Finite Group Databasebeta

← #26109540

Subgroups — #26109540

Order 12032.

414 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252100C211yesmaximal, characteristic
2#20243525C211yesmaximal, characteristic, complemented
3C2×#15688873C211yesmaximal, characteristic, complemented
4#20242065C211yesmaximal, characteristic
5#20241263C211yesmaximal, characteristic, complemented
6#20241262C211yesmaximal, characteristic, complemented
7C47×#1547C211yesmaximal, characteristic
8C2×#13094031—21no—
9#15689201—21no—
10#15689201C412yes—
11D4×C47⋊C8—21no—
12D4×C47⋊C8C412yes—
13#15689147C2211yescharacteristic
14#15689147C2211yescharacteristic
15C2×C4⋊(C47⋊C8)C2211yescharacteristic
16#15689090—21no—
17C2×C4×C47⋊C8C2211yescharacteristic
18#15688887—21no—
19#15688887—21no—
20#15688873—21no—
21#15688873—21no—
22#15688869—21no—
23#15688867—21no—
24C47×#532C2211yescharacteristic
25C47×#423C2211yescharacteristic
26C2×C4⋊C8×C47C2211yescharacteristic
27C47×#332—21no—
28C47×#331—21no—
29#13094031—42no—
30#13094031—42no—
31C2×#10625329—21no—
32C2×#10625329—22no—
33C2×#10625329—21no—
34C22×C47⋊C8—22no—
35C22×C47⋊C8—21no—
36C22×C47⋊C8D411yescharacteristic
37C22×C47⋊C8D411yescharacteristic
38C4⋊(C47⋊C8)—21no—
39C4⋊(C47⋊C8)—21no—
40C4⋊(C47⋊C8)—21no—
41C4⋊(C47⋊C8)—21no—
42C4⋊(C47⋊C8)—21no—
43C4⋊(C47⋊C8)C2×C412yes—
44C22⋊(C47⋊C8)—22no—
45C22⋊(C47⋊C8)—22no—
46C4×C47⋊C8C2×C411yescharacteristic
47C4×C47⋊C8C2×C411yescharacteristic
48C4×C47⋊C8D411yescharacteristic
49C4×C47⋊C8D411yescharacteristic
50C2×#41×C47—21no—
51C2×M16×C47—21no—
52C22×C8×C47—21no—
53C4⋊Q8×C47D411yescharacteristic
54#127×C47D411yescharacteristic
55#124×C47—21no—
56#121×C47—21no—
57#121×C47Q812yes—
58C4×D4×C47—21no—
59C4×D4×C47C2×C412yes—
60C2×C42×C47C2311yescharacteristic
61C4⋊C8×C47—21no—
62C4⋊C8×C47—21no—
63C4⋊C8×C47—21no—
64C4⋊C8×C47—21no—
65#41×C47—41no—
66#41×C47—41no—
67#41×C47—41no—
68#41×C47—41no—
69C2×Q8×C47—21no—
70C2×D4×C47—22no—
71C2×D4×C47—21no—
72C2×D4×C47—21no—
73C2×D4×C47C4⋊C412yes—
74C22×C4×C47—22no—
75C22×C4×C47C2×D411yescharacteristic
76C22×C4×C47C2×Q811yescharacteristic
77C22×C4×C47#4111yescharacteristic
78M16×C47—41no—
79M16×C47—41no—
80C2×C8×C47—41no—
81C2×C8×C47—41no—
82C2×C8×C47—81no—
83C4⋊C4×C47—21no—
84C4⋊C4×C47C4⋊C412yes—
85C22⋊C4×C47—22no—
86C22⋊C4×C47—22no—
87C42×C47C22×C411yescharacteristic
88C42×C47C2×D411yescharacteristic
89C42×C47C2×D411yescharacteristic
90C42×C47#4111yescharacteristic
91#10625329—44no—
92#10625329—41no—
93#10625329—41no—
94#10625329—41no—
95#10625329—41no—
96C2×C47⋊C8—21no—
97C2×C47⋊C8—21no—
98C2×C47⋊C8—21no—
99C2×C47⋊C8—21no—
100C2×C47⋊C8—22no—
101C2×C47⋊C8—44no—
102C2×C47⋊C8—41no—
103C2×C47⋊C8—41no—
104C2×C47⋊C8—81no—
105C2×C47⋊C8C22⋊C411yescharacteristic
106C2×C47⋊C8C22⋊C411yescharacteristic
