Finite Group Databasebeta

← #26108809

Subgroups — #26108809

Order 12032.

694 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 115 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×#15689751C211yesmaximal, characteristic
2#20242106C212yesmaximal, complemented
3#20242014C212yesmaximal, complemented
4#20241722C211yesmaximal, characteristic
5C47×#1464C211yesmaximal, characteristic
6#15689751—24no—
7#15689751—24no—
8C2×#13094024C411yescharacteristic
9C2×#13094024C2211yescharacteristic
10C2×#13094023—21no—
11C2×#13094016C411yescharacteristic
12C2×#13094016C2211yescharacteristic
13C2×#13094015—21no—
14C2×#13094001—21no—
15C2×#13094001—21no—
16C2×#13093983C411yescharacteristic
17C22×C4⋊Dic47C411yescharacteristic
18#15689180C2212yes—
19#15688870—24no—
20#15688864—24no—
21#15688849—21no—
22#15688847—21no—
23C2×#117×C47C2211yescharacteristic
24C47×#432C2212yes—
25C47×#324—21no—
26C47×#323—21no—
27C22×Dic94—21no—
28C22×Dic94—21no—
29C23×Dic47—21no—
30#13094024—22no—
31#13094024—22no—
32#13094024—22no—
33#13094024D414yes—
34#13094023—28no—
35#13094016—22no—
36#13094016—22no—
37#13094016—22no—
38#13094016D414yes—
39#13094015—28no—
40#13094001—28no—
41#13094001—28no—
42#13093983—22no—
43#13093983—22no—
44C2×C4⋊Dic47—21no—
45C2×C4⋊Dic47—21no—
46C2×C4⋊Dic47—21no—
47C2×C4⋊Dic47—22no—
48C2×C4⋊Dic47—22no—
49C2×C4⋊Dic47C2×C411yescharacteristic
50C2×C4⋊Dic47C2×C411yescharacteristic
51C2×C4⋊Dic47C2×C411yescharacteristic
52C2×C4⋊Dic47C2×C411yescharacteristic
53C2×Dic47⋊C4—21no—
54C2×Dic47⋊C4—21no—
55C2×Dic47⋊C4—21no—
56C2×Dic47⋊C4—21no—
57C2×C22⋊Dic47—21no—
58C2×C4×Dic47—21no—
59C4⋊(C47⋊C8)—22no—
60C22⋊(C47⋊C8)—22no—
61#13093935—22no—
62C23×C4×C47C2×C411yescharacteristic
63#117×C47—22no—
64#117×C47—22no—
65C2×C4⋊C4×C47D411yescharacteristic
66C2×C4⋊C4×C47D411yescharacteristic
67C2×C22⋊C4×C47D411yescharacteristic
68C2×C22⋊C4×C47D411yescharacteristic
69C2×C42×C47C2×C411yescharacteristic
70C2×C42×C47C2311yescharacteristic
71C4⋊C8×C47—22no—
72C22⋊C8×C47—22no—
73#97×C47—22no—
74C24×C47C22⋊C411yescharacteristic
75C22×C4×C47—22no—
76C22×C4×C47—22no—
77C22×C4×C47—21no—
78C22×C4×C47—21no—
79C22×C4×C47C22⋊C411yescharacteristic
80C22×C4×C47C22⋊C411yescharacteristic
81C22×C4×C47C22⋊C411yescharacteristic
82C22×C4×C47C22×C411yescharacteristic
83C22×C4×C47C2×D411yescharacteristic
84C22×C4×C47C2×D411yescharacteristic
85C22×Dic47—21no—
86C22×Dic47—21no—
87C22×Dic47—21no—
88C22×Dic47—21no—
89C22×Dic47—24no—
90C22×Dic47—82no—
91C2×Dic94—24no—
92C2×Dic94—24no—
93C2×Dic94—82no—
94C2×Dic94—82no—
95C2×C8×C47—42no—
96C4⋊C4×C47—22no—
97C4⋊C4×C47—22no—
98C22⋊C4×C47—22no—
99C22⋊C4×C47—22no—
100C42×C47—21no—
101C42×C47—21no—
102C42×C47C2×D412yes—
103C42×C47C2×D412yes—
104C22⋊Dic47—24no—
105C4⋊Dic47—24no—
106C4⋊Dic47—24no—
107C4⋊Dic47—21no—
