Finite Group Databasebeta

← #26379102

Subgroups — #26379102

Order 12032.

2238 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
1C22×#13094024C211yesmaximal, characteristic
2#20246568C214yesmaximal, complemented
3#20246477C214yesmaximal, complemented
4#20246259C214yesmaximal
5#20246255C214yesmaximal
6#20246136C214yesmaximal, complemented
7#20246134C214yesmaximal, complemented
8#20245973C214yesmaximal
9#20245786C211yesmaximal, characteristic
10C47×#2283C211yesmaximal, characteristic
11C23×Dic94C2212yes—
12C2×#13094024C4116yescomplemented
13C2×#13094024C2212yes—
14C2×#13094024C2212yes—
15C2×#13094024C2212yes—
16C2×#13094024C2212yes—
17C2×#13093980C2218yes—
18C22×C4⋊Dic47C2212yes—
19C22×C4⋊Dic47C2212yes—
20#15689027C22116yes—
21#15689027C22116yes—
22#15689021C2218yes—
23#15689019C2218yes—
24#15689008C22116yes—
25#15689006C22116yes—
26#15688991C2212yes—
27#15688989C2212yes—
28#15688986C2212yes—
29#15688984C2218yes—
30#15688983C2212yes—
31C4⋊C4×Dic47C2218yes—
32C4×C4⋊Dic47C2212yes—
33C4×C4⋊Dic47C2218yes—
34C4×C22⋊Dic47C2212yes—
35#15688969C2212yes—
36C22×C42×C47C2211yescharacteristic
37C47×#383C2212yes—
38C47×#382C2212yes—
39C47×#381C2212yes—
40C47×#380C2212yes—
41C4×C4⋊C4×C47C2212yes—
42C4×C22⋊C4×C47C2212yes—
43C47×#376C2212yes—
44C22×Dic94—48no—
45C22×Dic94C2×C4116yes—
46C22×Dic94C2314yes—
47C22×Dic94C2314yes—
48C23×Dic47C2314yes—
49#13094024C2×C4116yes—
50#13094024C2×C4116yes—
51#13094024C2×C4116yes—
52#13094024C2×C4116yes—
53#13093980—416no—
54C2×C4⋊Dic47C2×C4116yes—
55C2×C4⋊Dic47C2×C4116yes—
56C2×C4⋊Dic47C2314yes—
57C2×C4⋊Dic47C2314yes—
58C2×C4⋊Dic47C2314yes—
59C2×C4⋊Dic47C2314yes—
60C2×C4⋊Dic47C2314yes—
61C2×Dic47⋊C4C23116yes—
62C2×C22⋊Dic47C2314yes—
63C2×C4×Dic47C2314yes—
64C2×C4×Dic47C23116yes—
65C47⋊#95C2314yes—
66C47⋊#95C2314yes—
67C47⋊#95C2314yes—
68C47⋊#95C23116yes—
69C47⋊#95C2314yes—
70C47⋊#95C23116yes—
71C23×C4×C47C2311yescharacteristic
72C23×C4×C47C2311yescharacteristic
73C23×C4×C47C2311yescharacteristic
74C2×C4⋊C4×C47C2314yes—
75C2×C22⋊C4×C47C2314yes—
76C2×C42×C47C2×C418yes—
77C2×C42×C47C2314yes—
78C2×C42×C47C2311yescharacteristic
79C2×C42×C47C2311yescharacteristic
80C2×C42×C47C2311yescharacteristic
81C2×C42×C47C2311yescharacteristic
82#95×C47C2314yes—
83#95×C47C2314yes—
84#95×C47C2314yes—
85#95×C47C2314yes—
86C24×C47C2411yescharacteristic
87C22×C4×C47—24no—
88C22×C4×C47—24no—
89C22×C4×C47—24no—
90C22×C4×C47C22×C414yes—
91C22×C4×C47C22×C414yes—
92C22×C4×C47C22×C414yes—
93C22×C4×C47C2418yes—
94C22×C4×C47C2411yescharacteristic
95C22×C4×C47C2411yescharacteristic
96C22×C4×C47C2411yescharacteristic
97C22×C4×C47C2411yescharacteristic
98C22×C4×C47C2411yescharacteristic
99C22×C4×C47C2411yescharacteristic
100C22×Dic47—216no—
101C22×Dic47C22×C4116yes—
102C22×Dic47C2418yes—
103C22×Dic47C2418yes—
104C2×Dic94—432no—
105C2×Dic94—44no—
106C2×Dic94—48no—
107C2×Dic94—44no—
108C2×Dic94C22×C4116yes—
109C2×Dic94C22×C4116yes—
110C4⋊C4×C47—28no—