107C2×C47⋊C8C22⋊C411yescharacteristic
108C2×C47⋊C8C22⋊C411yescharacteristic
109C23×C47—22no—
110C23×C47C22⋊Q811yescharacteristic
111Q8×C47—41no—
112Q8×C47—41no—
113D4×C47—42no—
114D4×C47—42no—
115D4×C47—42no—
116D4×C47—41no—
117D4×C47—41no—
118D4×C47—41no—
119D4×C47—41no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—22no—
123C2×C4×C47—41no—
124C2×C4×C47—42no—
125C2×C4×C47—42no—
126C2×C4×C47C2×C22⋊C411yescharacteristic
127C2×C4×C47C2×C4⋊C411yescharacteristic
128C2×C4×C47#11711yescharacteristic
129C2×C4×C47#12111yescharacteristic
130C2×C4×C47#12111yescharacteristic
131C2×C4×C47C22⋊Q811yescharacteristic
132C2×C4×C47#13511yescharacteristic
133C2×C4×C47#13511yescharacteristic
134C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
135C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
136C8×C47—81no—
137C8×C47—81no—
138C47⋊C8—21no—
139C47⋊C8—21no—
140C47⋊C8—21no—
141C47⋊C8—21no—
142C47⋊C8—42no—
143C47⋊C8—81no—
144C47⋊C8—81no—
145#8686—471nomaximal, Sylow
146C22×C47—21no—
147C22×C47—21no—
148C22×C47—42no—
149C22×C47—42no—
150C22×C47—42no—
151C22×C47#38011yescharacteristic
152C22×C47#47811yescharacteristic
153C22×C47#47811yescharacteristic
154C4×C47—21no—
155C4×C47—21no—
156C4×C47—21no—
157C4×C47—42no—
158C4×C47#41611yescharacteristic
159C4×C47#41811yescharacteristic
160C4×C47#46311yescharacteristic
161C4×C47#46311yescharacteristic
162C4×C47#47111yescharacteristic
163C4×C47#47111yescharacteristic
164#2931—471no—
165#1837—471no—
166C2×#334—471no—
167#1547—471no—
168#1547D4711yescharacteristic
169#1292—471no—
170#1292—471no—
171C2×C47—21no—
172C2×C47—21no—
173C2×C47—82no—
174C2×C47#180411yescharacteristic
175C2×C47#181311yescharacteristic
176C2×C47#181511yescharacteristic
177C2×#131—941no—
178#532C2×D4711yescharacteristic
179#436—472no—
180#436—941no—
181C8×D4—472no—
182C8×D4—941no—
183#423—471no—
184#423—471no—
185#423C2×D4711yescharacteristic
186C2×C4⋊C8—471no—
187C2×C4⋊C8C2×D4711yescharacteristic
188#408—941no—
189C2×C4×C8—471no—
190#341—941no—
191#341—941no—
192#334—941no—
193#334—941no—
194#332—21no—
195#332—941no—
196#331—21no—
197#331—941no—
198C47#868611yescharacteristic, complemented, Sylow
199C2×#41—21no—
200#131—1882no—
201#131—1882no—
202C2×M16—21no—
203C2×M16—942no—
204C2×M16—941no—
205C2×M16—941no—
206C22×C8—21no—
207C22×C8—471no—
208C22×C8—471no—
209C22×C8—942no—
210C22×C8—941no—
211C4⋊Q8C22⋊D4711yescharacteristic
212#127C22⋊D4711yescharacteristic
213#124—21no—
214#121—21no—
215#121Dic9412yes—
216C4×D4—21no—
217C4×D4C4×D4712yes—
218C2×C42C22×D4711yescharacteristic
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—472no—
224C4⋊C8—941no—
225C4⋊C8—941no—
226C4⋊C8—941no—
227C4⋊C8—941no—
228C4⋊C8—941no—
229C22⋊C8—942no—
230C22⋊C8—942no—
231C4×C8—471no—
232C4×C8—471no—
233C4×C8—471no—
234C4×C8—471no—
