Finite Group Databasebeta

← #30617824

Subgroups — #30617824

Order 15616.

1194 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 1194 conjugacy classes of subgroups — too many to draw. Only its 216 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#21462499C212yesmaximal, complemented
2#21461963C214yesmaximal, complemented
3#21461962C214yesmaximal, complemented
4#21461656C212yesmaximal
5#21461653C212yesmaximal, complemented
6C2×#16840334C211yesmaximal, characteristic
7#16840786—28no—
8#16840765C2214yes—
9C2×#14519838C2218yes—
10C2×#14519838C2214yes—
11C2×#14519837C2214yes—
12#16840732C2214yes—
13C4×#11758217C2214yes—
14C22×#11758224C2211yescharacteristic
15C23×Dic122C2211yescharacteristic
16#16840334—28no—
17C2×#14519776C2212yes—
18C2×#14519763C2212yes—
19C2×#14519761C411yescharacteristic
20C2×#14519760C411yescharacteristic
21C2×#14519758C411yescharacteristic
22C2×#14519755C412yes—
23C2×#14519754C412yes—
24C2×#14519733C2211yescharacteristic
25C2×#123×C61C411yescharacteristic
26C2×#11758217C2318yes—
27#14519844—24no—
28C4⋊#92420—24no—
29#14519838—48no—
30#14519838—44no—
31#14519837—44no—
32C22⋊#92420—28no—
33#14519832—24no—
34C4×#92420—24no—
35C2×#11758224—24no—
36C2×#11758224—44no—
37C22×Dic122—24no—
38C22×Dic122—44no—
39C22×Dic122C2×C412yes—
40C22×Dic122C2312yes—
41C22×Dic122C2311yescharacteristic
42C22×C22⋊D61C2×C411yescharacteristic
43C22×C22⋊D61C2311yescharacteristic
44C23×Dic61C2×C411yescharacteristic
45C23×Dic61C2×C411yescharacteristic
46C23×Dic61C2311yescharacteristic
47C22×C4×D61C2×C411yescharacteristic
48#14519776—24no—
49#14519776—24no—
50#14519763—24no—
51#14519763—24no—
52#14519761—24no—
53#14519760—24no—
54#14519758—24no—
55#14519755—28no—
56#14519754—28no—
57#14519733D418yes—
58C2×C4⋊Dic61C2×C412yes—
59C2×#11758196C2×C412yes—
60C2×#11758196C2×C412yes—
61C2×#11758196C2×C412yes—
62C2×C22⋊Dic61C2×C412yes—
63C2×C22⋊Dic61C2×C411yescharacteristic
64C2×#11758198C2×C412yes—
65C2×#11758198C2×C412yes—
66C2×C4×Dic61C2312yes—
67C22×D4×C61C2×C411yescharacteristic
68C23×C4×C61C2×C411yescharacteristic
69#123×C61—24no—
70C2×C4⋊C4×C61C2×C412yes—
71C2×C22⋊C4×C61C2×C412yes—
72C2×C22⋊C4×C61C2×C411yescharacteristic
73C24×C61C22×C411yescharacteristic
74C24×C61C22×C411yescharacteristic
75C23×D61C22×C411yescharacteristic
76C2×D4×C61—22no—
77C2×D4×C61—42no—
78C22×C4×C61—22no—
79C22×C4×C61—42no—
80C22×C4×C61C22×C411yescharacteristic
81C22×C4×C61C22×C412yes—
82C22×C4×C61C22×C412yes—
83C2×C22⋊D61—22no—
84C2×C22⋊D61—22no—
85C2×C22⋊D61—24no—
86C2×C22⋊D61—42no—
87C22×Dic61—24no—
88C22×Dic61—22no—
89C22×Dic61—22no—
90C22×Dic61—42no—
91C22×Dic61—42no—
92C22×Dic61C22×C411yescharacteristic
93C22×Dic61C22×C412yes—
94C22×Dic61C22×C411yescharacteristic
95C22×Dic61C22×C412yes—
96C22×Dic61C2×D414yes—
97C22×Dic61C2411yescharacteristic
98#11758224—416no—
99C2×C4×D61—22no—
