Finite Group Databasebeta

← #26216781

Subgroups — #26216781

Order 12032.

854 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689184C212yesmaximal
2#20243540C212yesmaximal, complemented
3#20243508C212yesmaximal, complemented
4#20243372C212yesmaximal
5#20243368C212yesmaximal
6#20243245C212yesmaximal, complemented
7#20243241C211yesmaximal, characteristic, complemented
8C4×C4⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
9C4×#117×C47C211yesmaximal, characteristic
10#15689184—28no—
11C2×C4⋊(C47⋊C8)C2212yes—
12C2×C4⋊(C47⋊C8)C2212yes—
13C2×C4⋊(C47⋊C8)C2214yes—
14C2×C4⋊(C47⋊C8)C2214yes—
15C2×C22⋊(C47⋊C8)C2214yes—
16C2×#13093935C2212yes—
17C2×C4×C47⋊C8C2212yes—
18#15688878C2212yes—
19#15688878C2212yes—
20#15688878C2214yes—
21C2×#117×C47C2211yescharacteristic
22C2×#117×C47C2211yescharacteristic
23C22×C42×C47C2211yescharacteristic
24C4×C4⋊C4×C47C412yes—
25C4×C4⋊C4×C47C412yes—
26C4×C22⋊C4×C47C412yes—
27C4×C22⋊C4×C47C412yes—
28C47×#376C2211yescharacteristic
29C47×#376C2211yescharacteristic
30C43×C47C2211yescharacteristic
31C43×C47C2211yescharacteristic
32C22×C47⋊C8C2314yes—
33C22×C47⋊C8C2314yes—
34C4⋊(C47⋊C8)—28no—
35C4⋊(C47⋊C8)—24no—
36C4⋊(C47⋊C8)—24no—
37C4⋊(C47⋊C8)—28no—
38C22⋊(C47⋊C8)—28no—
39#13093935—24no—
40C4×C47⋊C8—24no—
41C23×C4×C47C2×C411yescharacteristic
42C23×C4×C47C2×C411yescharacteristic
43C23×C4×C47C2311yescharacteristic
44#117×C47—24no—
45#117×C47D414yes—
46#117×C47Q814yes—
47C2×C4⋊C4×C47C2×C412yes—
48C2×C4⋊C4×C47C2×C412yes—
49C2×C22⋊C4×C47C2×C412yes—
50C2×C22⋊C4×C47C2×C412yes—
51C2×C42×C47—24no—
52C2×C42×C47C2×C411yescharacteristic
53C2×C42×C47C2×C411yescharacteristic
54C2×C42×C47C2×C412yes—
55C2×C42×C47C2×C412yes—
56C2×C42×C47C2×C412yes—
57C2×C42×C47C2×C412yes—
58C2×C42×C47C2311yescharacteristic
59C2×C42×C47C2311yescharacteristic
60C2×C42×C47C2311yescharacteristic
61C2×C42×C47C2311yescharacteristic
62C2×C42×C47C2311yescharacteristic
63C2×C42×C47C2311yescharacteristic
64#95×C47C2×C412yes—
65#95×C47C2×C412yes—
66#95×C47C2×C412yes—
67#95×C47C2×C412yes—
68C24×C47C22×C411yescharacteristic
69C22×C4×C47—22no—
70C22×C4×C47—22no—
71C22×C4×C47—22no—
72C22×C4×C47—22no—
73C22×C4×C47—24no—
74C22×C4×C47—42no—
75C22×C4×C47C22×C412yes—
76C22×C4×C47C22×C411yescharacteristic
77C22×C4×C47C22×C411yescharacteristic
78C22×C4×C47C22×C412yes—
79C22×C4×C47C22×C412yes—
80C22×C4×C47C22×C411yescharacteristic
81C22×C4×C47C22×C411yescharacteristic
82C22×C4×C47C22×C412yes—
83C22×C4×C47C22×C411yescharacteristic
84C22×C4×C47C2×D412yes—
85C22×C4×C47C2×Q812yes—
86C22×C4×C47C2411yescharacteristic
87C4⋊C4×C47—24no—
88C4⋊C4×C47C4⋊C418yes—
89C22⋊C4×C47—24no—
90C22⋊C4×C47C4⋊C418yes—
91C42×C47—22no—
92C42×C47—22no—
93C42×C47—24no—
94C42×C47—24no—
95C42×C47—22no—
96C42×C47—22no—
97C42×C47—42no—
98C42×C47—42no—
99C42×C47C2×D412yes—
100C42×C47C2×D412yes—
101C42×C47C2×Q812yes—
102C42×C47C2×Q812yes—
103C42×C47#4114yes—
104C42×C47#4114yes—
105C2×C47⋊C8—24no—
106C2×C47⋊C8—24no—
107C2×C47⋊C8—24no—
