Finite Group Databasebeta

← #26109049

Subgroups — #26109049

Order 12032.

540 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243241C211yesmaximal, characteristic, complemented
2#20243241C211yesmaximal, characteristic, complemented
3C2×#15688863C212yesmaximal
4#20241269C212yesmaximal, complemented
5C4×#124×C47C211yesmaximal, characteristic
6C2×C4⋊(C47⋊C8)—21no—
7C2×C4⋊(C47⋊C8)—21no—
8C2×#13093935C2212yes—
9C2×#13093935C2212yes—
10#15688878—21no—
11#15688878—21no—
12#15688863—24no—
13C2×#124×C47C2211yescharacteristic
14C2×C4×Q8×C47C411yescharacteristic
15C2×C4×D4×C47C411yescharacteristic
16C47×#391C411yescharacteristic
17C47×#390C411yescharacteristic
18C47×#388—21no—
19C47×#388—21no—
20C47×#382C2211yescharacteristic
21C4×C22⋊C4×C47—21no—
22C4×C22⋊C4×C47—21no—
23C43×C47C2211yescharacteristic
24C22×C47⋊C8—22no—
25C22×C47⋊C8—22no—
26C4⋊(C47⋊C8)—24no—
27C4⋊(C47⋊C8)—24no—
28#13093935—22no—
29#13093935—22no—
30#13093935—22no—
31#13093935—22no—
32C22×Q8×C47C2×C411yescharacteristic
33C22×D4×C47C2×C411yescharacteristic
34C23×C4×C47—21no—
35#124×C47—21no—
36#124×C47—21no—
37#124×C47D412yes—
38#124×C47Q812yes—
39C4×Q8×C47—42no—
40C4×D4×C47—42no—
41C2×C4⋊C4×C47—21no—
42C2×C4⋊C4×C47C2×C411yescharacteristic
43C2×C4⋊C4×C47C2×C411yescharacteristic
44C2×C22⋊C4×C47—21no—
45C2×C22⋊C4×C47—21no—
46C2×C22⋊C4×C47—21no—
47C2×C22⋊C4×C47—21no—
48C2×C22⋊C4×C47—21no—
49C2×C42×C47—21no—
50C2×C42×C47C2×C411yescharacteristic
51C2×C42×C47C2×C411yescharacteristic
52C2×C42×C47D411yescharacteristic
53C2×C42×C47D411yescharacteristic
54C2×C42×C47D411yescharacteristic
55C2×C42×C47D411yescharacteristic
56C2×C42×C47C2311yescharacteristic
57#95×C47—21no—
58#95×C47—21no—
59#95×C47—21no—
60#95×C47—21no—
61C24×C47—21no—
62C2×Q8×C47—21no—
63C2×Q8×C47—41no—
64C2×Q8×C47—41no—
65C2×Q8×C47C4⋊C412yes—
66C2×D4×C47—21no—
67C2×D4×C47—41no—
68C2×D4×C47—41no—
69C2×D4×C47C4⋊C412yes—
70C22×C4×C47—21no—
71C22×C4×C47—21no—
72C22×C4×C47—21no—
73C22×C4×C47—42no—
74C22×C4×C47—42no—
75C22×C4×C47—42no—
76C22×C4×C47C22⋊C411yescharacteristic
77C22×C4×C47C22⋊C411yescharacteristic
78C22×C4×C47C22⋊C411yescharacteristic
79C22×C4×C47C22⋊C411yescharacteristic
80C22×C4×C47C22×C411yescharacteristic
81C22×C4×C47C2×D411yescharacteristic
82C22×C4×C47C2×D411yescharacteristic
83C4⋊C4×C47—41no—
84C4⋊C4×C47—41no—
85C4⋊C4×C47—42no—
86C22⋊C4×C47—22no—
87C22⋊C4×C47—22no—
88C22⋊C4×C47—22no—
89C22⋊C4×C47—22no—
90C22⋊C4×C47—42no—
91C22⋊C4×C47—42no—
92C22⋊C4×C47—42no—
93C42×C47—22no—
94C42×C47—22no—
95C42×C47—22no—
96C42×C47—41no—
97C42×C47—41no—
98C42×C47—41no—
99C42×C47—42no—
100C42×C47C2×D411yescharacteristic
101C42×C47C2×Q811yescharacteristic
102C42×C47#4111yescharacteristic
103C42×C47#4111yescharacteristic
104C2×C47⋊C8—24no—
105C2×C47⋊C8—24no—
106C2×C47⋊C8—42no—
107C2×C47⋊C8—42no—
108C2×C47⋊C8—42no—
109C2×C47⋊C8—42no—
110C23×C47—22no—
111C23×C47—41no—
112C23×C47—41no—
113C23×C47—41no—
114C23×C47—41no—
115C23×C47—41no—