100C2×C4×D61—42no—
101C2×Dic122—24no—
102C2×Dic122—22no—
103C2×Dic122—22no—
104C2×Dic122—22no—
105C2×Dic122—28no—
106C2×Dic122—24no—
107C2×Dic122—44no—
108C2×Dic122—416no—
109#11758217—28no—
110#11758217—28no—
111C2×#92420—28no—
112C4⋊C4×C61—24no—
113C22⋊C4×C61—24no—
114C22⋊C4×C61C22⋊C414yes—
115C22⋊Dic61—24no—
116C22⋊Dic61C22⋊C414yes—
117#11758198—24no—
118#11758198—24no—
119C4⋊Dic61—24no—
120#11758196—24no—
121#11758196—24no—
122#11758196C22⋊C418yes—
123C4×Dic61C2×D414yes—
124C4×Dic61C2×D414yes—
125C23×C61—22no—
126C23×C61—21no—
127C23×C61—22no—
128C23×C61—42no—
129C23×C61—42no—
130C23×C61C2×C22⋊C412yes—
131C23×C61C23×C411yescharacteristic
132C22×D61—21no—
133C22×D61—22no—
134C22×D61—42no—
135D4×C61—44no—
136C2×C4×C61—21no—
137C2×C4×C61—22no—
138C2×C4×C61—24no—
139C2×C4×C61—22no—
140C2×C4×C61—42no—
141C2×C4×C61—44no—
142C2×C4×C61C2×C22⋊C414yes—
143C22⋊D61—44no—
144C22⋊D61—44no—
145C2×Dic61—22no—
146C2×Dic61—28no—
147C2×Dic61—21no—
148C2×Dic61—22no—
149C2×Dic61—24no—
150C2×Dic61—22no—
151C2×Dic61—42no—
152C2×Dic61—44no—
153C2×Dic61—44no—
154C2×Dic61C2×C22⋊C414yes—
155C2×Dic61C2×C22⋊C412yes—
156C2×Dic61C22×D412yes—
157C2×Dic61H(5,𝔽2)14yes—
158C4×D61—44no—
159Dic122—48no—
160Dic122—44no—
161Dic122—44no—
162Dic122—44no—
163#92420—48no—
164#21802—611nomaximal, Sylow
165C22×C61—22no—
166C22×C61—22no—
167C22×C61—24no—
168C22×C61—44no—
169C22×C61—41no—
170C22×C61—42no—
171C22×C61—82no—
172C22×C61C22×C22⋊C411yescharacteristic
173C22×C61#51812yes—
174C2×D61—41no—
175C2×D61—42no—
176C2×D61—82no—
177C4×C61—24no—
178C4×C61—42no—
179C4×C61—41no—
180Dic61—22no—
181Dic61—24no—
182Dic61—42no—
183Dic61—41no—
184Dic61#53514yes—
185C2×#536—611no—
186#2921—612no—
187#1967—614no—
188#1966—614no—
189#1785—612no—
190#1784—612no—
191C2×C61—22no—
192C2×C61—42no—
193C2×C61—81no—
194C2×C61#230911yescharacteristic
195C2×C61#288912yes—
196D61—81no—
197C22×#41—611no—
198C23×Q8—611no—
199#536—1228no—
200C2×C4⋊Q8—612no—
201C2×#124—612no—
202C2×#123—611no—
203C2×#123—611no—
204C2×#123—611no—
205C2×#123#692311yescharacteristic
206C2×C22⋊Q8—612no—
207C2×C22⋊Q8—612no—
208C2×#117—611no—
209#433—1228no—
210#423—614no—
211C2×#101—614no—
212C2×#101—618no—
213C2×#100—614no—
214#407—614no—
215C4×M16—614no—
216C61#2180211yescharacteristic, complemented, Sylow
217C2×#41—1224no—
218C2×#41—2444no—
219C22×Q8—612no—
220C22×Q8—611no—
221C22×Q8—612no—
222C22×Q8—1224no—
223C22×Q8—2444no—
224C22×D4—611no—
225C22×D4—611no—
226C22×D4C2×#692311yescharacteristic
227C23×C4—611no—
228C23×C4—611no—
229C23×C4—611no—
230C23×C4—611no—
231C23×C4C2×#692311yescharacteristic
232C2×M16—618no—
233C4⋊Q8—1224no—
234C4⋊Q8—1224no—
235#124—1224no—
236#124—1224no—
237#123—24no—
238#123—1224no—
239#123—1224no—
240#123—1224no—
241C22⋊Q8—1228no—
242C22⋊Q8—1228no—
243#117—618no—
244C2×C4⋊C4—612no—
245C2×C4⋊C4—612no—
246C2×C4⋊C4—612no—
