Finite Group Databasebeta

← #26112239

Subgroups — #26112239

Order 12032.

532 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 99 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#20252417C211yesmaximal, characteristic
2#20249305C211yesmaximal, characteristic
3#20241907C212yesmaximal, complemented
4#20241241C212yesmaximal, complemented
5C4×#99×C47C211yesmaximal, characteristic
6C2×#13094003C2211yescharacteristic
7#15689195—24no—
8#15689180—21no—
9#15689087C2212yes—
10C2×C22⋊(C47⋊C8)—21no—
11#15689053C411yescharacteristic, complemented
12#15689051—21no—
13#15689048C411yescharacteristic, complemented
14#15689043C411yescharacteristic, complemented
15#15689037—21no—
16#15689035C411yescharacteristic, complemented
17#15689007—21no—
18#15688996—21no—
19#15688986C2211yescharacteristic
20#15688859—24no—
21C2×C4×D4×C47C2211yescharacteristic
22C2×#99×C47—21no—
23C4×C22⋊C4×C47C2212yes—
24C47×#342—21no—
25C22×Dic94C2×C411yescharacteristic
26C2×#10625329—22no—
27C22×C47⋊C8—22no—
28#13094003—22no—
29#13094003—22no—
30C2×C4⋊Dic47C2×C411yescharacteristic
31C2×Dic47⋊C4—21no—
32C2×Dic47⋊C4—21no—
33C2×Dic47⋊C4—21no—
34C2×C22⋊Dic47—21no—
35C2×C22⋊Dic47—21no—
36C2×C22⋊Dic47—21no—
37C2×C22⋊Dic47—21no—
38C4⋊(C47⋊C8)—22no—
39C22⋊(C47⋊C8)—22no—
40C22⋊(C47⋊C8)—22no—
41C22⋊(C47⋊C8)D414yes—
42#13093935—22no—
43C47⋊#95—21no—
44C47⋊#95—21no—
45C47⋊#95—21no—
46C47⋊#95—21no—
47C47⋊#95—21no—
48C47⋊#95C2×C411yescharacteristic
49C47⋊#95C2×C411yescharacteristic
50C22×D4×C47D411yescharacteristic
51C23×C4×C47D411yescharacteristic
52C23×C4×C47C2311yescharacteristic
53C4×D4×C47—22no—
54C4×D4×C47—22no—
55C2×C4⋊C4×C47C2×C411yescharacteristic
56C2×C22⋊C4×C47—22no—
57C2×C22⋊C4×C47C2×C411yescharacteristic
58C2×C22⋊C4×C47D411yescharacteristic
59C2×C42×C47—22no—
60C2×C42×C47D411yescharacteristic
61#99×C47—24no—
62#95×C47—22no—
63C24×C47C22⋊C411yescharacteristic
64C24×C47C2×D411yescharacteristic
65C2×D4×C47—22no—
66C2×D4×C47—22no—
67C2×D4×C47—41no—
68C22×C4×C47—21no—
69C22×C4×C47—21no—
70C22×C4×C47—42no—
71C22×C4×C47—81no—
72C22×C4×C47—81no—
73C22×C4×C47C22⋊C411yescharacteristic
74C22×C4×C47C22⋊C411yescharacteristic
75C22×C4×C47C22⋊C411yescharacteristic
76C22×C4×C47C22×C411yescharacteristic
77C22×C4×C47C2×D412yes—
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47#4112yes—
80C22×Dic47—21no—
81C22×Dic47—21no—
82C22×Dic47—21no—
83C22×Dic47—21no—
84C2×Dic94—22no—
85C2×Dic94—81no—
86C4⋊C4×C47—22no—
87C22⋊C4×C47—22no—
88C22⋊C4×C47—22no—
89C22⋊C4×C47—24no—
90C22⋊C4×C47—42no—
91C42×C47—21no—
92C42×C47—21no—
93C42×C47—82no—
94C22⋊Dic47—42no—
95C22⋊Dic47—42no—
96C22⋊Dic47—81no—
97C22⋊Dic47—81no—
98C4⋊Dic47—22no—
99Dic47⋊C4—42no—
100Dic47⋊C4—41no—
101Dic47⋊C4—41no—
