Finite Group Databasebeta

← #26201621

Subgroups — #26201621

Order 12032.

644 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254518C211yesmaximal, characteristic
2C2×#15689418C211yesmaximal, characteristic, complemented
3#20252860C211yesmaximal, characteristic
4#20242804C214yesmaximal, complemented
5#20242770C214yesmaximal, complemented
6#20242159C211yesmaximal, characteristic, complemented
7#20242068C211yesmaximal, characteristic, complemented
8#20242035C211yesmaximal, characteristic
9C47×#2982C211yesmaximal, characteristic, complemented
10C2×#13094055—21no—
11#15689673C2211yescharacteristic
12C2×#13094024C2211yescharacteristic
13#15689521C2212yes—
14#15689470C2212yes—
15#15689442—22no—
16#15689418—22no—
17#15689418C2214yes—
18#15689414—22no—
19#15689345—22no—
20#15689333C2212yes—
21#15689323C2212yes—
22C2×#13093957—21no—
23#15689147C2211yescharacteristic
24C2×C4⋊(C47⋊C8)C2211yescharacteristic
25#15689122—21no—
26C2×#13093948—21no—
27#15688874C2214yes—
28#15688871C2214yes—
29#15688868C2212yes—
30#15688866C2212yes—
31C2×C4×Q8×C47C2211yescharacteristic
32C8×Q8×C47—22no—
33C8×Q8×C47C2214yes—
34C47×#431—22no—
35C47×#431C2211yescharacteristic
36C47×#431C2211yescharacteristic
37C2×#10625350—21no—
38C22×Dic94—21no—
39#13094055—44no—
40C2×#10625337—22no—
41C2×#10625336—22no—
42C2×#10625329—21no—
43C22×C47⋊C8—21no—
44#13094024—21no—
45#13094024—21no—
46#13094024—21no—
47#13094024C2311yescharacteristic
48#13094024C2312yes—
49#13094021C2311yescharacteristic
50#13094012—21no—
51#13093998—21no—
52#13093998—21no—
53C4×D188—21no—
54C4×D188—21no—
55C2×C4⋊Dic47—21no—
56C2×C4⋊Dic47—21no—
57#13093957—24no—
58#13093956—24no—
59C4⋊(C47⋊C8)—21no—
60C4⋊(C47⋊C8)—21no—
61C4⋊(C47⋊C8)C2312yes—
62C4⋊(C47⋊C8)C2312yes—
63#13093948—24no—
64#13093947—22no—
65#13093946—24no—
66#13093944—22no—
67C22×Q8×C47D411yescharacteristic
68C4×Q8×C47—21no—
69C4×Q8×C47—21no—
70C4×Q8×C47D412yes—
71C4×Q8×C47C2312yes—
72C2×C4⋊C4×C47—21no—
73C2×C4⋊C4×C47D411yescharacteristic
74C2×C42×C47—21no—
75C2×C42×C47C2311yescharacteristic
76C4⋊C8×C47—24no—
77C4⋊C8×C47D412yes—
78C4⋊C8×C47C2312yes—
79C22⋊C8×C47—21no—
80C22⋊C8×C47D412yes—
81C4×C8×C47—24no—
82C4×C8×C47—21no—
83C4×C8×C47C2312yes—
84C2×Q8×C47—21no—
85C2×Q8×C47—22no—
86C2×Q8×C47SD1614yes—
87C2×Q8×C47C2×D412yes—
88C22×C4×C47—21no—
89C22×C4×C47—21no—
90C22×C4×C47C2×D411yescharacteristic
91C22×C4×C47C2×D411yescharacteristic
92C22×C4×C47#4111yescharacteristic
93C2×C22⋊D47—41no—
94C22×Dic47—41no—
95#10625350—42no—
96#10625350—42no—
97C2×D188—21no—
98C2×D188—21no—
99C2×D188—21no—
100C2×C4×D47—41no—
101C2×Dic94—21no—
102C2×Dic94—22no—
103C2×Dic94—41no—
104C2×Dic94—82no—
105C2×C8×C47—22no—
106C2×C8×C47C2×D414yes—
107C4⋊C4×C47—22no—
108C4⋊C4×C47—21no—
109C4⋊C4×C47—22no—
110C4⋊C4×C47C2×D412yes—
111C42×C47—22no—
112C42×C47—22no—
113C42×C47C2×D411yescharacteristic
114C42×C47C2×D411yescharacteristic
115C42×C47#4111yescharacteristic
116C42×C47C2411yescharacteristic
117#10625337—44no—
