Finite Group Databasebeta

← #26119879

Subgroups — #26119879

Order 12032.

440 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 97 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×#15689328C211yesmaximal, characteristic
2#20242160C211yesmaximal, characteristic, complemented
3#20242061C211yesmaximal, characteristic, complemented
4#20242061C211yesmaximal, characteristic, complemented
5#20241779C211yesmaximal, characteristic
6#20241283C211yesmaximal, characteristic, complemented
7C47×#1800C211yesmaximal, characteristic
8C22×#10625335—21no—
9C2×#13094024C2211yescharacteristic
10#15689328—22no—
11#15689328—22no—
12#15689176C2211yescharacteristic
13#15689176C2211yescharacteristic
14C2×C4⋊(C47⋊C8)C2211yescharacteristic
15C2×#13093947—21no—
16#15688873—22no—
17#15688870—22no—
18#15688870—22no—
19D4⋊(C47⋊C8)—22no—
20C2×C4×D4×C47C2211yescharacteristic
21C47×#423C2211yescharacteristic
22C47×#423C2211yescharacteristic
23C2×#100×C47—21no—
24C47×#343—21no—
25C22×Dic94—21no—
26C2×#10625335—24no—
27C2×#10625335—42no—
28C2×#10625335—42no—
29C22×C47⋊C8—21no—
30#13094024—21no—
31#13094024—21no—
32#13094024—21no—
33#13094024D412yes—
34C2×C4⋊Dic47—21no—
35C2×C4⋊Dic47—21no—
36C4⋊(C47⋊C8)—21no—
37C4⋊(C47⋊C8)—21no—
38C4⋊(C47⋊C8)D412yes—
39C4⋊(C47⋊C8)D412yes—
40#13093947—24no—
41C22⋊(C47⋊C8)—21no—
42C22⋊(C47⋊C8)—21no—
43C4×C47⋊C8—21no—
44C4×C47⋊C8—21no—
45C22×D4×C47D411yescharacteristic
46C23×C4×C47—21no—
47C2×M16×C47—21no—
48C2×M16×C47—21no—
49C4×D4×C47—21no—
50C4×D4×C47—21no—
51C4×D4×C47—21no—
52C4×D4×C47—21no—
53C2×C4⋊C4×C47D411yescharacteristic
54C2×C22⋊C4×C47—21no—
55C2×C42×C47C2311yescharacteristic
56C4⋊C8×C47—21no—
57C4⋊C8×C47—21no—
58C4⋊C8×C47—21no—
59C4⋊C8×C47—21no—
60#100×C47—42no—
61C24×C47—21no—
62C2×D4×C47—21no—
63C2×D4×C47—21no—
64C2×D4×C47—21no—
65C2×D4×C47—21no—
66C2×D4×C47—21no—
67C2×D4×C47SD1612yes—
68C22×C4×C47—21no—
69C22×C4×C47—41no—
70C22×C4×C47—41no—
71C22×C4×C47—82no—
72C22×C4×C47C2×D411yescharacteristic
73C22×C4×C47#4111yescharacteristic
74C22×C4×C47#4111yescharacteristic
75C22×Dic47—41no—
76C2×Dic94—22no—
77C2×Dic94—41no—
78C2×Dic94—41no—
79C2×Dic94—41no—
80C2×Dic94—41no—
81C2×Dic94—41no—
82M16×C47—41no—
83M16×C47—41no—
84M16×C47—41no—
85M16×C47—41no—
86C2×C8×C47—41no—
87C2×C8×C47—41no—
88C4⋊C4×C47—21no—
89C4⋊C4×C47—21no—
90C22⋊C4×C47—41no—
91C22⋊C4×C47—41no—
92C42×C47C2×D411yescharacteristic
93C42×C47C2×D411yescharacteristic
94C42×C47C2×D411yescharacteristic
95C42×C47#4111yescharacteristic
96#10625335—48no—
97C4⋊Dic47—22no—
98C4⋊Dic47—22no—
99C4⋊Dic47—41no—
100C4⋊Dic47—41no—
101C2×C47⋊C8—22no—
102C2×C47⋊C8—22no—
103C2×C47⋊C8—41no—
104C2×C47⋊C8—41no—
105C2×C47⋊C8—41no—
106C23×C47—22no—
107C23×C47—41no—
108C23×C47—41no—
109C23×C47—81no—
110C23×C47—81no—
111C23×C47#12311yescharacteristic
112D4×C47—22no—
113D4×C47—22no—
114D4×C47—41no—
115D4×C47—41no—
116C2×C4×C47—22no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
119C2×C4×C47—82no—
120C2×C4×C47—82no—
121C2×C4×C47—81no—
122C2×C4×C47#12011yescharacteristic
123C2×C4×C47#12111yescharacteristic
124C2×C4×C47#12111yescharacteristic
125C2×C4×C47#12111yescharacteristic
126C2×C4×C47#12311yescharacteristic
127C2×C4×C47#12411yescharacteristic
128C2×C4×C47C2×SD1611yescharacteristic
129C2×C4×C47#13511yescharacteristic
130C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
131C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
132C2×Dic47—42no—
133C2×Dic47—81no—
134C2×Dic47—81no—
135Dic94—42no—
136Dic94—42no—
137Dic94—81no—
138Dic94—81no—
139C8×C47—81no—
140C8×C47—81no—
141C47⋊C8—42no—
142C47⋊C8—42no—
143C47⋊C8—81no—
144#10445—471nomaximal, Sylow
145C22×C47—21no—
146C22×C47—21no—
147C22×C47—42no—
148C22×C47—82no—
149C22×C47—82no—
150C22×C47—81no—
151C22×C47—81no—
152C22×C47#39411yescharacteristic
153C22×C47#48111yescharacteristic
154C22×C47#48211yescharacteristic
155C4×C47—21no—
156C4×C47—21no—
157C4×C47—21no—
158C4×C47—81no—
159C4×C47#44911yescharacteristic
160C4×C47#45011yescharacteristic
161C4×C47#46511yescharacteristic
162C4×C47#46911yescharacteristic
163C4×C47#47112yes—
164Dic47—82no—