102Dic47⋊C4—81no—
103#10625329—82no—
104C2×C47⋊C8—24no—
105C2×C47⋊C8—82no—
106C23×C47—21no—
107C23×C47—21no—
108C23×C47—21no—
109C23×C47—21no—
110C23×C47—41no—
111C23×C47—41no—
112C23×C47—41no—
113C23×C47—41no—
114C23×C47#10412yes—
115C23×C47C2×C22⋊C411yescharacteristic
116C23×C47#12112yes—
117D4×C47—44no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—42no—
124C2×C4×C47—42no—
125C2×C4×C47—41no—
126C2×C4×C47—41no—
127C2×C4×C47—42no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47—44no—
131C2×C4×C47—81no—
132C2×C4×C47—82no—
133C2×C4×C47—82no—
134C2×C4×C47C4×D412yes—
135C2×C4×C47#12011yescharacteristic
136C2×C4×C47#12311yescharacteristic
137C2×Dic47—41no—
138C2×Dic47—41no—
139C2×Dic47—41no—
140C2×Dic47—41no—
141C2×Dic47—41no—
142C2×Dic47—41no—
143C2×Dic47—41no—
144C2×Dic47—41no—
145C2×Dic47—41no—
146C2×Dic47—41no—
147C2×Dic47—41no—
148C2×Dic47—41no—
149Dic94—82no—
150C47⋊C8—44no—
151#9183—471nomaximal, Sylow
152C22×C47—21no—
153C22×C47—21no—
154C22×C47—22no—
155C22×C47—22no—
156C22×C47—22no—
157C22×C47—22no—
158C22×C47—42no—
159C22×C47—41no—
160C22×C47—41no—
161C22×C47—81no—
162C22×C47—81no—
163C22×C47—81no—
164C22×C47#38611yescharacteristic
165C22×C47C2×#10411yescharacteristic
166C22×C47#42111yescharacteristic
167C4×C47—21no—
168C4×C47—22no—
169C4×C47—21no—
170C4×C47—41no—
171C4×C47—41no—
172C4×C47—41no—
173C4×C47—84no—
174Dic47—81no—
175Dic47—81no—
176Dic47—81no—
177Dic47—81no—
178#2973—471no—
179#2638—471no—
180C4×#99D4711yescharacteristic
181#1508—472no—
182#1284—472no—
183C2×C47—21no—
184C2×C47—21no—
185C2×C47—42no—
186C2×C47—42no—
187C2×C47#187111yescharacteristic
188C2×C47#190911yescharacteristic
189C2×C47#191311yescharacteristic
190C2×C22⋊Q8—471no—
191C2×C4×D4C2×D4711yescharacteristic
192#435—944no—
193#432—941no—
194C2×#99—21no—
195#407—472no—
196C2×C22⋊C8—941no—
197#398—471no—
198#397—471no—
199#397—941no—
200#395—471no—
201#394—941no—
202#393—471no—
203#387—941no—
204#385—941no—
205#381—471no—
206C4×C22⋊C4C2×D4712yes—
207#342—21no—
208#328—944no—
209C47#918311yescharacteristic, complemented, Sylow
210C22×Q8—471no—
211C22×D4C22⋊D4711yescharacteristic
212C23×C4D18811yescharacteristic
213C23×C4C22×D4711yescharacteristic
214C2×M16—942no—
215C22×C8—942no—
216C22⋊Q8—942no—
217C22⋊Q8—942no—
218C4×D4—22no—
219C4×D4—22no—
220C2×C4⋊C4—471no—
221C2×C4⋊C4—941no—
222C2×C4⋊C4—941no—
223C2×C4⋊C4—941no—
224C2×C4⋊C4C4×D4711yescharacteristic
225C2×C22⋊C4—22no—
226C2×C22⋊C4—941no—
227C2×C22⋊C4—941no—
228C2×C22⋊C4—941no—
229C2×C22⋊C4—941no—
230C2×C22⋊C4C4×D4711yescharacteristic
231C2×C22⋊C4C22⋊D4711yescharacteristic
232C2×C42—22no—
233C2×C42D18811yescharacteristic
234C4⋊C8—942no—
235#99—24no—
236C22⋊C8—474no—
237C22⋊C8—942no—
238C22⋊C8—942no—