108C4⋊Dic47—21no—
109C4⋊Dic47—21no—
110C4⋊Dic47—21no—
111C4⋊Dic47—24no—
112C4⋊Dic47C22⋊C412yes—
113C4⋊Dic47C22⋊C412yes—
114C4⋊Dic47C22⋊C412yes—
115C4⋊Dic47C22⋊C412yes—
116Dic47⋊C4—24no—
117Dic47⋊C4—24no—
118Dic47⋊C4—24no—
119Dic47⋊C4—24no—
120C4×Dic47—24no—
121C2×C47⋊C8—42no—
122C23×C47—21no—
123C23×C47—21no—
124C23×C47—41no—
125C23×C47—41no—
126C23×C47#10112yes—
127C23×C47C2×C22⋊C411yescharacteristic
128C2×C4×C47—24no—
129C2×C4×C47—24no—
130C2×C4×C47—21no—
131C2×C4×C47—21no—
132C2×C4×C47—21no—
133C2×C4×C47—21no—
134C2×C4×C47—21no—
135C2×C4×C47—21no—
136C2×C4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47#10112yes—
141C2×C4×C47C2×C22⋊C411yescharacteristic
142C2×C4×C47C2×C22⋊C411yescharacteristic
143C2×C4×C47#12012yes—
144C2×C4×C47#12012yes—
145C2×Dic47—22no—
146C2×Dic47—22no—
147C2×Dic47—22no—
148C2×Dic47—22no—
149C2×Dic47—42no—
150C2×Dic47—42no—
151C2×Dic47—42no—
152C2×Dic47—42no—
153C2×Dic47—84no—
154Dic94—84no—
155Dic94—84no—
156C8×C47—82no—
157C47⋊C8—82no—
158#8525—471nomaximal, Sylow
159C22×C47—21no—
160C22×C47—21no—
161C22×C47—22no—
162C22×C47—42no—
163C22×C47—42no—
164C22×C47—41no—
165C22×C47—41no—
166C22×C47#37911yescharacteristic
167C22×C47C2×#10111yescharacteristic
168C22×C47C2×#10111yescharacteristic
169C4×C47—22no—
170C4×C47—22no—
171C4×C47—22no—
172C4×C47—21no—
173C4×C47—21no—
174C4×C47—41no—
175C4×C47—41no—
176Dic47—42no—
177Dic47—42no—
178Dic47—42no—
179Dic47—42no—
180C2×#544—471no—
181#1556—472no—
182#1534—472no—
183#1464—471no—
184#1464D4711yescharacteristic
185C2×C47—21no—
186C2×C47—21no—
187C2×C47—42no—
188C2×C47#178411yescharacteristic
189C2×C47#179511yescharacteristic
190C2×C47#179611yescharacteristic
191#544—944no—
192#544—944no—
193C2×C4⋊Q8—471no—
194C2×C4⋊Q8—471no—
195C2×C4⋊Q8—941no—
196C2×#125—471no—
197C2×#125—471no—
198C2×#125—941no—
199C2×C22⋊Q8—941no—
200C2×C22⋊Q8—941no—
201C2×#117—471no—
202C2×#117C2×D4711yescharacteristic
203C22×C4⋊C4—471no—
204#432—472no—
205#432C2×D4712yes—
206#332—944no—
207#330—944no—
208#324—21no—
209#324—941no—
210#323—21no—
211#323—941no—
212C47#852511yescharacteristic, complemented, Sylow
213C22×Q8—941no—
214C22×Q8—941no—
215C23×C4—941no—
216C23×C4C4×D4711yescharacteristic
217C4⋊Q8—474no—
218C4⋊Q8—942no—
219C4⋊Q8—942no—
220C4⋊Q8—948no—
221C4⋊Q8—942no—
222#125—474no—
223#125—942no—
224#125—942no—
225#125—948no—
226#125—942no—
227C22⋊Q8—948no—
228C22⋊Q8—948no—
229#117—22no—
230#117—22no—
231#117—942no—
232#117—942no—
233C2×C4⋊C4—471no—
234C2×C4⋊C4—471no—
235C2×C4⋊C4—471no—
236C2×C4⋊C4—471no—
237C2×C4⋊C4—942no—
238C2×C4⋊C4—942no—
239C2×C4⋊C4—941no—
240C2×C4⋊C4—941no—
241C2×C4⋊C4—941no—
242C2×C4⋊C4—941no—
243C2×C4⋊C4—941no—
244C2×C4⋊C4—941no—
245C2×C4⋊C4—941no—
246C2×C4⋊C4D18811yescharacteristic
247C2×C4⋊C4C22⋊D4711yescharacteristic
