Finite Group Databasebeta

← #26199736

Subgroups — #26199736

Order 12032.

584 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254725C212yesmaximal
2#20253974C211yesmaximal, characteristic, complemented
3#20242906C211yesmaximal, characteristic, complemented
4#20242901C211yesmaximal, characteristic, complemented
5#20242882C211yesmaximal, characteristic, complemented
6#20242877C211yesmaximal, characteristic, complemented
7#20242818C212yesmaximal, complemented
8#20242811C212yesmaximal, complemented
9C2×#15688874C211yesmaximal, characteristic, complemented
10#20242089C212yesmaximal, complemented
11C47×#2966C211yesmaximal, characteristic
12#15689680C2212yes—
13#15689528C2211yescharacteristic
14#15689503C2211yescharacteristic
15#15689474C2212yes—
16#15689427—22no—
17#15689408C2212yes—
18#15689403C2212yes—
19#15689364—21no—
20#15689364—21no—
21#15689362—22no—
22#15689350—22no—
23#15689342—22no—
24#15689333C2212yes—
25#15689328C2212yes—
26C4×Dic188—21no—
27C4×Dic188—21no—
28C2×C4⋊(C47⋊C8)C2212yes—
29#15688874C2212yes—
30#15688874C2211yescharacteristic
31#15688874C2211yescharacteristic
32#15688870C2212yes—
33#15688870C2212yes—
34#15688868C2212yes—
35#15688868C2212yes—
36C47×#529C2211yescharacteristic
37C47×#446C2211yescharacteristic
38C8×Q8×C47C2211yescharacteristic
39C47×#436C2211yescharacteristic
40C47×#435—21no—
41C8×D4×C47—21no—
42C8×D4×C47C2211yescharacteristic
43C47×#433—21no—
44C47×#432C2211yescharacteristic
45C47×#405—21no—
46C2×C4×C8×C47C2211yescharacteristic
47C2×Dic188—21no—
48C2×Dic188—21no—
49C2×#10625337—22no—
50C2×#10625335—22no—
51C22×C47⋊C8—22no—
52#13094024C2312yes—
53#13094024C2312yes—
54#13094016—22no—
55#13094003—22no—
56#13094003—22no—
57C4×Dic94—22no—
58C4×Dic94—22no—
59#13093983—22no—
60#13093958—21no—
61#13093958—21no—
62#13093957—22no—
63#13093955—22no—
64C4⋊(C47⋊C8)—22no—
65C4⋊(C47⋊C8)C2312yes—
66C4⋊(C47⋊C8)C2312yes—
67#13093949—22no—
68#13093949—22no—
69#13093948—22no—
70#13093947—22no—
71#13093947—22no—
72#13093945—22no—
73C2×M16×C47—21no—
74C22×C8×C47—21no—
75C22×C8×C47D411yescharacteristic
76C22×C8×C47D411yescharacteristic
77#126×C47—21no—
78#125×C47D411yescharacteristic
79#124×C47D411yescharacteristic
80#123×C47—21no—
81C22⋊Q8×C47D411yescharacteristic
82#121×C47D411yescharacteristic
83C4×Q8×C47C2311yescharacteristic
84C4×D4×C47—21no—
85C4×D4×C47C2311yescharacteristic
86#117×C47—21no—
87C2×C42×C47C2311yescharacteristic
88C4⋊C8×C47—21no—
89C4⋊C8×C47—21no—
90C4⋊C8×C47C2311yescharacteristic
91C4⋊C8×C47C2311yescharacteristic
92C22⋊C8×C47—21no—
93C22⋊C8×C47—21no—
94C22⋊C8×C47—21no—
95#97×C47—21no—
96#97×C47—21no—
97C4×C8×C47—21no—
98C4×C8×C47D411yescharacteristic
99C4×C8×C47D411yescharacteristic
100C4×C8×C47C2311yescharacteristic
101C4×C8×C47C2311yescharacteristic
102C2×Q8×C47C2×D411yescharacteristic
103C2×D4×C47—21no—
104C2×D4×C47C2×D411yescharacteristic
105C22×C4×C47—21no—
106C22×C4×C47C2×D411yescharacteristic
107C22×C4×C47C2×D411yescharacteristic
108C22×C4×C47#4111yescharacteristic
109C2×Dic94—22no—
110C2×Dic94—22no—
111M16×C47—42no—
112C2×C8×C47—22no—
113C2×C8×C47—21no—
114C2×C8×C47—21no—
115C2×C8×C47—41no—
116C2×C8×C47—42no—
117C2×C8×C47C2×D411yescharacteristic
118C2×C8×C47C2×D411yescharacteristic
119C2×C8×C47C2×D411yescharacteristic
120C2×C8×C47C2×D411yescharacteristic
121C4⋊C4×C47—21no—
122C4⋊C4×C47—21no—
123C4⋊C4×C47—21no—
124C4⋊C4×C47—21no—
125C4⋊C4×C47C2×D411yescharacteristic