239#97—942no—
240#95—22no—
241#95—471no—
242#95—471no—
243#95—941no—
244#95—941no—
245#95—941no—
246#95—941no—
247#95—941no—
248C24(C2×D47)⋊C411yescharacteristic
249C24C2×C22⋊D4711yescharacteristic
250C2×Q8—942no—
251C2×Q8—3761no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—41no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—42no—
258C22×C4—81no—
259C22×C4—81no—
260C22×C4—941no—
261C22×C4—941no—
262C22×C4—941no—
263C22×C4—941no—
264C22×C4(C2×D47)⋊C411yescharacteristic
265C22×C4(C2×D47)⋊C411yescharacteristic
266C22×C4(C2×D47)⋊C411yescharacteristic
267C22×C4C2×C4×D4711yescharacteristic
268C22×C4C2×D18811yescharacteristic
269C22×C4D4×D4712yes—
270C22×C4#1062535212yes—
271M16—3762no—
272C2×C8—944no—
273C2×C8—3762no—
274C4⋊C4—22no—
275C4⋊C4—942no—
276C4⋊C4—3761no—
277C4⋊C4—1882no—
278C4⋊C4—1881no—
279C4⋊C4—1881no—
280C22⋊C4—22no—
281C22⋊C4—22no—
282C22⋊C4—24no—
283C22⋊C4—42no—
284C22⋊C4—3761no—
285C22⋊C4—3761no—
286C22⋊C4—1882no—
287C22⋊C4—1882no—
288C42—21no—
289C42—21no—
290C42—82no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23#1309395212yes—
300C23C2×(C2×D47)⋊C411yescharacteristic
301C23#1309399612yes—
302Q8—3762no—
303D4—44no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—44no—
317C2×C4—81no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—1881no—
321C2×C4—1881no—
322C2×C4—1881no—
323C2×C4—1881no—
324C2×C4—1881no—
325C2×C4—1881no—
326C2×C4—1881no—
327C2×C4—1881no—
328C2×C4—1881no—
329C2×C4—1881no—
330C2×C4—1881no—
331C2×C4—1881no—
332C2×C4(C22⋊D47)⋊C412yes—
333C2×C4#1309399411yescharacteristic
334C2×C4#1309400811yescharacteristic
335C8—1884no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—42no—
343C22—41no—
344C22—41no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22#1568900011yescharacteristic
349C22C2×#1309395211yescharacteristic
350C22#1568914111yescharacteristic
351C4—21no—
352C4—21no—
353C4—22no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—84no—
358C4—3761no—
359C4—3761no—
360C4—3761no—
361C4—3761no—
362C2—21no—
363C2—21no—
364C2—42no—
365C2—42no—
366C2#2024368811yescharacteristic
367C2#2024390811yescharacteristic
368C2#2024392811yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC4×#99×C47C2
socleC22×C47#386
prime radicalC22×D4×C47D4
cosocleC23×C4×C47C23
hypercenterC4×#99D47
hypocenterC47#9183
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
3. #20241907C2semidirect
4. #20241241C2semidirect
11. #15689053C4semidirect
13. #15689048C4semidirect
14. #15689043C4semidirect
16. #15689035C4semidirect
209. C47#9183semidirect

Maximal subgroups

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

subgroupindex
#202524172
#202493052
#202419072
#202412412
C4×#99×C472
#918347