116C23×C47—41no—
117C23×C47C2×C22⋊C411yescharacteristic
118Q8×C47—42no—
119Q8×C47—42no—
120D4×C47—42no—
121D4×C47—42no—
122C2×C4×C47—21no—
123C2×C4×C47—21no—
124C2×C4×C47—21no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—21no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—21no—
132C2×C4×C47—21no—
133C2×C4×C47—21no—
134C2×C4×C47—22no—
135C2×C4×C47—21no—
136C2×C4×C47—21no—
137C2×C4×C47—21no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47—41no—
141C2×C4×C47—41no—
142C2×C4×C47—41no—
143C2×C4×C47—41no—
144C2×C4×C47—41no—
145C2×C4×C47—41no—
146C2×C4×C47—84no—
147C2×C4×C47#10012yes—
148C2×C4×C47#10412yes—
149C2×C4×C47#10412yes—
150C2×C4×C47C2×C4⋊C411yescharacteristic
151C2×C4×C47#11711yescharacteristic
152C2×C4×C47#12111yescharacteristic
153C2×C4×C47#12111yescharacteristic
154C2×C4×C47C22⋊Q811yescharacteristic
155C2×C4×C47C22⋊Q811yescharacteristic
156C47⋊C8—44no—
157C47⋊C8—44no—
158#8581—471nomaximal, Sylow
159C22×C47—42no—
160C22×C47—42no—
161C22×C47—42no—
162C22×C47—81no—
163C22×C47—81no—
164C22×C47—81no—
165C22×C47—81no—
166C22×C47#38011yescharacteristic
167C22×C47C2×#10011yescharacteristic
168C22×C47#41311yescharacteristic
169C22×C47C2×#10411yescharacteristic
170C22×C47C2×#10411yescharacteristic
171C22×C47#42111yescharacteristic
172C22×C47#42111yescharacteristic
173C4×C47—21no—
174C4×C47—21no—
175C4×C47—21no—
176C4×C47—21no—
177C4×C47—21no—
178C4×C47—21no—
179C4×C47—22no—
180C4×C47—22no—
181C4×C47—22no—
182C4×C47—42no—
183C4×C47—41no—
184C4×C47—81no—
185C4×C47—81no—
186C4×#124D4711yescharacteristic
187#1767—471no—
188#1767—471no—
189C2×#329—472no—
190#1295—472no—
191C2×C47—82no—
192C2×C47#152411yescharacteristic
193C2×C47#152711yescharacteristic
194C2×C47#153011yescharacteristic
195C2×C47#153511yescharacteristic
196C2×C47#180011yescharacteristic
197C2×C47#180611yescharacteristic
198C2×C47#180611yescharacteristic
199C2×#124C2×D4711yescharacteristic
200C2×C4×Q8Dic4711yescharacteristic
201C2×C4×D4Dic4711yescharacteristic
202C2×C4⋊C8—941no—
203C2×C4⋊C8—941no—
204C2×#97—472no—
205C2×#97—472no—
206#391Dic4711yescharacteristic
207#390Dic4711yescharacteristic
208#388—21no—
209#388—21no—
210#382C2×D4711yescharacteristic
211C4×C22⋊C4—21no—
212C4×C22⋊C4—21no—
213C43C2×D4711yescharacteristic
214#336—941no—
215#336—941no—
216#329—944no—
217C47#858111yescharacteristic, complemented, Sylow
218C22×Q8C2×Dic4711yescharacteristic
219C22×D4C2×Dic4711yescharacteristic
220C23×C4—21no—
221C22×C8—942no—
222C22×C8—942no—
223#124—21no—
224#124—21no—
225#124Dic9412yes—
226#124D18812yes—
227C4×Q8—42no—
228C4×D4—42no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4C2×Dic4711yescharacteristic
231C2×C4⋊C4C2×Dic4711yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C42—21no—
238C2×C42C2×Dic4711yescharacteristic
239C2×C42C2×Dic4711yescharacteristic
240C2×C42C22⋊D4711yescharacteristic
241C2×C42C22⋊D4711yescharacteristic
242C2×C42C22⋊D4711yescharacteristic
