
374C4—1222no—
375C4—9762no—
376C4—2441no—
377C4—2444no—
378C2—21no—
379C2—21no—
380C2—42no—
381C2—4881no—
382C2—2442no—
383C2#2146164511yescharacteristic
384C2#2146167112yes—
subgroupG/H
centerC22#16840703
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC23×Q8×C61C4
socleC22×C61#379
prime radicalC22×#11758222C22
cosocleC22×C4×D61C23
hypercenterC23×Q8#6923
hypocenterC61#8520
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
3. #21461229C2semidirect
18. C2×#14519753C4semidirect
20. C2×#14519736C4semidirect
21. C23×Q8×C61C4semidirect
198. C23×Q8#6923semidirect
214. C61#8520semidirect

Maximal subgroups

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

subgroupindex
#214616552
#214616482
#214612292
C2×#168403362
#852061