126C4⋊C4×C47C2×D411yescharacteristic
127C22⋊C4×C47—21no—
128C22⋊C4×C47—21no—
129C22⋊C4×C47—21no—
130C22⋊C4×C47—21no—
131C22⋊C4×C47—21no—
132C42×C47—21no—
133C42×C47C2×D411yescharacteristic
134C42×C47C2×D411yescharacteristic
135C42×C47#4111yescharacteristic
136C42×C47C2411yescharacteristic
137C22⋊Dic47—42no—
138#10625337—82no—
139#10625335—82no—
140C4⋊Dic47—22no—
141C4⋊Dic47—22no—
142C4⋊Dic47—22no—
143C4⋊Dic47—22no—
144Dic47⋊C4—42no—
145Dic47⋊C4—42no—
146C4×Dic47—42no—
147C2×C47⋊C8—22no—
148C2×C47⋊C8—22no—
149C2×C47⋊C8—82no—
150Dic188—42no—
151Dic188—42no—
152C23×C47—21no—
153C23×C47#12111yescharacteristic
154Q8×C47—22no—
155D4×C47—22no—
156D4×C47—42no—
157C2×C4×C47—22no—
158C2×C4×C47—21no—
159C2×C4×C47—21no—
160C2×C4×C47—22no—
161C2×C4×C47—21no—
162C2×C4×C47—21no—
163C2×C4×C47—21no—
164C2×C4×C47—41no—
165C2×C4×C47—41no—
166C2×C4×C47#12111yescharacteristic
167C2×C4×C47#12111yescharacteristic
168C2×C4×C47#12111yescharacteristic
169C2×C4×C47C22×D411yescharacteristic
170C2×C4×C47C22×D411yescharacteristic
171C2×C4×C47C2×#4111yescharacteristic
172C2×Dic47—42no—
173C2×Dic47—42no—
174Dic94—42no—
175Dic94—42no—
176Dic94—42no—
177Dic94—42no—
178C8×C47—21no—
179C8×C47—21no—
180C8×C47—21no—
181C8×C47—41no—
182C8×C47—41no—
183C8×C47#13512yes—
184C47⋊C8—42no—
185C47⋊C8—42no—
186#19228—471nomaximal, Sylow
187C22×C47—21no—
188C22×C47—21no—
189C22×C47—41no—
190C22×C47—41no—
191C22×C47—41no—
192C22×C47#47312yes—
193C22×C47C2×#12111yescharacteristic
194C4×C47—21no—
195C4×C47—21no—
196C4×C47—21no—
197C4×C47—21no—
198C4×C47—41no—
199C4×C47—41no—
200C4×C47—41no—
201C4×C47C2×#13511yescharacteristic
202C4×C47#57511yescharacteristic
203C4×C47#57712yes—
204Dic47—82no—
205Dic47—82no—
206#3248—472no—
207#3170—471no—
208#2966D4711yescharacteristic
209#1694—471no—
210#1693—471no—
211#1690—471no—
212#1689—471no—
213#1680—472no—
214#1679—472no—
215C2×#334—471no—
216#1552—472no—
217C2×C47—22no—
218C2×C47—81no—
219C2×C47#303511yescharacteristic
220C2×C47#306011yescharacteristic
221C2×C47C2×#47311yescharacteristic
222#533—472no—
223#529C2×D4711yescharacteristic
224#500—471no—
225#494—471no—
226#487—472no—
227#479—942no—
228#475—472no—
229#474—472no—
230#467—941no—
231#467—941no—
232#467—942no—
233#465—942no—
234#464—942no—
235#462—472no—
236#461—472no—
237#446C2×D4711yescharacteristic
238C8×Q8C2×D4711yescharacteristic
239C4×Q16—941no—
240C4×Q16—941no—
241#436C2×D4711yescharacteristic
242#435—21no—
243C8×D4—21no—
244C8×D4C2×D4711yescharacteristic
245#433—21no—
246#432C2×D4711yescharacteristic
247C2×C4⋊C8—472no—
248#405—21no—
249C2×C4×C8C2×D4711yescharacteristic
250#334—472no—
251#334—471no—
252#334—471no—
253#332—472no—
254#332—472no—
255#332—472no—
256#332—472no—
257C47#1922811yescharacteristic, complemented, Sylow
258C2×Q16—942no—
259C2×Q16—941no—
260C2×Q16—941no—
261C2×SD16—942no—
262C2×M16—21no—
263C22×C8—21no—
264C22×C8—942no—
265C22×C8C22⋊D4711yescharacteristic
266C22×C8C22⋊D4711yescharacteristic
267C4⋊Q8—472no—
268C4⋊Q8—472no—
269#126—21no—
270#125—942no—
271#125D18811yescharacteristic
272#124D18811yescharacteristic
273#123—21no—
274C22⋊Q8—942no—
275C22⋊Q8—942no—
276C22⋊Q8D18811yescharacteristic
277#121D18811yescharacteristic
278C4×Q8—942no—
279C4×Q8—942no—
280C4×Q8C22×D4711yescharacteristic
281C4×D4—21no—
282C4×D4C22×D4711yescharacteristic
283#117—21no—
284#117—942no—
285C2×C42C22×D4711yescharacteristic
286#107—941no—
287#107—941no—
288#107—942no—