247C2×C4⋊C4—612no—
248C2×C4⋊C4C2×#692312yes—
249C2×C22⋊C4—611no—
250C2×C22⋊C4—612no—
251C2×C22⋊C4—612no—
252C2×C22⋊C4—612no—
253C2×C22⋊C4C2×#692312yes—
254C2×C22⋊C4C2×#692311yescharacteristic
255C2×C42—612no—
256C4⋊C8—1224no—
257C4⋊C8—1224no—
258#101—2448no—
259#101—2444no—
260#100—2444no—
261C22⋊C8—1228no—
262#97—1224no—
263C4×C8—1224no—
264C24—611no—
265C24C22×#692311yescharacteristic
266C24C22×#692311yescharacteristic
267#41—24416no—
268C2×Q8—1228no—
269C2×Q8—1224no—
270C2×Q8—1224no—
271C2×Q8—1222no—
272C2×Q8—1222no—
273C2×Q8—1222no—
274C2×Q8—2444no—
275C2×Q8—24416no—
276C2×D4—22no—
277C2×D4—42no—
278C2×D4—1224no—
279C2×D4—1222no—
280C2×D4—1222no—
281C2×D4—2442no—
282C22×C4—22no—
283C22×C4—42no—
284C22×C4—611no—
285C22×C4—614no—
286C22×C4—611no—
287C22×C4—612no—
288C22×C4—612no—
289C22×C4—611no—
290C22×C4—1222no—
291C22×C4—1224no—
292C22×C4—1222no—
293C22×C4—1222no—
294C22×C4—2442no—
295C22×C4—2442no—
296C22×C4—2442no—
297C22×C4C22×#692312yes—
298C22×C4C22×#692311yescharacteristic
299C22×C4C22×#692312yes—
300M16—1228no—
301M16—1228no—
302C2×C8—1228no—
303C4⋊C4—24no—
304C4⋊C4—618no—
305C4⋊C4—1224no—
306C4⋊C4—1224no—
307C4⋊C4—1224no—
308C22⋊C4—24no—
309C22⋊C4—614no—
310C22⋊C4—1224no—
311C22⋊C4—1224no—
312C22⋊C4—1224no—
313C22⋊C4C22⋊#692314yes—
314C42—614no—
315C42—614no—
316C23—22no—
317C23—21no—
318C23—22no—
319C23—42no—
320C23—42no—
321C23—1222no—
322C23—1221no—
323C23—2442no—
324C23C2×C22⋊#692312yes—
325C23C23×#692311yescharacteristic
326Q8—2444no—
327Q8—2444no—
328Q8—2444no—
329Q8—2448no—
330D4—44no—
331D4—2444no—
332D4—2444no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—612no—
340C2×C4—614no—
341C2×C4—614no—
342C2×C4—612no—
343C2×C4—1228no—
344C2×C4—1222no—
345C2×C4—1222no—
346C2×C4—1221no—
347C2×C4—1222no—
348C2×C4—1224no—
349C2×C4—2444no—
350C2×C4—2442no—
351C2×C4—2444no—
352C2×C4—2444no—
353C2×C4C2×C22⋊#692314yes—
354C8—2448no—
355C22—22no—
356C22—22no—
357C22—24no—
358C22—44no—
359C22—41no—
360C22—42no—
361C22—82no—
362C22—4882no—
363C22—2441no—
364C22—2442no—
365C22C22×C22⋊#692311yescharacteristic
366C22#1684082912yes—
367C4—24no—
368C4—42no—
369C4—41no—
370C4—614no—
371C4—1222no—
372C4—1224no—
373C4—2441no—
374C4—2442no—
375C2—22no—
376C2—42no—
377C2—81no—
378C2—4881no—
379C2#2146229211yescharacteristic
380C2#2146243112yes—
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×#123×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC22×#11758224C22
cosocleC22×Dic61C24
hypercenterC2×#123#6923
hypocenterC61#21802
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. #21462499C2semidirect
2. #21461963C2semidirect
3. #21461962C2semidirect
5. #21461653C2semidirect
216. C61#21802semidirect

Maximal subgroups

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

subgroupindex
#214624992
#214619632
#214619622
#214616562
#214616532
C2×#168403342
#2180261