Finite Group Databasebeta

← #26131795

Subgroups — #26131795

Order 12032.

962 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 962 conjugacy classes of subgroups — too many to draw. Only its 139 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
1C2×#15688922C212yesmaximal, complemented
2C2×#15688920C212yesmaximal, complemented
3#20243312C212yesmaximal, complemented
4C2×C47×#536C211yesmaximal, characteristic
5C2×#13093943—22no—
6C2×#13093943C2214yes—
7#15689087—22no—
8#15688922—24no—
9#15688920—24no—
10C22×#41×C47C411yescharacteristic
11C23×Q8×C47C2211yescharacteristic
12C47×#536—24no—
13C47×#536—24no—
14C2×C4⋊Q8×C47C411yescharacteristic
15C2×C4⋊Q8×C47C2211yescharacteristic
16C2×#124×C47C411yescharacteristic
17C2×#124×C47C2211yescharacteristic
18C2×#123×C47—22no—
19C2×C22⋊Q8×C47—22no—
20C2×#117×C47C411yescharacteristic
21C2×#10625329—24no—
22#13093943—24no—
23#13093943—24no—
24#13093943—44no—
25C22⋊(C47⋊C8)—44no—
26C2×#41×C47—22no—
27C2×#41×C47—22no—
28C2×#41×C47—42no—
29C2×#41×C47—42no—
30C22×Q8×C47—22no—
31C22×Q8×C47—22no—
32C22×Q8×C47—42no—
33C22×Q8×C47—42no—
34C22×Q8×C47C2×C411yescharacteristic
35C22×Q8×C47C2×C411yescharacteristic
36C22×Q8×C47D412yes—
37C22×Q8×C47C2311yescharacteristic
38C22×D4×C47—21no—
39C22×D4×C47C2×C411yescharacteristic
40C23×C4×C47—21no—
41C23×C4×C47C2×C411yescharacteristic
42C23×C4×C47D412yes—
43C4⋊Q8×C47—22no—
44C4⋊Q8×C47—22no—
45C4⋊Q8×C47—22no—
46C4⋊Q8×C47D414yes—
47#124×C47—22no—
48#124×C47—22no—
49#124×C47—22no—
50#124×C47D414yes—
51#123×C47—216no—
52C22⋊Q8×C47—216no—
53#117×C47—22no—
54#117×C47—22no—
55C2×C4⋊C4×C47—21no—
56C2×C4⋊C4×C47—22no—
57C2×C4⋊C4×C47—22no—
58C2×C22⋊C4×C47—21no—
59C2×C22⋊C4×C47—22no—
60C2×C22⋊C4×C47—22no—
61C2×C42×C47C2×C411yescharacteristic
62C2×C42×C47C2×C411yescharacteristic
63C24×C47—21no—
64C24×C47C22⋊C411yescharacteristic
65#41×C47—48no—
66#41×C47—48no—
67C2×Q8×C47—24no—
68C2×Q8×C47—21no—
69C2×Q8×C47—21no—
70C2×Q8×C47—21no—
71C2×Q8×C47—24no—
72C2×Q8×C47—41no—
73C2×Q8×C47—41no—
74C2×Q8×C47—42no—
75C2×Q8×C47—41no—
76C2×Q8×C47—41no—
77C2×Q8×C47—48no—
78C2×Q8×C47—48no—
79C2×Q8×C47—41no—
80C2×Q8×C47—41no—
81C2×Q8×C47C22⋊C412yes—
82C2×Q8×C47C2×D412yes—
83C2×Q8×C47C2×D412yes—
84C2×D4×C47—24no—
85C2×D4×C47—21no—
86C2×D4×C47—41no—
87C2×D4×C47—41no—
88C2×D4×C47—42no—
89C2×D4×C47—42no—
90C2×D4×C47C22⋊C412yes—
91C22×C4×C47—22no—
92C22×C4×C47—22no—
93C22×C4×C47—21no—
94C22×C4×C47—21no—
95C22×C4×C47—24no—
96C22×C4×C47—24no—
97C22×C4×C47—21no—
98C22×C4×C47—21no—
99C22×C4×C47—21no—
100C22×C4×C47—24no—
101C22×C4×C47—42no—
102C22×C4×C47—42no—
103C22×C4×C47—42no—
104C22×C4×C47C22⋊C412yes—
105C22×C4×C47C22⋊C411yescharacteristic
106C22×C4×C47C22×C411yescharacteristic
107C22×C4×C47C2×D412yes—