111C22⋊C4×C47—28no—
112C42×C47—28no—
113C42×C47C22×C414yes—
114C42×C47C22×C414yes—
115C42×C47C22×C414yes—
116C42×C47C22×C414yes—
117C22⋊Dic47—28no—
118C4⋊Dic47—28no—
119C4⋊Dic47C22×C4116yes—
120C4⋊Dic47C22×C4116yes—
121C4⋊Dic47C22×C4116yes—
122C4⋊Dic47C22×C4116yes—
123Dic47⋊C4—416no—
124C4×Dic47—28no—
125C4×Dic47—416no—
126C23×C47—22no—
127C23×C47—22no—
128C23×C47C23×C412yes—
129C23×C47H(5,𝔽2)14yes—
130C23×C47C2511yescharacteristic
131C2×C4×C47—28no—
132C2×C4×C47—22no—
133C2×C4×C47—22no—
134C2×C4×C47—22no—
135C2×C4×C47—22no—
136C2×C4×C47—28no—
137C2×C4×C47—28no—
138C2×C4×C47—28no—
139C2×C4×C47—28no—
140C2×C4×C47—28no—
141C2×C4×C47C23×C412yes—
142C2×C4×C47C23×C412yes—
143C2×C4×C47C23×C412yes—
144C2×C4×C47C23×C412yes—
145C2×C4×C47C23×C412yes—
146C2×C4×C47C23×C412yes—
147C2×C4×C472−1+414yes—
148C2×C4×C472−1+414yes—
149C2×Dic47—232no—
150C2×Dic47—28no—
151C2×Dic47—28no—
152C2×Dic47—28no—
153C2×Dic47—28no—
154C2×Dic47—28no—
155C2×Dic47C23×C4116yes—
156Dic94—416no—
157Dic94—416no—
158#34365—471nomaximal, Sylow
159C22×C47—22no—
160C22×C47—22no—
161C22×C47—28no—
162C22×C47#51814yes—
163C22×C47C24×C411yescharacteristic
164C22×C47C2×H(5,𝔽2)12yes—
165C22×C47C2×2−1+412yes—
166C22×C47C2×2−1+412yes—
167C4×C47—22no—
168C4×C47—22no—
169C4×C47—22no—
170C4×C47—22no—
171C4×C47—48no—
172C4×C47#51914yes—
173C4×C47#51914yes—
174Dic47—216no—
175Dic47—48no—
176C22×C4⋊Q8—471no—
177#2381—474no—
178#2369—474no—
179#2344—474no—
180#2344—474no—
181#2331—474no—
182#2331—474no—
183#2310—474no—
184#2283—471no—
185#2283D4711yescharacteristic
186C2×C47—24no—
187C2×C47C2×#51811yescharacteristic
188C2×C47C2×#51911yescharacteristic
189C2×C47C2×#51911yescharacteristic
190C2×C47#354414yes—
191C23×Q8—472no—
192C2×C4⋊Q8—4716no—
193C2×C4⋊Q8—472no—
194C2×C4⋊Q8—472no—
195C2×C4⋊Q8—472no—
196C2×C4⋊Q8—472no—
197C2×#117—478no—
198C22×C4⋊C4—472no—
199C22×C4⋊C4—472no—
200C22×C42C2×D4711yescharacteristic
201#391—4716no—
202#391—4716no—
203#389—478no—
204#389—478no—
205#387—4716no—
206#387—4716no—
207#383—472no—
208#383C2×D4712yes—
209#382—472no—
210#382C2×D4712yes—
211#381—472no—
212#381—478no—
213#381C2×D4712yes—
214#380—472no—
215#380C2×D4712yes—
216C4×C4⋊C4—478no—
217C4×C4⋊C4—478no—
218C4×C4⋊C4—472no—
219C4×C4⋊C4C2×D4712yes—
220C4×C22⋊C4—472no—
221C4×C22⋊C4C2×D4712yes—
222#376—472no—
223#376C2×D4712yes—
224C47#3436511yescharacteristic, complemented, Sylow
225C22×Q8—4716no—
226C22×Q8—474no—
227C22×Q8—474no—
228C22×Q8—1888no—
229C23×C4—474no—
230C23×C4C22×D4711yescharacteristic
231C23×C4C22×D4711yescharacteristic
232C23×C4C22×D4711yescharacteristic
233C4⋊Q8—4716no—
234C4⋊Q8—4716no—
235C4⋊Q8—4716no—
236C4⋊Q8—4716no—
237#117—18816no—
238C2×C4⋊C4—4716no—
239C2×C4⋊C4—4716no—
240C2×C4⋊C4—474no—
241C2×C4⋊C4—474no—
242C2×C4⋊C4—474no—
243C2×C4⋊C4—474no—
244C2×C4⋊C4—474no—
245C2×C4⋊C4—4716no—
246C2×C4⋊C4C22×D4714yes—
247C2×C22⋊C4—474no—
248C2×C22⋊C4C22×D4714yes—
249C2×C42—474no—
250C2×C42—4716no—