235#41—41no—
236#41—41no—
237#41—41no—
238#41—41no—
239C2×Q8—21no—
240C2×D4—22no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4Dic47⋊C412yes—
244C22×C4—22no—
245C22×C4C2×Dic9411yescharacteristic
246C22×C4D4×D4711yescharacteristic
247C22×C4#1062535211yescharacteristic
248M16—41no—
249M16—41no—
250M16—1884no—
251M16—1881no—
252M16—1881no—
253M16—1881no—
254M16—1881no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—81no—
258C2×C8—471no—
259C2×C8—471no—
260C2×C8—471no—
261C2×C8—471no—
262C2×C8—941no—
263C2×C8—941no—
264C2×C8—941no—
265C2×C8—941no—
266C2×C8—942no—
267C2×C8—3761no—
268C2×C8—1884no—
269C2×C8—1881no—
270C2×C8—1881no—
271C4⋊C4—21no—
272C4⋊C4Dic47⋊C412yes—
273C22⋊C4—22no—
274C22⋊C4—22no—
275C42C2×C4×D4711yescharacteristic
276C42D4×D4711yescharacteristic
277C42#1062535211yescharacteristic
278C42C2×C22⋊D4711yescharacteristic
279C23—22no—
280C23#1309400111yescharacteristic
281Q8—41no—
282Q8—41no—
283D4—42no—
284D4—42no—
285D4—42no—
286D4—41no—
287D4—41no—
288D4—41no—
289D4—41no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4C22⋊C4×D4711yescharacteristic
297C2×C4C2×Dic47⋊C411yescharacteristic
298C2×C4#1309398011yescharacteristic
299C2×C4#1309399611yescharacteristic
300C2×C4#1309399611yescharacteristic
301C2×C4#1309400111yescharacteristic
302C2×C4#1309404311yescharacteristic
303C2×C4#1309404511yescharacteristic
304C2×C4#1309404811yescharacteristic
305C2×C4#1309405111yescharacteristic
306C8—81no—
307C8—81no—
308C8—941no—
309C8—941no—
310C8—941no—
311C8—941no—
312C8—3761no—
313C8—3761no—
314C8—1882no—
315C22—21no—
316C22—21no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22#1568898111yescharacteristic
321C22#1568942111yescharacteristic
322C22#1568942511yescharacteristic
323C4—21no—
324C4—21no—
325C4—21no—
326C4—42no—
327C4#1568911611yescharacteristic
328C4#1568912611yescharacteristic
329C4#1568933611yescharacteristic
330C4#1568934011yescharacteristic
331C4#1568938411yescharacteristic
332C4#1568938811yescharacteristic
333C2—21no—
334C2—21no—
335C2—82no—
336C2#2024339611yescharacteristic
337C2#2024343611yescharacteristic
338C2#2024344511yescharacteristic
subgroupG/H
centerC22#15688981
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1547C2
socleC22×C47#380
prime radicalC47×#532C22
cosocleC2×C42×C47C23
hypercenter#1547D47
hypocenterC47#8686
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1547

Complements

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

NH ≅ G/Ntype
2. #20243525C2semidirect
3. C2×#15688873C2semidirect
5. #20241263C2semidirect
6. #20241262C2semidirect
198. C47#8686semidirect

Maximal subgroups

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

subgroupindex
#202521002
#202435252
C2×#156888732
#202420652
#202412632
#202412622
C47×#15472
#868647