Finite Group Databasebeta

← #26185407

Subgroups — #26185407

Order 12032.

602 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252499C211yesmaximal, characteristic
2#20252032C211yesmaximal, characteristic
3#20242179C212yesmaximal, complemented
4#20242174C212yesmaximal, complemented
5#20242169C212yesmaximal, complemented
6#20242160C212yesmaximal, complemented
7#20242141C211yesmaximal, characteristic, complemented
8#20242138C211yesmaximal, characteristic, complemented
9#20242134C211yesmaximal, characteristic, complemented
10C2×#15688873C211yesmaximal, characteristic, complemented
11C47×#2989C211yesmaximal, characteristic
12Q8×C47⋊C8—21no—
13D4×C47⋊C8—22no—
14D4×C47⋊C8—21no—
15#15689184C414yes—
16#15689176C2212yes—
17#15689176C2211yescharacteristic
18#15689176C2212yes—
19#15689176C2212yes—
20#15689150C412yes—
21C2×C4⋊(C47⋊C8)C2211yescharacteristic
22C2×C4⋊(C47⋊C8)C2212yes—
23#15689090C412yes—
24C4×#10625329C2211yescharacteristic
25C2×C4×C47⋊C8C2211yescharacteristic
26#15688875C2211yescharacteristic
27#15688875C2212yes—
28#15688875C2211yescharacteristic
29#15688875C2212yes—
30#15688873C2212yes—
31#15688873C2212yes—
32#15688873C2212yes—
33#15688873C2211yescharacteristic
34#15688873C2211yescharacteristic
35#15688873C2212yes—
36C4×#41×C47C2211yescharacteristic
37C47×#446—21no—
38C47×#436—21no—
39C47×#435—22no—
40C47×#432C2212yes—
41C47×#431C2211yescharacteristic
42C47×#423C2212yes—
43C2×C4⋊C8×C47C2211yescharacteristic
44C2×#10625329C2×C412yes—
45C22×C47⋊C8—21no—
46C22×C47⋊C8—22no—
47C22×C47⋊C8C2×C412yes—
48C4⋊(C47⋊C8)—21no—
49C4⋊(C47⋊C8)—22no—
50C4⋊(C47⋊C8)C2×C414yes—
51C4⋊(C47⋊C8)C2×C414yes—
52C4⋊(C47⋊C8)C2312yes—
53C4⋊(C47⋊C8)C2311yescharacteristic
54C4⋊(C47⋊C8)C2311yescharacteristic
55C22⋊(C47⋊C8)—21no—
56C22⋊(C47⋊C8)—22no—
57C22⋊(C47⋊C8)C2×C414yes—
58C22⋊(C47⋊C8)C2×C414yes—
59#13093935C2×C412yes—
60C4×C47⋊C8—21no—
61C4×C47⋊C8—22no—
62C4×C47⋊C8C2×C412yes—
63C4×C47⋊C8C2311yescharacteristic
64C4×C47⋊C8C2311yescharacteristic
65C4×C47⋊C8C2312yes—
66C2×#41×C47C2×C411yescharacteristic
67C2×M16×C47—22no—
68C22×C8×C47—21no—
69C4×Q8×C47D411yescharacteristic
70C4×Q8×C47Q811yescharacteristic
71C4×D4×C47D412yes—
72C4×D4×C47D411yescharacteristic
73C4×D4×C47Q812yes—
74C4×D4×C47Q811yescharacteristic
75#117×C47C2×C411yescharacteristic
76#117×C47C2×C412yes—
77C2×C42×C47C2311yescharacteristic
78C2×C42×C47C2312yes—
79C4⋊C8×C47—21no—
80C4⋊C8×C47—22no—
81C4⋊C8×C47C2312yes—
82C4⋊C8×C47C2311yescharacteristic
83C4⋊C8×C47C2311yescharacteristic
84C22⋊C8×C47—21no—
85C22⋊C8×C47—22no—
86#97×C47—22no—
87C4×C8×C47—21no—
88#41×C47—24no—
89C2×Q8×C47C4⋊C411yescharacteristic
90C2×D4×C47C4⋊C412yes—
91C2×D4×C47C4⋊C411yescharacteristic
92C22×C4×C47C22×C412yes—
93C22×C4×C47C22×C411yescharacteristic
94C22×C4×C47C2×D411yescharacteristic
95C22×C4×C47C2×D412yes—
96C22×C4×C47C2×Q812yes—
97C22×C4×C47C2×Q811yescharacteristic
98M16×C47—82no—
99C2×C8×C47—22no—
100C2×C8×C47—21no—
101C2×C8×C47—21no—
102C2×C8×C47—81no—
103C4⋊C4×C47C4⋊C412yes—
104C4⋊C4×C47C4⋊C411yescharacteristic
105C4⋊C4×C47C4⋊C412yes—
106C4⋊C4×C47C4⋊C411yescharacteristic
107C22⋊C4×C47C4⋊C412yes—