248C2×C22⋊C4—941no—
249C2×C22⋊C4D18811yescharacteristic
250C2×C22⋊C4C22⋊D4711yescharacteristic
251C2×C42—941no—
252C2×C42C4×D4711yescharacteristic
253C2×C42C22×D4711yescharacteristic
254C4⋊C8—22no—
255C4⋊C8—942no—
256C22⋊C8—22no—
257C22⋊C8—942no—
258#97—22no—
259#97—942no—
260C24(C2×D47)⋊C411yescharacteristic
261C2×Q8—944no—
262C2×Q8—944no—
263C2×Q8—3762no—
264C2×Q8—3762no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—944no—
270C22×C4—941no—
271C22×C4—941no—
272C22×C4—941no—
273C22×C4—941no—
274C22×C4—3762no—
275C22×C4(C2×D47)⋊C411yescharacteristic
276C22×C4(C2×D47)⋊C411yescharacteristic
277C22×C4(C2×D47)⋊C411yescharacteristic
278C22×C4C2×C4×D4711yescharacteristic
279C22×C4C2×D18811yescharacteristic
280C22×C4C2×C22⋊D4711yescharacteristic
281C2×C8—42no—
282C2×C8—1882no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—472no—
286C4⋊C4—472no—
287C4⋊C4—472no—
288C4⋊C4—472no—
289C4⋊C4—944no—
290C4⋊C4—944no—
291C4⋊C4—944no—
292C4⋊C4—944no—
293C4⋊C4—941no—
294C4⋊C4—941no—
295C4⋊C4—941no—
296C4⋊C4—941no—
297C4⋊C4—944no—
298C4⋊C4—944no—
299C4⋊C4—944no—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—944no—
303C42—21no—
304C42—21no—
305C42—944no—
306C42D4×D4712yes—
307C42D4×D4712yes—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23#1309394212yes—
313C23C2×(C2×D47)⋊C411yescharacteristic
314Q8—3764no—
315Q8—3764no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—942no—
329C2×C4—942no—
330C2×C4—942no—
331C2×C4—942no—
332C2×C4—3764no—
333C2×C4—1882no—
334C2×C4—1882no—
335C2×C4—1882no—
336C2×C4—1882no—
337C2×C4#1309394212yes—
338C2×C4C22⋊C4×D4711yescharacteristic
339C2×C4C22⋊C4×D4711yescharacteristic
340C2×C4#1309399212yes—
341C2×C4#1309399412yes—
342C8—82no—
343C8—3762no—
344C22—21no—
345C22—21no—
346C22—22no—
347C22—42no—
348C22—42no—
349C22—41no—
350C22—41no—
351C22#1568897811yescharacteristic
352C22C2×#1309394211yescharacteristic
353C22C2×#1309394211yescharacteristic
354C4—22no—
355C4—21no—
356C4—21no—
357C4—22no—
358C4—22no—
359C4—41no—
360C4—41no—
361C4—1882no—
362C4—1882no—
363C4—1882no—
364C4—1882no—
365C2—21no—
366C2—21no—
367C2—42no—
368C2#2024331111yescharacteristic
369C2#2024335611yescharacteristic
370C2#2024336411yescharacteristic
subgroupG/H
centerC22#15688978
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1464C2
socleC22×C47#379
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1464D47
hypocenterC47#8525
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1464

Complements

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

NH ≅ G/Ntype
2. #20242106C2semidirect
3. #20242014C2semidirect
212. C47#8525semidirect

Maximal subgroups

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

subgroupindex
C2×#156897512
#202421062
#202420142
#202417222
C47×#14642
#852547