Finite Group Databasebeta

← #26107916

Subgroups — #26107916

Order 12032.

808 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13093943C211yesmaximal, characteristic, complemented
2#20241986C212yesmaximal, complemented
3#20241831C212yesmaximal, complemented
4#20241732C211yesmaximal, characteristic, complemented
5C2×C47×#559C211yesmaximal, characteristic
6#15689180C2212yes—
7C2×#13093943—21no—
8C4×#10625329C2212yes—
9#15688892—21no—
10#15688863—24no—
11D4⋊(C47⋊C8)—24no—
12#15688849—21no—
13#15688849—21no—
14C47×#559—24no—
15C47×#559—24no—
16C2×#127×C47C411yescharacteristic
17C2×#124×C47—21no—
18C2×#124×C47C411yescharacteristic
19C2×#124×C47C2211yescharacteristic
20C2×#123×C47—21no—
21C2×#123×C47—21no—
22C2×C4×Q8×C47C411yescharacteristic
23C2×C4×Q8×C47C2211yescharacteristic
24C2×C4×D4×C47—21no—
25C2×C4×D4×C47C411yescharacteristic
26C2×C4×D4×C47C2211yescharacteristic
27C2×#10625329—22no—
28C4⋊(C47⋊C8)—22no—
29#13093943—24no—
30C22⋊(C47⋊C8)—22no—
31#13093935—22no—
32#13093935—22no—
33C4×C47⋊C8—22no—
34C22×Q8×C47C2×C411yescharacteristic
35C22×Q8×C47D411yescharacteristic
36C22×D4×C47—21no—
37C22×D4×C47—21no—
38C22×D4×C47C2×C411yescharacteristic
39C22×D4×C47C2×C411yescharacteristic
40C23×C4×C47—21no—
41C23×C4×C47—21no—
42#127×C47—22no—
43#127×C47—22no—
44#124×C47—28no—
45#124×C47—22no—
46#124×C47—22no—
47#124×C47—22no—
48#124×C47D414yes—
49#123×C47—28no—
50#123×C47—28no—
51C4×Q8×C47—22no—
52C4×Q8×C47—22no—
53C4×Q8×C47—22no—
54C4×Q8×C47—22no—
55C4×D4×C47—28no—
56C4×D4×C47—22no—
57C4×D4×C47—22no—
58C4×D4×C47—22no—
59C4×D4×C47D414yes—
60C2×C4⋊C4×C47—21no—
61C2×C4⋊C4×C47C2×C411yescharacteristic
62C2×C4⋊C4×C47D411yescharacteristic
63C2×C4⋊C4×C47D411yescharacteristic
64C2×C4⋊C4×C47D411yescharacteristic
65C2×C22⋊C4×C47—21no—
66C2×C22⋊C4×C47—21no—
67C2×C22⋊C4×C47—21no—
68C2×C22⋊C4×C47—21no—
69C2×C22⋊C4×C47—21no—
70C2×C22⋊C4×C47—21no—
71C2×C42×C47—21no—
72C2×C42×C47C2×C411yescharacteristic
73C2×C42×C47C2×C411yescharacteristic
74C2×C42×C47C2311yescharacteristic
75C24×C47—21no—
76C24×C47—21no—
77C2×Q8×C47—22no—
78C2×Q8×C47—21no—
79C2×Q8×C47—22no—
80C2×Q8×C47—41no—
81C2×Q8×C47—42no—
82C2×Q8×C47C22⋊C412yes—
83C2×D4×C47—24no—
84C2×D4×C47—24no—
85C2×D4×C47—21no—
86C2×D4×C47—21no—
87C2×D4×C47—42no—
88C2×D4×C47—42no—
89C2×D4×C47—42no—
90C2×D4×C47—42no—
91C2×D4×C47—82no—
92C2×D4×C47C22⋊C412yes—
93C2×D4×C47C22⋊C412yes—
94C22×C4×C47—21no—
95C22×C4×C47—21no—
96C22×C4×C47—24no—
97C22×C4×C47—24no—
98C22×C4×C47—82no—
99C22×C4×C47—82no—
100C22×C4×C47C22⋊C411yescharacteristic
101C22×C4×C47C22⋊C411yescharacteristic
102C22×C4×C47C22⋊C411yescharacteristic
103C22×C4×C47C22⋊C411yescharacteristic
104C22×C4×C47C22×C411yescharacteristic
105C22×C4×C47C2×D411yescharacteristic
106C22×C4×C47C2×D411yescharacteristic
107C4⋊C4×C47—24no—
108C4⋊C4×C47—21no—
109C4⋊C4×C47—22no—
110C4⋊C4×C47—22no—
111C4⋊C4×C47—22no—
112C4⋊C4×C47C22⋊C412yes—
113C22⋊C4×C47—24no—
114C22⋊C4×C47—24no—
115C22⋊C4×C47—24no—
116C22⋊C4×C47—24no—