108C22⋊C4×C47C4⋊C412yes—
109C22⋊C4×C47C4⋊C412yes—
110C42×C47C22×C411yescharacteristic
111C42×C47C22×C412yes—
112C42×C47C2×D411yescharacteristic
113C42×C47C2×D412yes—
114C42×C47C2×Q812yes—
115C42×C47C2×Q811yescharacteristic
116C42×C47C2411yescharacteristic
117#10625329—24no—
118C2×C47⋊C8—24no—
119C2×C47⋊C8—21no—
120C2×C47⋊C8—21no—
121C2×C47⋊C8—22no—
122C2×C47⋊C8—81no—
123C2×C47⋊C8—82no—
124C2×C47⋊C8C22×C412yes—
125C2×C47⋊C8C22×C412yes—
126C2×C47⋊C8C22×C414yes—
127C23×C47C2×C4⋊C412yes—
128C23×C47C2×C4⋊C411yescharacteristic
129Q8×C47—22no—
130D4×C47—24no—
131D4×C47—22no—
132C2×C4×C47—24no—
133C2×C4×C47—21no—
134C2×C4×C47—22no—
135C2×C4×C47—21no—
136C2×C4×C47—22no—
137C2×C4×C47—41no—
138C2×C4×C47—42no—
139C2×C4×C47C2×C4⋊C411yescharacteristic
140C2×C4×C47C2×C4⋊C412yes—
141C2×C4×C47C2×C4⋊C411yescharacteristic
142C2×C4×C47C2×C4⋊C412yes—
143C2×C4×C47C2×C4⋊C412yes—
144C2×C4×C47C2×C4⋊C411yescharacteristic
145C2×C4×C47#13114yes—
146C2×C4×C47C23×C411yescharacteristic
147C2×C4×C47C22×D411yescharacteristic
148C2×C4×C47C22×Q811yescharacteristic
149C8×C47—41no—
150C8×C47—42no—
151C8×C47—41no—
152C47⋊C8—22no—
153C47⋊C8—22no—
154C47⋊C8—24no—
155C47⋊C8—41no—
156C47⋊C8—41no—
157C47⋊C8—42no—
158#17811—471nomaximal, Sylow
159C22×C47—21no—
160C22×C47—21no—
161C22×C47—22no—
162C22×C47—22no—
163C22×C47—22no—
164C22×C47#42412yes—
165C22×C47C22×C4⋊C411yescharacteristic
166C4×C47—22no—
167C4×C47—21no—
168C4×C47—22no—
169C4×C47—21no—
170C4×C47—21no—
171C4×C47—22no—
172C4×C47#42412yes—
173C4×C47C2×#13112yes—
174C4×C47#57711yescharacteristic
175C4×C47#57711yescharacteristic
176#2989D4711yescharacteristic
177#2988—471no—
178#2922—471no—
179#1574—472no—
180#1573—472no—
181#1572—472no—
182#1570—472no—
183#1565—471no—
184#1565—471no—
185#1564—471no—
186C2×#334—471no—
187C2×C47—22no—
188C2×C47—42no—
189C2×C47C2×#42411yescharacteristic
190C2×C47#291211yescharacteristic
191C2×C47#291611yescharacteristic
192C4×#41C2×D4711yescharacteristic
193#446—21no—
194C8×Q8—941no—
195#436—21no—
196#435—22no—
197C8×D4—942no—
198C8×D4—941no—
199#433—474no—
200#432C2×D4712yes—
201#431—472no—
202#431—472no—
203#431—471no—
204#431—472no—
205#431C2×D4711yescharacteristic
206#424—472no—
207#423C2×D4712yes—
208C2×C4⋊C8—471no—
209C2×C4⋊C8—472no—
210C2×C4⋊C8C2×D4711yescharacteristic
211#408—472no—
212C4×M16—471no—
213C2×C4×C8—471no—
214#335—471no—
215#335—471no—
216#335—472no—
217#335—472no—
218#334—472no—
219#334—472no—
220#334—471no—
221#334—471no—
222#334—472no—
223#334—472no—
224C47#1781111yescharacteristic, complemented, Sylow
225C2×#41C4×D4711yescharacteristic
226C2×M16—22no—
227C2×M16—472no—
228C22×C8—21no—
229C22×C8—472no—
230C22×C8—941no—
231C22×C8—942no—
232C4×Q8Dic9411yescharacteristic
233C4×Q8C22⋊D4711yescharacteristic
234C4×D4Dic9412yes—
235C4×D4Dic9411yescharacteristic
236C4×D4C22⋊D4712yes—
237C4×D4C22⋊D4711yescharacteristic
238#117C4×D4711yescharacteristic
239#117C4×D4712yes—
240C2×C42C22×D4711yescharacteristic
241C2×C42C22×D4712yes—
242C4⋊C8—21no—
243C4⋊C8—22no—
244C4⋊C8—474no—
245C4⋊C8—474no—
246C4⋊C8—472no—
247C4⋊C8—471no—
248C4⋊C8—471no—
249C4⋊C8—941no—