108C4⋊C4×C47—24no—
109C4⋊C4×C47—28no—
110C4⋊C4×C47—28no—
111C22⋊C4×C47—24no—
112C22⋊C4×C47—28no—
113C22⋊C4×C47—28no—
114C42×C47—21no—
115C42×C47—21no—
116C42×C47C22⋊C412yes—
117C42×C47C22⋊C412yes—
118#10625329—44no—
119#10625329—44no—
120C2×C47⋊C8—44no—
121C23×C47—21no—
122C23×C47—21no—
123C23×C47—22no—
124C23×C47—42no—
125C23×C47—44no—
126C23×C47—41no—
127C23×C47—41no—
128C23×C47#9912yes—
129C23×C47C2×C22⋊C411yescharacteristic
130Q8×C47—42no—
131Q8×C47—42no—
132Q8×C47—48no—
133Q8×C47—42no—
134Q8×C47—42no—
135Q8×C47—42no—
136Q8×C47—42no—
137D4×C47—42no—
138D4×C47—42no—
139D4×C47—48no—
140C2×C4×C47—24no—
141C2×C4×C47—24no—
142C2×C4×C47—21no—
143C2×C4×C47—22no—
144C2×C4×C47—22no—
145C2×C4×C47—21no—
146C2×C4×C47—21no—
147C2×C4×C47—21no—
148C2×C4×C47—22no—
149C2×C4×C47—22no—
150C2×C4×C47—22no—
151C2×C4×C47—22no—
152C2×C4×C47—22no—
153C2×C4×C47—22no—
154C2×C4×C47—48no—
155C2×C4×C47—48no—
156C2×C4×C47—42no—
157C2×C4×C47—42no—
158C2×C4×C47—41no—
159C2×C4×C47—41no—
160C2×C4×C47#9912yes—
161C2×C4×C47C2×C22⋊C411yescharacteristic
162C2×C4×C47C2×C22⋊C411yescharacteristic
163C2×C4×C47#12014yes—
164C47⋊C8—84no—
165#12702—471nomaximal, Sylow
166C22×C47—22no—
167C22×C47—24no—
168C22×C47—22no—
169C22×C47—42no—
170C22×C47—42no—
171C22×C47—44no—
172C22×C47—42no—
173C22×C47—82no—
174C22×C47—82no—
175C22×C47#37911yescharacteristic
176C22×C47C2×#9911yescharacteristic
177C22×C47C2×#9911yescharacteristic
178C4×C47—24no—
179C4×C47—22no—
180C4×C47—21no—
181C4×C47—21no—
182C4×C47—42no—
183C4×C47—42no—
184C4×C47—42no—
185C4×C47—42no—
186C2×#536D4711yescharacteristic
187C2×#356—472no—
188C2×#355—472no—
189#1784—472no—
190C2×C47—21no—
191C2×C47—21no—
192C2×C47—42no—
193C2×C47—82no—
194C2×C47#178111yescharacteristic
195C2×C47#213312yes—
196C22×#41Dic4711yescharacteristic
197C23×Q8C2×D4711yescharacteristic
198#536—24no—
199#536—24no—
200C2×C4⋊Q8Dic4711yescharacteristic
201C2×C4⋊Q8C2×D4711yescharacteristic
202C2×#124Dic4711yescharacteristic
203C2×#124C2×D4711yescharacteristic
204C2×#123—22no—
205C2×C22⋊Q8—22no—
206C2×#117Dic4711yescharacteristic
207C2×#101—474no—
208C2×#101—942no—
209#407—942no—
210#356—944no—
211#355—944no—
212C47#1270211yescharacteristic, complemented, Sylow
213C2×#41—22no—
214C2×#41—22no—
215C2×#41—42no—
216C2×#41—42no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—42no—
220C22×Q8—42no—
221C22×Q8C2×Dic4711yescharacteristic
222C22×Q8C2×Dic4711yescharacteristic
223C22×Q8C22⋊D4712yes—
224C22×Q8C22×D4711yescharacteristic
225C22×D4—21no—
226C22×D4C2×Dic4711yescharacteristic
227C23×C4—21no—
228C23×C4C2×Dic4711yescharacteristic
229C23×C4C22⋊D4712yes—
230C2×M16—944no—
231C4⋊Q8—22no—
232C4⋊Q8—22no—
233C4⋊Q8—22no—
234C4⋊Q8C22⋊D4714yes—
235#124—22no—