117C22⋊C4×C47—24no—
118C22⋊C4×C47—24no—
119C42×C47—24no—
120C42×C47—21no—
121C42×C47—21no—
122C42×C47—21no—
123C42×C47—21no—
124C42×C47C2×D412yes—
125C42×C47C2×D412yes—
126#10625329—24no—
127#10625329—42no—
128C2×C47⋊C8—42no—
129C2×C47⋊C8—42no—
130C23×C47—22no—
131C23×C47—22no—
132C23×C47—42no—
133C23×C47—42no—
134C23×C47—42no—
135C23×C47—42no—
136C23×C47—41no—
137C23×C47—41no—
138C23×C47—41no—
139C23×C47—41no—
140C23×C47C2×C22⋊C411yescharacteristic
141Q8×C47—44no—
142Q8×C47—42no—
143Q8×C47—42no—
144D4×C47—24no—
145D4×C47—42no—
146D4×C47—42no—
147D4×C47—48no—
148D4×C47—42no—
149D4×C47—84no—
150C2×C4×C47—21no—
151C2×C4×C47—21no—
152C2×C4×C47—21no—
153C2×C4×C47—21no—
154C2×C4×C47—22no—
155C2×C4×C47—22no—
156C2×C4×C47—21no—
157C2×C4×C47—21no—
158C2×C4×C47—22no—
159C2×C4×C47—21no—
160C2×C4×C47—21no—
161C2×C4×C47—21no—
162C2×C4×C47—41no—
163C2×C4×C47—42no—
164C2×C4×C47—42no—
165C2×C4×C47—41no—
166C2×C4×C47—41no—
167C2×C4×C47—84no—
168C2×C4×C47—84no—
169C2×C4×C47#10412yes—
170C2×C4×C47C2×C22⋊C411yescharacteristic
171C2×C4×C47C2×C22⋊C411yescharacteristic
172C2×C4×C47#12012yes—
173C2×C4×C47#12012yes—
174C47⋊C8—44no—
175C47⋊C8—82no—
176#8370—471nomaximal, Sylow
177C22×C47—21no—
178C22×C47—21no—
179C22×C47—42no—
180C22×C47—42no—
181C22×C47—42no—
182C22×C47—42no—
183C22×C47—42no—
184C22×C47—42no—
185C22×C47—82no—
186C22×C47—82no—
187C22×C47—82no—
188C22×C47—82no—
189C22×C47#37911yescharacteristic
190C22×C47C2×#10411yescharacteristic
191C22×C47#42111yescharacteristic
192C4×C47—22no—
193C4×C47—21no—
194C4×C47—22no—
195C4×C47—22no—
196C4×C47—42no—
197C4×C47—41no—
198C4×C47—41no—
199C4×C47—42no—
200C4×C47—41no—
201C4×C47#41312yes—
202C2×#559D4711yescharacteristic
203C4×#101—471no—
204#1527—472no—
205#1490—472no—
206#1467—471no—
207C2×C47—21no—
208C2×C47—21no—
209C2×C47—82no—
210C2×C47—82no—
211C2×C47#178511yescharacteristic
212C2×C47#178911yescharacteristic
213C2×C47#179511yescharacteristic
214#559—24no—
215#559—24no—
216C2×#127Dic4711yescharacteristic
217C2×#124—21no—
218C2×#124Dic4711yescharacteristic
219C2×#124C2×D4711yescharacteristic
220C2×#123—21no—
221C2×#123—21no—
222C2×C4×Q8Dic4711yescharacteristic
223C2×C4×Q8C2×D4711yescharacteristic
224C2×C4×D4—21no—
225C2×C4×D4Dic4711yescharacteristic
226C2×C4×D4C2×D4711yescharacteristic
227#432—472no—
228C2×#101—941no—
229C4×M16—472no—
230#343—941no—
231#329—944no—
232D4⋊C8—944no—
233#324—941no—
234#324—941no—
235C47#837011yescharacteristic, complemented, Sylow
236C22×Q8C2×Dic4711yescharacteristic
237C22×Q8C22⋊D4711yescharacteristic
238C22×D4—21no—
239C22×D4—21no—
240C22×D4C2×Dic4711yescharacteristic
241C22×D4C2×Dic4711yescharacteristic
242C23×C4—21no—
243C23×C4—21no—
244C2×M16—942no—
245#127—22no—
246#127—22no—
247#124—28no—
248#124—22no—
249#124—22no—
250#124—22no—
251#124C22⋊D4714yes—
252#123—28no—
253#123—28no—
254C4×Q8—22no—
255C4×Q8—22no—
256C4×Q8—22no—
257C4×Q8—22no—
258C4×D4—28no—
259C4×D4—22no—
260C4×D4—22no—
261C4×D4—22no—
262C4×D4C22⋊D4714yes—