250C4⋊C8—942no—
251C4⋊C8C22×D4712yes—
252C4⋊C8C22×D4711yescharacteristic
253C4⋊C8C22×D4711yescharacteristic
254C22⋊C8—21no—
255C22⋊C8—22no—
256C22⋊C8—474no—
257C22⋊C8—474no—
258C22⋊C8—941no—
259C22⋊C8—942no—
260#97—22no—
261#97—472no—
262C4×C8—21no—
263C4×C8—472no—
264C4×C8—471no—
265C4×C8—471no—
266C4×C8—472no—
267C4×C8—941no—
268C4×C8—942no—
269#41—24no—
270C2×Q8Dic47⋊C411yescharacteristic
271C2×D4Dic47⋊C412yes—
272C2×D4Dic47⋊C411yescharacteristic
273C22×C4C2×Dic9412yes—
274C22×C4C2×Dic9411yescharacteristic
275C22×C4C2×C4×D4712yes—
276C22×C4C2×C4×D4711yescharacteristic
277C22×C4C2×C22⋊D4711yescharacteristic
278C22×C4C2×C22⋊D4712yes—
279M16—82no—
280M16—944no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—22no—
284C2×C8—81no—
285C2×C8—472no—
286C2×C8—472no—
287C2×C8—474no—
288C2×C8—944no—
289C2×C8—941no—
290C2×C8—941no—
291C2×C8—942no—
292C2×C8—3761no—
293C2×C8—3762no—
294C4⋊C4Dic47⋊C412yes—
295C4⋊C4Dic47⋊C411yescharacteristic
296C4⋊C4Dic47⋊C412yes—
297C4⋊C4Dic47⋊C411yescharacteristic
298C22⋊C4Dic47⋊C412yes—
299C22⋊C4Dic47⋊C412yes—
300C22⋊C4Dic47⋊C412yes—
301C42C2×Dic9412yes—
302C42C2×Dic9411yescharacteristic
303C42C2×C4×D4711yescharacteristic
304C42C2×C4×D4712yes—
305C42C2×C22⋊D4711yescharacteristic
306C42C2×C22⋊D4712yes—
307C42C23×D4711yescharacteristic
308C23C2×Dic47⋊C412yes—
309C23C2×Dic47⋊C411yescharacteristic
310Q8—22no—
311D4—24no—
312D4—22no—
313C2×C4—24no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4C2×Dic47⋊C411yescharacteristic
321C2×C4C2×Dic47⋊C412yes—
322C2×C4C2×Dic47⋊C412yes—
323C2×C4C2×Dic47⋊C411yescharacteristic
324C2×C4C2×Dic47⋊C412yes—
325C2×C4C2×Dic47⋊C411yescharacteristic
326C2×C4#1309403214yes—
327C2×C4C22×C4×D4711yescharacteristic
328C2×C4C22×C22⋊D4711yescharacteristic
329C2×C4C22×Dic9411yescharacteristic
330C8—41no—
331C8—42no—
332C8—41no—
333C8—942no—
334C8—942no—
335C8—944no—
336C8—1881no—
337C8—1882no—
338C8—1881no—
339C22—21no—
340C22—21no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22#1568915112yes—
345C22C22×Dic47⋊C411yescharacteristic
346C4—22no—
347C4—21no—
348C4—22no—
349C4—21no—
350C4—21no—
351C4—22no—
352C4#1568915112yes—
353C4C2×#1309403212yes—
354C4#1568993111yescharacteristic
355C4#1568993411yescharacteristic
356C2—22no—
357C2—42no—
358C2C2×#1568915111yescharacteristic
359C2#2025198111yescharacteristic
360C2#2025199711yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#2989C2
socleC22×C47C22×C4⋊C4
prime radicalC2×#41×C47C2×C4
cosocleC42×C47C24
hypercenter#2989D47
hypocenterC47#17811
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ #2989

Complements

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

NH ≅ G/Ntype
3. #20242179C2semidirect
4. #20242174C2semidirect
5. #20242169C2semidirect
6. #20242160C2semidirect
7. #20242141C2semidirect
8. #20242138C2semidirect
9. #20242134C2semidirect
10. C2×#15688873C2semidirect
224. C47#17811semidirect

Maximal subgroups

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

subgroupindex
#202524992
#202520322
#202421792
#202421742
#202421692
#202421602
#202421412
#202421382
#202421342
C2×#156888732
C47×#29892
#1781147