108C2×C47⋊C8—24no—
109C2×C47⋊C8—24no—
110C2×C47⋊C8—24no—
111C23×C47—21no—
112C23×C47—21no—
113C23×C47—22no—
114C23×C47—22no—
115C23×C47C2×C4⋊C412yes—
116C23×C47C23×C411yescharacteristic
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—22no—
120C2×C4×C47—22no—
121C2×C4×C47—22no—
122C2×C4×C47—22no—
123C2×C4×C47—22no—
124C2×C4×C47—22no—
125C2×C4×C47—22no—
126C2×C4×C47—24no—
127C2×C4×C47—24no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—22no—
131C2×C4×C47—22no—
132C2×C4×C47—22no—
133C2×C4×C47—22no—
134C2×C4×C47—22no—
135C2×C4×C47—44no—
136C2×C4×C47—44no—
137C2×C4×C47C2×C4⋊C414yes—
138C2×C4×C47C2×C4⋊C414yes—
139C2×C4×C47C2×C4⋊C412yes—
140C2×C4×C47#11712yes—
141C2×C4×C47#11714yes—
142C2×C4×C47#11712yes—
143C2×C4×C47#13114yes—
144C2×C4×C47C22×D411yescharacteristic
145C2×C4×C47C22×Q811yescharacteristic
146C2×C4×C47C2×#4112yes—
147C2×C4×C47C2×#4112yes—
148C47⋊C8—44no—
149C47⋊C8—44no—
150#20812—471nomaximal, Sylow
151C22×C47—24no—
152C22×C47—22no—
153C22×C47—41no—
154C22×C47—41no—
155C22×C47—42no—
156C22×C47C22×C4⋊C411yescharacteristic
157C22×C47C2×#11711yescharacteristic
158C22×C47C2×#11711yescharacteristic
159C22×C47C2×#13112yes—
160C22×C47#56812yes—
161C4×C47—24no—
162C4×C47—24no—
163C4×C47—22no—
164C4×C47—24no—
165C4×C47—22no—
166C4×C47—41no—
167C4×C47—41no—
168C4×C47—42no—
169C4×C47#43314yescomplemented
170C4×C47#53212yes—
171C4×C47#53312yes—
172C2×#433—472no—
173C4×#117D4711yescharacteristic
174#1842—472no—
175#1832—472no—
176#1798—472no—
177#1797—472no—
178#1768—472no—
179#1767—471no—
180C4×C4⋊C8—471no—
181C2×C47—42no—
182C2×C47#228511yescharacteristic
183C2×C47#290612yes—
184C2×C47C2×#43311yescharacteristic
185C2×C47#292111yescharacteristic
186C2×C47#292512yes—
187C2×#117C2×D4711yescharacteristic
188C2×#117C2×D4711yescharacteristic
189C22×C42C2×D4711yescharacteristic
190#433—948no—
191C2×C4⋊C8—472no—
192C2×C4⋊C8—472no—
193C2×C4⋊C8—474no—
194C2×C4⋊C8—474no—
195C2×C22⋊C8—474no—
196C2×#97—472no—
197C2×C4×C8—472no—
198C4×C4⋊C4Dic4712yes—
199C4×C4⋊C4Dic4712yes—
200C4×C22⋊C4Dic4712yes—
201C4×C22⋊C4Dic4712yes—
202#376C2×D4711yescharacteristic
203#376C2×D4711yescharacteristic
204C43C2×D4711yescharacteristic
205C43C2×D4711yescharacteristic
206#336—474no—
207#336—472no—
208#336—472no—
209C47#2081211yescharacteristic, complemented, Sylow
210C23×C4C2×Dic4711yescharacteristic
211C23×C4C2×Dic4711yescharacteristic
212C23×C4C22×D4711yescharacteristic
213C22×C8—474no—
214C22×C8—474no—
215#117—24no—
216#117Dic9414yes—
217#117D18814yes—
218C2×C4⋊C4C2×Dic4712yes—
219C2×C4⋊C4C2×Dic4712yes—
220C2×C22⋊C4C2×Dic4712yes—
221C2×C22⋊C4C2×Dic4712yes—
222C2×C42—24no—
223C2×C42C2×Dic4712yes—
224C2×C42C2×Dic4712yes—
225C2×C42C2×Dic4711yescharacteristic
226C2×C42C2×Dic4711yescharacteristic
227C2×C42C2×Dic4712yes—
228C2×C42C2×Dic4712yes—
229C2×C42C22×D4711yescharacteristic
230C2×C42C22×D4711yescharacteristic
231C2×C42C22×D4711yescharacteristic
232C2×C42C22×D4711yescharacteristic