263C2×C4⋊C4—21no—
264C2×C4⋊C4C2×Dic4711yescharacteristic
265C2×C4⋊C4C22⋊D4711yescharacteristic
266C2×C4⋊C4C22⋊D4711yescharacteristic
267C2×C4⋊C4C22⋊D4711yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C42—21no—
275C2×C42C2×Dic4711yescharacteristic
276C2×C42C2×Dic4711yescharacteristic
277C2×C42C22×D4711yescharacteristic
278C4⋊C8—942no—
279#101—944no—
280C22⋊C8—942no—
281#97—942no—
282#97—942no—
283C4×C8—942no—
284C24—21no—
285C24—21no—
286C2×Q8—22no—
287C2×Q8—21no—
288C2×Q8—22no—
289C2×Q8—41no—
290C2×Q8—42no—
291C2×Q8C22⋊Dic4712yes—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—82no—
301C2×D4C22⋊Dic4712yes—
302C2×D4C22⋊Dic4712yes—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4C22⋊Dic4711yescharacteristic
310C22×C4C22⋊Dic4711yescharacteristic
311C22×C4C22⋊Dic4711yescharacteristic
312C22×C4C22⋊Dic4711yescharacteristic
313C22×C4C22×Dic4711yescharacteristic
314C22×C4C2×C22⋊D4711yescharacteristic
315C22×C4C2×C22⋊D4711yescharacteristic
316M16—944no—
317M16—1882no—
318C2×C8—1882no—
319C2×C8—1882no—
320C4⋊C4—24no—
321C4⋊C4—21no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4C22⋊Dic4712yes—
326C22⋊C4—24no—
327C22⋊C4—24no—
328C22⋊C4—24no—
329C22⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—24no—
332C42—24no—
333C42—21no—
334C42—21no—
335C42—21no—
336C42—21no—
337C42C2×C22⋊D4712yes—
338C42C2×C22⋊D4712yes—
339C23—22no—
340C23—22no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—42no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23C2×C22⋊Dic4711yescharacteristic
350Q8—44no—
351Q8—42no—
352Q8—42no—
353D4—24no—
354D4—42no—
355D4—42no—
356D4—48no—
357D4—42no—
358D4—84no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—84no—
377C2×C4—84no—
378C2×C4#1309395112yes—
379C2×C4C2×C22⋊Dic4711yescharacteristic
380C2×C4C2×C22⋊Dic4711yescharacteristic
381C2×C4#1309399312yes—
382C2×C4#1309399312yes—
383C8—3762no—
384C8—1884no—
385C22—21no—
386C22—21no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—82no—
394C22—82no—
395C22—82no—
396C22—82no—
397C22#1568897911yescharacteristic
398C22C2×#1309395111yescharacteristic
399C22#1568914011yescharacteristic
400C4—21no—
401C4—22no—
402C4—22no—
403C4—22no—
404C4—42no—
405C4—41no—
406C4—42no—
407C4—41no—
408C4—41no—
409C4#1568910712yes—
410C2—21no—
411C2—21no—
412C2—82no—
413C2—82no—
414C2#2024331611yescharacteristic
415C2#2024333211yescharacteristic
416C2#2024335811yescharacteristic
subgroupG/H
centerC22#15688979
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC2×C47×#559C2
socleC22×C47#379
prime radicalC2×#127×C47C4
cosocleC2×C42×C47C23
hypercenterC2×#559D47
hypocenterC47#8370
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
1. C4×#13093943C2semidirect
2. #20241986C2semidirect
3. #20241831C2semidirect
4. #20241732C2semidirect
235. C47#8370semidirect

Maximal subgroups

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

subgroupindex
C4×#130939432
#202419862
#202418312
#202417322
C2×C47×#5592
#837047