251C2×C42C4×D4718yes—
252C2×C42C22×D4714yes—
253C2×C42C22×D4711yescharacteristic
254C2×C42C22×D4711yescharacteristic
255C2×C42C22×D4711yescharacteristic
256C2×C42C22×D4711yescharacteristic
257#95—474no—
258#95—474no—
259#95—474no—
260#95—474no—
261#95—4716no—
262#95—4716no—
263#95C22×D4714yes—
264#95C22×D4714yes—
265#95C22×D4714yes—
266#95C22×D4714yes—
267C24C23×D4711yescharacteristic
268C2×Q8—4716no—
269C2×Q8—4716no—
270C2×Q8—18832no—
271C2×Q8—1884no—
272C2×Q8—1888no—
273C2×Q8—1884no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—4716no—
278C22×C4—478no—
279C22×C4—478no—
280C22×C4—9416no—
281C22×C4C2×C4×D4714yes—
282C22×C4C2×C4×D4714yes—
283C22×C4C2×C4×D4714yes—
284C22×C4C23×D4718yes—
285C22×C4C23×D4711yescharacteristic
286C22×C4C23×D4711yescharacteristic
287C22×C4C23×D4711yescharacteristic
288C22×C4C23×D4711yescharacteristic
289C22×C4C23×D4711yescharacteristic
290C22×C4C23×D4711yescharacteristic
291C4⋊C4—28no—
292C4⋊C4—4716no—
293C4⋊C4—4716no—
294C4⋊C4—4716no—
295C4⋊C4—4716no—
296C4⋊C4—948no—
297C4⋊C4—18816no—
298C22⋊C4—28no—
299C22⋊C4—948no—
300C42—28no—
301C42—948no—
302C42—18816no—
303C42C2×C4×D4714yes—
304C42C2×C4×D4714yes—
305C42C2×C4×D4714yes—
306C42C2×C4×D4714yes—
307C23—22no—
308C23—22no—
309C23C22×C4×D4712yes—
310C23#1309407014yes—
311C23C24×D4711yescharacteristic
312Q8—18816no—
313Q8—18816no—
314C2×C4—28no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—28no—
320C2×C4—28no—
321C2×C4—28no—
322C2×C4—28no—
323C2×C4—28no—
324C2×C4—4716no—
325C2×C4—9432no—
326C2×C4—948no—
327C2×C4—948no—
328C2×C4—948no—
329C2×C4—948no—
330C2×C4—948no—
331C2×C4C22×C4×D4712yes—
332C2×C4C22×C4×D4712yes—
333C2×C4C22×C4×D4712yes—
334C2×C4C22×C4×D4712yes—
335C2×C4C22×C4×D4712yes—
336C2×C4C22×C4×D4712yes—
337C2×C4#1309407114yes—
338C2×C4#1309407114yes—
339C22—22no—
340C22—22no—
341C22—28no—
342C22#1568959714yes—
343C22C23×C4×D4711yescharacteristic
344C22C2×#1309407012yes—
345C22C2×#1309407112yes—
346C22C2×#1309407112yes—
347C4—22no—
348C4—22no—
349C4—22no—
350C4—22no—
351C4—48no—
352C4—9416no—
353C4—1888no—
354C4#1568960014yes—
355C4#1568960014yes—
356C2—24no—
357C2C2×#1568959711yescharacteristic
358C2C2×#1568960011yescharacteristic
359C2C2×#1568960011yescharacteristic
360C2#2025772814yes—
subgroupG/H
center, FrattiniC23C24×D47
commutatorC22×C47C24×C4
FittingC47×#2283C2
socle, cosocleC23×C47C25
prime radicalC24×C47C24
hypercenter#2283D47
hypocenterC47#34365
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ #2283

Complements

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

NH ≅ G/Ntype
2. #20246568C2semidirect
3. #20246477C2semidirect
6. #20246136C2semidirect
7. #20246134C2semidirect
12. C2×#13094024C4semidirect
224. C47#34365semidirect

Maximal subgroups

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

subgroupindex
C22×#130940242
#202465682
#202464772
#202462592
#202462552
#202461362
#202461342
#202459732
#202457862
C47×#22832
#3436547