118#10625336—44no—
119(C2×D47)⋊C4—41no—
120(C2×D47)⋊C4—41no—
121C4⋊Dic47—22no—
122C4⋊Dic47—22no—
123C4⋊Dic47—21no—
124C4⋊Dic47—21no—
125C4⋊Dic47—41no—
126C4⋊Dic47—41no—
127#10625329—42no—
128C2×C47⋊C8—22no—
129C2×C47⋊C8—22no—
130C2×C47⋊C8—42no—
131C23×C47#12111yescharacteristic
132C22×D47—41no—
133Q8×C47—22no—
134Q8×C47—24no—
135Q8×C47C2×SD1614yes—
136C2×C4×C47—22no—
137C2×C4×C47—24no—
138C2×C4×C47—22no—
139C2×C4×C47—22no—
140C2×C4×C47—41no—
141C2×C4×C47—81no—
142C2×C4×C47#12111yescharacteristic
143C2×C4×C47#12111yescharacteristic
144C2×C4×C47#12111yescharacteristic
145C2×C4×C47C2×SD1612yes—
146C2×C4×C47#13512yes—
147C2×C4×C47C22×D411yescharacteristic
148C2×C4×C47C22×D411yescharacteristic
149C2×C4×C47C2×#4111yescharacteristic
150C22⋊D47—82no—
151C2×Dic47—42no—
152C2×Dic47—41no—
153C2×Dic47—82no—
154D188—41no—
155D188—41no—
156D188—42no—
157D188—81no—
158C4×D47—82no—
159Dic94—41no—
160Dic94—41no—
161Dic94—82no—
162Dic94—82no—
163C8×C47—24no—
164C8×C47—42no—
165C47⋊C8—42no—
166C47⋊C8—42no—
167#19403—471nomaximal, Sylow
168C22×C47—21no—
169C22×C47—21no—
170C22×C47#46512yes—
171C22×C47C2×#12111yescharacteristic
172C2×D47—81no—
173C2×D47—81no—
174C2×D47—81no—
175C4×C47—21no—
176C4×C47—24no—
177C4×C47—21no—
178C4×C47—21no—
179C4×C47—42no—
180C4×C47#46512yes—
181C4×C47C22×SD1611yescharacteristic
182C4×C47C2×#13511yescharacteristic
183C4×C47#57812yes—
184Dic47—81no—
185Dic47—82no—
186#3227—471no—
187C2×#477—471no—
188#3037—471no—
189#2982D4711yescharacteristic, complemented
190#1678—474no—
191#1672—474no—
192#1570—471no—
193#1548—471no—
194#1540—471no—
195C2×C47—22no—
196C2×C47#304411yescharacteristic
197C2×C47C2×#46511yescharacteristic
198C2×C47#306811yescharacteristic
199D47—161no—
200C2×#137—941no—
201#532—471no—
202C2×C4⋊Q8—471no—
203C2×C4×Q8C2×D4711yescharacteristic
204#498—472no—
205#486—472no—
206#481—942no—
207#477—474no—
208#477—942no—
209#476—942no—
210#464—942no—
211#462—472no—
212#460—472no—
213C8×Q8—22no—
214C8×Q8C2×D4714yes—
215#431—22no—
216#431C2×D4711yescharacteristic
217#431C2×D4711yescharacteristic
218C2×#107—941no—
219#423—471no—
220C2×C4⋊C8—471no—
221#417—941no—
222C2×#103—941no—
223#334—474no—
224#333—474no—
225#332—472no—
226#331—472no—
227C47#1940311yescharacteristic, complemented, Sylow
228C2×#41—941no—
229C22×Q8—941no—
230C22×Q8D18811yescharacteristic
231#137—1884no—
232C2×Q16—942no—
233C2×SD16—942no—
234C2×M16—941no—
235C22×C8—941no—
236C4⋊Q8—472no—
237C4⋊Q8—471no—
238C4⋊Q8—941no—
239C4⋊Q8—941no—
240C4⋊Q8—941no—
241#127—471no—
242#124—941no—
243#121—941no—
244#121—941no—
245C4×Q8—21no—
246C4×Q8—21no—
247C4×Q8D18812yes—
248C4×Q8C22×D4712yes—
249C4×D4—941no—
250C4×D4—941no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—941no—
253C2×C4⋊C4—941no—
254C2×C4⋊C4D18811yescharacteristic
255C2×C42—21no—
256C2×C42C22×D4711yescharacteristic
257#107—944no—
258#106—944no—
259C4⋊C8—24no—
260C4⋊C8—472no—
261C4⋊C8—472no—
262C4⋊C8—941no—
263C4⋊C8—941no—