289#106—942no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—472no—
293C4⋊C8—472no—
294C4⋊C8—942no—
295C4⋊C8C22×D4711yescharacteristic
296C4⋊C8C22×D4711yescharacteristic
297#103—942no—
298#103—942no—
299#103—942no—
300#103—942no—
301#103—942no—
302#102—942no—
303C22⋊C8—21no—
304C22⋊C8—21no—
305C22⋊C8—21no—
306#97—21no—
307#97—21no—
308C4×C8—21no—
309C4×C8C22⋊D4711yescharacteristic
310C4×C8C22⋊D4711yescharacteristic
311C4×C8C22×D4711yescharacteristic
312C4×C8C22×D4711yescharacteristic
313C2×Q8—942no—
314C2×Q8—942no—
315C2×Q8C2×D18811yescharacteristic
316C2×D4—21no—
317C2×D4C2×D18811yescharacteristic
318C22×C4—21no—
319C22×C4C2×D18811yescharacteristic
320C22×C4#1062535011yescharacteristic
321C22×C4C2×C22⋊D4711yescharacteristic
322Q16—3762no—
323Q16—1882no—
324Q16—1882no—
325SD16—3762no—
326M16—42no—
327C2×C8—22no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—41no—
331C2×C8—42no—
332C2×C8—942no—
333C2×C8—942no—
334C2×C8—3762no—
335C2×C8C2×C22⋊D4711yescharacteristic
336C2×C8C2×C22⋊D4711yescharacteristic
337C2×C8C2×C22⋊D4711yescharacteristic
338C2×C8C2×C22⋊D4711yescharacteristic
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—942no—
344C4⋊C4—942no—
345C4⋊C4—942no—
346C4⋊C4—942no—
347C4⋊C4—1882no—
348C4⋊C4—1882no—
349C4⋊C4C2×D18811yescharacteristic
350C4⋊C4C2×D18811yescharacteristic
351C22⋊C4—21no—
352C22⋊C4—21no—
353C22⋊C4—21no—
354C22⋊C4—21no—
355C22⋊C4—21no—
356C22⋊C4—1882no—
357C42—21no—
358C42—1882no—
359C42C2×D18811yescharacteristic
360C42#1062535011yescharacteristic
361C42C2×C22⋊D4711yescharacteristic
362C42C23×D4711yescharacteristic
363C23—21no—
364C23#1309399811yescharacteristic
365Q8—22no—
366Q8—1882no—
367Q8—1882no—
368Q8—1882no—
369Q8—1882no—
370D4—22no—
371D4—42no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—1882no—
382C2×C4—1882no—
383C2×C4#1309399811yescharacteristic
384C2×C4#1309399811yescharacteristic
385C2×C4#1309399811yescharacteristic
386C2×C4C22×C22⋊D4711yescharacteristic
387C2×C4C22×D18811yescharacteristic
388C2×C4C2×#1062535011yescharacteristic
389C8—21no—
390C8—21no—
391C8—21no—
392C8—41no—
393C8—41no—
394C8—1882no—
395C8—1882no—
396C8#1309404412yes—
397C22—21no—
398C22—21no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22#1568939812yes—
403C22C2×#1309399811yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—41no—
409C4—41no—
410C4—41no—
411C4—3762no—
412C4—3762no—
413C4C2×#1309404411yescharacteristic
414C4#1568992111yescharacteristic
415C4#1568993512yes—
416C2—22no—
417C2—81no—
418C2#2025284311yescharacteristic
419C2#2025305011yescharacteristic
420C2C2×#1568939811yescharacteristic
subgroupG/H
centerC22C2×#13093998
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#2966C2
socleC22×C47C2×#121
prime radical#121×C47D4
hypercenter#2966D47
hypocenterC47#19228
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2966

Complements

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

NH ≅ G/Ntype
2. #20253974C2semidirect
3. #20242906C2semidirect
4. #20242901C2semidirect
5. #20242882C2semidirect
6. #20242877C2semidirect
7. #20242818C2semidirect
8. #20242811C2semidirect
9. C2×#15688874C2semidirect
10. #20242089C2semidirect
257. C47#19228semidirect

Maximal subgroups

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

subgroupindex
#202547252
#202539742
#202429062
#202429012
#202428822
#202428772
#202428182
#202428112
C2×#156888742
#202420892
C47×#29662
#1922847