236#124—22no—
237#124—22no—
238#124C22⋊D4714yes—
239#123—216no—
240C22⋊Q8—216no—
241#117—22no—
242#117—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C42C2×Dic4711yescharacteristic
250C2×C42C2×Dic4711yescharacteristic
251#101—944no—
252#101—944no—
253#101—1884no—
254C22⋊C8—1884no—
255C24—21no—
256C24C22⋊Dic4711yescharacteristic
257#41—48no—
258#41—48no—
259C2×Q8—24no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—24no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—42no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—48no—
270C2×Q8—48no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8C22⋊Dic4712yes—
274C2×Q8C2×C22⋊D4712yes—
275C2×Q8C2×C22⋊D4712yes—
276C2×D4—24no—
277C2×D4—21no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4C22⋊Dic4712yes—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—24no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4C22⋊Dic4712yes—
297C22×C4C22⋊Dic4711yescharacteristic
298C22×C4C22×Dic4711yescharacteristic
299C22×C4C2×C22⋊D4712yes—
300M16—1884no—
301M16—1884no—
302C2×C8—1884no—
303C4⋊C4—24no—
304C4⋊C4—28no—
305C4⋊C4—28no—
306C22⋊C4—24no—
307C22⋊C4—28no—
308C22⋊C4—28no—
309C42—21no—
310C42—21no—
311C42C22⋊Dic4712yes—
312C42C22⋊Dic4712yes—
313C23—21no—
314C23—21no—
315C23—22no—
316C23—42no—
317C23—44no—
318C23—41no—
319C23—41no—
320C23#1309393912yes—
321C23C2×C22⋊Dic4711yescharacteristic
322Q8—42no—
323Q8—42no—
324Q8—48no—
325Q8—42no—
326Q8—42no—
327Q8—42no—
328Q8—42no—
329D4—42no—
330D4—42no—
331D4—48no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—48no—
347C2×C4—48no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4#1309393912yes—
353C2×C4C2×C22⋊Dic4711yescharacteristic
354C2×C4C2×C22⋊Dic4711yescharacteristic
355C2×C4#1309399314yes—
356C8—3764no—
357C22—22no—
358C22—24no—
359C22—22no—
360C22—42no—
361C22—42no—
362C22—44no—
363C22—42no—
364C22—82no—
365C22—82no—
366C22#1568897911yescharacteristic
367C22C2×#1309393911yescharacteristic
368C22C2×#1309393911yescharacteristic
369C4—24no—
370C4—22no—
371C4—21no—
372C4—21no—
373C4—42no—
374C4—42no—
375C4—42no—
376C4—42no—
377C2—21no—
378C2—21no—
379C2—42no—
380C2—82no—
381C2#2024330311yescharacteristic
382C2#2024507712yes—
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC22×Q8C22×D47
FittingC2×C47×#536C2
socleC22×C47C2×#99
prime radicalC22×#41×C47C4
cosocleC22×Q8×C47C23
hypercenterC2×#536D47
hypocenterC47#12702
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
1. C2×#15688922C2semidirect
2. C2×#15688920C2semidirect
3. #20243312C2semidirect
212. C47#12702semidirect

Maximal subgroups

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

subgroupindex
C2×#156889222
C2×#156889202
#202433122
C2×C47×#5362
#1270247