264C4⋊C8C22⋊D4712yes—
265C4⋊C8C22×D4712yes—
266#103—944no—
267#103—942no—
268#102—944no—
269#102—942no—
270C22⋊C8—21no—
271C22⋊C8C22⋊D4712yes—
272C4×C8—24no—
273C4×C8—21no—
274C4×C8C22×D4712yes—
275#41—1882no—
276#41—1882no—
277C2×Q8—21no—
278C2×Q8—22no—
279C2×Q8—941no—
280C2×Q8—942no—
281C2×Q8—3762no—
282C2×Q8—1881no—
283C2×Q8#1062532514yes—
284C2×Q8C2×D18812yes—
285C2×D4—941no—
286C2×D4—941no—
287C2×D4—941no—
288C2×D4—1881no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—1881no—
292C22×C4—1881no—
293C22×C4C2×D18811yescharacteristic
294C22×C4#1062535011yescharacteristic
295C22×C4C2×C22⋊D4711yescharacteristic
296Q16—1884no—
297SD16—1884no—
298M16—1882no—
299C2×C8—22no—
300C2×C8—942no—
301C2×C8—942no—
302C2×C8—1882no—
303C2×C8C2×C22⋊D4714yes—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—942no—
308C4⋊C4—941no—
309C4⋊C4—942no—
310C4⋊C4—941no—
311C4⋊C4—1881no—
312C4⋊C4—1881no—
313C4⋊C4C2×D18812yes—
314C22⋊C4—1881no—
315C22⋊C4—1881no—
316C42—22no—
317C42—22no—
318C42C2×D18811yescharacteristic
319C42#1062535011yescharacteristic
320C42C2×C22⋊D4711yescharacteristic
321C42C23×D4711yescharacteristic
322C23—1881no—
323C23#1309399811yescharacteristic
324Q8—24no—
325Q8—22no—
326Q8—3762no—
327Q8—3762no—
328Q8—1881no—
329Q8—1881no—
330Q8C2×#1062532514yes—
331D4—3761no—
332D4—3762no—
333D4—1881no—
334D4—1881no—
335D4—1882no—
336C2×C4—24no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—81no—
342C2×C4—3762no—
343C2×C4—3762no—
344C2×C4—1881no—
345C2×C4—1882no—
346C2×C4#1309399811yescharacteristic
347C2×C4#1309399811yescharacteristic
348C2×C4#1309399811yescharacteristic
349C2×C4C2×#1062532512yes—
350C2×C4#1309404612yes—
351C2×C4C22×C22⋊D4711yescharacteristic
352C2×C4C22×D18811yescharacteristic
353C2×C4C2×#1062535011yescharacteristic
354C8—24no—
355C8—42no—
356C8—1882no—
357C8—1882no—
358C22—21no—
359C22—21no—
360C22—3761no—
361C22—3761no—
362C22—3761no—
363C22#1568935212yes—
364C22C2×#1309399811yescharacteristic
365C4—24no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—42no—
370C4—3761no—
371C4—3762no—
372C4#1568935212yes—
373C4C22×#1062532511yescharacteristic
374C4C2×#1309404611yescharacteristic
375C4#1568993812yes—
376C2—22no—
377C2—7521no—
378C2#2025291811yescharacteristic
379C2C2×#1568935211yescharacteristic
380C2#2025311611yescharacteristic
subgroupG/H
centerC22C2×#13093998
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#2982C2
socleC22×C47C2×#121
prime radical#15689673C22
hypercenter#2982D47
hypocenterC47#19403
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. C2×#15689418C2semidirect
4. #20242804C2semidirect
5. #20242770C2semidirect
6. #20242159C2semidirect
7. #20242068C2semidirect
9. C47×#2982C2semidirect
189. #2982D47semidirect
227. C47#19403semidirect

Maximal subgroups

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

subgroupindex
#202545182
C2×#156894182
#202528602
#202428042
#202427702
#202421592
#202420682
#202420352
C47×#29822
#1940347