233C2×C42C22×D4711yescharacteristic
234C2×C42C22×D4711yescharacteristic
235C4⋊C8—948no—
236C4⋊C8—944no—
237C4⋊C8—944no—
238C4⋊C8—948no—
239C22⋊C8—948no—
240#97—944no—
241C4×C8—944no—
242#95C2×Dic4712yes—
243#95C2×Dic4712yes—
244#95C2×Dic4712yes—
245#95C2×Dic4712yes—
246C24C22×Dic4711yescharacteristic
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—42no—
253C22×C4C2×Dic9412yes—
254C22×C4C2×D18812yes—
255C22×C4C22×Dic4711yescharacteristic
256C22×C4C22×Dic4711yescharacteristic
257C22×C4C22×Dic4712yes—
258C22×C4C22×Dic4712yes—
259C22×C4C22×Dic4712yes—
260C22×C4C22×Dic4711yescharacteristic
261C22×C4C22×Dic4712yes—
262C22×C4C22×Dic4711yescharacteristic
263C22×C4C22×Dic4711yescharacteristic
264C22×C4C23×D4711yescharacteristic
265C2×C8—944no—
266C2×C8—944no—
267C2×C8—944no—
268C2×C8—944no—
269C2×C8—944no—
270C2×C8—944no—
271C4⋊C4—24no—
272C4⋊C4C4⋊Dic4718yes—
273C22⋊C4—24no—
274C22⋊C4C4⋊Dic4718yes—
275C42—22no—
276C42—22no—
277C42—24no—
278C42—24no—
279C42—22no—
280C42—22no—
281C42—42no—
282C42—42no—
283C42C2×Dic9412yes—
284C42C2×Dic9412yes—
285C42C2×D18812yes—
286C42C2×D18812yes—
287C42#1062535014yes—
288C42#1062535014yes—
289C23—21no—
290C23—21no—
291C23—22no—
292C23—22no—
293C23C2×C4⋊Dic4712yes—
294C23C23×Dic4711yescharacteristic
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4C2×C4⋊Dic4714yes—
316C2×C4C2×C4⋊Dic4714yes—
317C2×C4C2×C4⋊Dic4712yes—
318C2×C4#1309398312yes—
319C2×C4#1309398314yes—
320C2×C4#1309398312yes—
321C2×C4#1309403114yes—
322C2×C4C22×D18811yescharacteristic
323C2×C4C22×Dic9411yescharacteristic
324C2×C4C2×#1062535012yes—
325C2×C4C2×#1062535012yes—
326C8—1884no—
327C8—1884no—
328C22—22no—
329C22—24no—
330C22—41no—
331C22—41no—
332C22—42no—
333C22C22×C4⋊Dic4711yescharacteristic
334C22C2×#1309398311yescharacteristic
335C22C2×#1309398311yescharacteristic
336C22C2×#1309403112yes—
337C22#1568989012yes—
338C4—22no—
339C4—24no—
340C4—24no—
341C4—22no—
342C4—24no—
343C4—41no—
344C4—41no—
345C4—42no—
346C4#1568918414yescomplemented
347C4#1568967312yes—
348C4#1568968012yes—
349C2—42no—
350C2#2024579511yescharacteristic
351C2#2025195612yes—
352C2C2×#1568918411yescharacteristic
353C2#2025202611yescharacteristic
354C2#2025205412yes—
subgroupG/H
center, FrattiniC22×C4C23×D47
commutator, socleC23×C47C23×C4
FittingC4×#117×C47C2
prime radicalC24×C47C22×C4
cosocleC22×C4×C47C24
hypercenterC4×#117D47
hypocenterC47#20812
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C4×#117

Complements

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

NH ≅ G/Ntype
2. #20243540C2semidirect
3. #20243508C2semidirect
6. #20243245C2semidirect
7. #20243241C2semidirect
8. C4×C4⋊(C47⋊C8)C2semidirect
169. C4×C47#433semidirect
209. C47#20812semidirect
346. C4#15689184semidirect

Maximal subgroups

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

subgroupindex
C2×#156891842
#202435402
#202435082
#202433722
#202433682
#202432452
#202432412
C4×C4⋊(C47⋊C8)2
C4×#117×C472
#2081247