165C2×#461—471no—
166#1800D4711yescharacteristic
167#1570—471no—
168#1546—471no—
169#1546—471no—
170#1478—471no—
171#1301—471no—
172C2×C47—21no—
173C2×C47—21no—
174C2×C47—82no—
175C2×C47#203411yescharacteristic
176C2×C47#204611yescharacteristic
177C2×C47#204911yescharacteristic
178C22×SD16—941no—
179C2×C4⋊Q8—471no—
180C2×C4×D4C2×D4711yescharacteristic
181#461—942no—
182#461—942no—
183#431—471no—
184#431—471no—
185#423C2×D4711yescharacteristic
186#423C2×D4711yescharacteristic
187C2×C4⋊C8—471no—
188C2×#103—941no—
189C2×#100—21no—
190#343—21no—
191#334—942no—
192#332—942no—
193#332—942no—
194D4⋊C8—942no—
195C47#1044511yescharacteristic, complemented, Sylow
196C22×Q8—941no—
197C22×D4C22⋊D4711yescharacteristic
198C23×C4—21no—
199C2×SD16—944no—
200C2×SD16—1882no—
201C2×SD16—1882no—
202C2×M16—21no—
203C2×M16—21no—
204C22×C8—941no—
205C4⋊Q8—472no—
206C4⋊Q8—941no—
207C4⋊Q8—941no—
208C4⋊Q8—941no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4—21no—
212C4×D4—21no—
213C2×C4⋊C4—941no—
214C2×C4⋊C4—941no—
215C2×C4⋊C4C22⋊D4711yescharacteristic
216C2×C22⋊C4—21no—
217C2×C42C22×D4711yescharacteristic
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—472no—
223C4⋊C8—472no—
224C4⋊C8—941no—
225C4⋊C8—941no—
226#103—944no—
227#100—42no—
228C22⋊C8—941no—
229C22⋊C8—941no—
230C4×C8—941no—
231C4×C8—941no—
232C24—21no—
233C2×Q8—942no—
234C2×Q8—1881no—
235C2×Q8—1881no—
236C2×Q8—1881no—
237C2×Q8—1881no—
238C2×Q8—1881no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4#1062533612yes—
245C22×C4—21no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—82no—
249C22×C4—1881no—
250C22×C4#1062535011yescharacteristic
251C22×C4#1062535011yescharacteristic
252C22×C4C2×C22⋊D4711yescharacteristic
253SD16—1888no—
254M16—41no—
255M16—41no—
256M16—41no—
257M16—41no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—942no—
261C2×C8—942no—
262C2×C8—1881no—
263C2×C8—1881no—
264C2×C8—1881no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—942no—
268C4⋊C4—942no—
269C4⋊C4—1881no—
270C4⋊C4—1881no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C42D4×D4711yescharacteristic
274C42D4×D4711yescharacteristic
275C42D4×D4711yescharacteristic
276C42#1062535211yescharacteristic
277C23—22no—
278C23—41no—
279C23—41no—
280C23—81no—
281C23—81no—
282C23#1309400711yescharacteristic
283Q8—3761no—
284Q8—3761no—
285Q8—1882no—
286Q8—1882no—
287D4—22no—
288D4—22no—
289D4—41no—
290D4—41no—
291C2×C4—22no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—81no—
297C2×C4—3761no—
298C2×C4—3761no—
299C2×C4—1882no—
300C2×C4#1309399211yescharacteristic
301C2×C4#1309399611yescharacteristic
302C2×C4#1309399711yescharacteristic
303C2×C4#1309399711yescharacteristic
304C2×C4#1309400611yescharacteristic
305C2×C4#1309401011yescharacteristic
306C2×C4C2×#1062533611yescharacteristic
307C2×C4#1309404411yescharacteristic
308C2×C4#1309404911yescharacteristic
309C2×C4#1309405011yescharacteristic
310C8—81no—
311C8—81no—
312C8—3761no—
313C8—1882no—
314C8—1882no—
315C22—21no—
316C22—21no—
317C22—42no—
318C22—82no—
319C22—82no—
320C22—81no—
321C22—81no—
322C22#1568903811yescharacteristic
323C22#1568944111yescharacteristic
324C22#1568944711yescharacteristic
325C4—21no—
326C4—21no—
327C4—21no—
328C4—81no—
329C4—3762no—
330C4#1568926611yescharacteristic
331C4#1568927311yescharacteristic
332C4#1568935411yescharacteristic
333C4#1568937411yescharacteristic
334C4#1568938512yes—
335C2—21no—
336C2—21no—
337C2—82no—
338C2#2024457711yescharacteristic
339C2#2024464911yescharacteristic
340C2#2024466811yescharacteristic
subgroupG/H
centerC22#15689038
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1800C2
socleC22×C47#394
prime radicalC22×D4×C47D4
hypercenter#1800D47
hypocenterC47#10445
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
2. #20242160C2semidirect
3. #20242061C2semidirect
4. #20242061C2semidirect
6. #20241283C2semidirect
195. C47#10445semidirect

Maximal subgroups

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

subgroupindex
C2×#156893282
#202421602
#202420612
#202417792
#202412832
C47×#18002
#1044547