243C2×C42C22⋊D4711yescharacteristic
244C2×C42C22×D4711yescharacteristic
245C4⋊C8—944no—
246C4⋊C8—944no—
247#97—942no—
248#97—942no—
249#97—942no—
250#97—942no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255C24—21no—
256C2×Q8—21no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8C4⋊Dic4712yes—
260C2×D4—21no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4C4⋊Dic4712yes—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4C22⋊Dic4711yescharacteristic
271C22×C4C22⋊Dic4711yescharacteristic
272C22×C4C22⋊Dic4711yescharacteristic
273C22×C4C22⋊Dic4711yescharacteristic
274C22×C4C22×Dic4711yescharacteristic
275C22×C4C2×C22⋊D4711yescharacteristic
276C22×C4C2×C22⋊D4711yescharacteristic
277C2×C8—944no—
278C2×C8—944no—
279C2×C8—1882no—
280C2×C8—1882no—
281C2×C8—1882no—
282C2×C8—1882no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—42no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—22no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—41no—
297C42—41no—
298C42—41no—
299C42—42no—
300C42C2×Dic9411yescharacteristic
301C42C2×D18811yescharacteristic
302C42#1062535011yescharacteristic
303C42#1062535011yescharacteristic
304C23—22no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23C2×C22⋊Dic4711yescharacteristic
312Q8—42no—
313Q8—42no—
314D4—42no—
315D4—42no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—84no—
341C2×C4#1309394112yes—
342C2×C4#1309395112yes—
343C2×C4#1309395112yes—
344C2×C4C2×C4⋊Dic4711yescharacteristic
345C2×C4#1309398311yescharacteristic
346C2×C4#1309399811yescharacteristic
347C2×C4#1309399811yescharacteristic
348C2×C4#1309400311yescharacteristic
349C2×C4#1309400311yescharacteristic
350C8—1884no—
351C8—1884no—
352C22—42no—
353C22—42no—
354C22—42no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22#1568898311yescharacteristic
360C22C2×#1309394111yescharacteristic
361C22#1568910711yescharacteristic
362C22C2×#1309395111yescharacteristic
363C22C2×#1309395111yescharacteristic
364C22#1568914011yescharacteristic
365C22#1568914011yescharacteristic
366C4—22no—
367C4—22no—
368C4—22no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—41no—
376C4—42no—
377C4—81no—
378C4—81no—
379C2—82no—
380C2#2024197611yescharacteristic
381C2#2024198611yescharacteristic
382C2#2024199911yescharacteristic
383C2#2024201911yescharacteristic
384C2#2024338011yescharacteristic
385C2#2024340811yescharacteristic
386C2#2024340811yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC4×#124×C47C2
socleC23×C47C2×C22⋊C4
prime radicalC22×D4×C47C2×C4
cosocleC2×C42×C47C23
hypercenterC4×#124D47
hypocenterC47#8581
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C43 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
1. #20243241C2semidirect
2. #20243241C2semidirect
4. #20241269C2semidirect
217. C47#8581semidirect

Maximal subgroups

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

subgroupindex
#202432412
C2×#156888632
#202412692
C4×#124×C472
#858147