Finite Group Databasebeta

← #26115393

Subgroups — #26115393

Order 12032.

590 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251746C211yesmaximal, characteristic
2#20243504C211yesmaximal, characteristic, complemented
3#20243243C211yesmaximal, characteristic, complemented
4C2×#15688869C211yesmaximal, characteristic, complemented
5C2×#15688866C211yesmaximal, characteristic
6C2×#15688866C211yesmaximal, characteristic
7C2×C47×#332C211yesmaximal, characteristic, complemented
8C2×#13094021C2211yescharacteristic
9C2×#13094013—22no—
10C2×C4⋊(C47⋊C8)C2211yescharacteristic
11C2×C4⋊(C47⋊C8)C2211yescharacteristic
12C2×#13093934—21no—
13#15689024—21no—
14#15689024—21no—
15#15689024C414yes—
16C42×Dic47C2211yescharacteristic
17#15688877—21no—
18#15688877—22no—
19#15688869—22no—
20#15688866—22no—
21#15688866—22no—
22C2×C4⋊Q8×C47C2211yescharacteristic
23C2×C4⋊C8×C47C2211yescharacteristic
24C2×C4⋊C8×C47C2211yescharacteristic
25C47×#332—22no—
26C22×D188—21no—
27C22×D188—21no—
28C22×D188C2×C412yes—
29C22×C47⋊C8—21no—
30C22×C47⋊C8—21no—
31#13094021—21no—
32#13094021—21no—
33#13094021—21no—
34#13094021—21no—
35#13094013—44no—
36#13094013—44no—
37C2×(C2×D47)⋊C4—24no—
38C2×(C2×D47)⋊C4—24no—
39C2×C4×Dic47C2×C412yes—
40C2×C4×Dic47D411yescharacteristic
41C2×C4×Dic47D411yescharacteristic
42C2×C4×Dic47D412yes—
43C4⋊(C47⋊C8)—21no—
44C4⋊(C47⋊C8)—21no—
45C4⋊(C47⋊C8)—21no—
46C4⋊(C47⋊C8)D412yes—
47#13093934—42no—
48C22×Q8×C47—21no—
49C22×C8×C47—21no—
50C22×C8×C47—21no—
51C4⋊Q8×C47—21no—
52C4⋊Q8×C47—21no—
53C4⋊Q8×C47—21no—
54C4⋊Q8×C47—21no—
55C2×C4⋊C4×C47—21no—
56C2×C4⋊C4×C47C2×C412yes—
57C2×C42×C47C2311yescharacteristic
58C4⋊C8×C47—21no—
59C4⋊C8×C47—21no—
60C4⋊C8×C47—21no—
61C4⋊C8×C47D412yes—
62C23×D47—22no—
63C2×Q8×C47—41no—
64C2×Q8×C47—41no—
65C2×Q8×C47—42no—
66C2×Q8×C47—42no—
67C22×C4×C47—22no—
68C22×C4×C47C22×C411yescharacteristic
69C22×C4×C47C2×D411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C22×Dic47C22⋊C412yes—
72C22×Dic47C22⋊C412yes—
73C2×D188—22no—
74C2×D188—22no—
75C2×D188—44no—
76C2×D188—41no—
77C2×D188—41no—
78C2×D188—41no—
79C2×D188—41no—
80C2×D188—41no—
81C2×D188—41no—
82C2×D188—42no—
83C2×D188—42no—
84C2×D188—41no—
85C2×D188—41no—
86C2×C8×C47—22no—
87C2×C8×C47—41no—
88C2×C8×C47—41no—
89C2×C8×C47—41no—
90C2×C8×C47—42no—
91C4⋊C4×C47—22no—
92C4⋊C4×C47—22no—
93C4⋊C4×C47—41no—
94C4⋊C4×C47—41no—
95C42×C47C2×D411yescharacteristic
96C42×C47C2×D411yescharacteristic
97C42×C47#4111yescharacteristic
98C42×C47#4111yescharacteristic
99(C2×D47)⋊C4—48no—
100(C2×D47)⋊C4—48no—
101C4×Dic47—21no—
102C4×Dic47—21no—
103C4×Dic47—21no—
104C4×Dic47—22no—
105C4×Dic47—22no—
106C4×Dic47—21no—
107C4×Dic47—42no—
108C2×C47⋊C8—22no—
109C2×C47⋊C8—41no—
110C2×C47⋊C8—41no—
111C2×C47⋊C8—41no—
112C2×C47⋊C8—42no—
113C23×C47C2×C22⋊C411yescharacteristic
114C22×D47—42no—
115C22×D47—44no—
116C22×D47—42no—
117C22×D47—42no—
118C22×D47—44no—
119Q8×C47—82no—
120Q8×C47—81no—
121Q8×C47—81no—
122C2×C4×C47—22no—
123C2×C4×C47—42no—
124C2×C4×C47—44no—
125C2×C4×C47C4×D411yescharacteristic
126C2×C4×C47C4×D411yescharacteristic
127C2×C4×C47#12111yescharacteristic
128C2×C4×C47#12111yescharacteristic
129C2×C4×C47#12411yescharacteristic
130C2×C4×C47#12711yescharacteristic
131C2×C4×C47#13511yescharacteristic
132C2×C4×C47#13511yescharacteristic
133C2×C4×C47AGL(1,ℤ/8)12yes—
134C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
135C2×C4×C47AGL(1,ℤ/8)12yes—
136C2×C4×C47#13711yescharacteristic
137C2×Dic47—22no—
138C2×Dic47—22no—
139C2×Dic47—22no—
140C2×Dic47—22no—
141C2×Dic47—22no—
142C2×Dic47—22no—
143D188—82no—
144D188—82no—
145D188—81no—
146D188—81no—
147D188—81no—
148D188—81no—
149D188—82no—
150D188—81no—
151D188—81no—
152C8×C47—42no—
153C8×C47—81no—
154C47⋊C8—42no—
155C47⋊C8—81no—
156#9710—471nomaximal, Sylow
157C22×C47#39011yescharacteristic
158C22×C47#41312yes—
159C22×C47#41611yescharacteristic
160C22×C47#41711yescharacteristic
161C22×C47#41812yes—
162C2×D47—84no—
163C2×D47—84no—
164C2×D47—82no—
165C2×D47—82no—
166C2×D47—82no—
167C4×C47—21no—
168C4×C47—21no—
169C4×C47—82no—
170C4×C47#46311yescharacteristic
171C4×C47#46311yescharacteristic
172C4×C47#46411yescharacteristic
173C4×C47#46411yescharacteristic
174C4×C47#49012yes—
175C4×C47#49611yescharacteristic
176C4×C47#49611yescharacteristic
177Dic47—42no—
178Dic47—42no—
179#2858—471no—
180#1831—471no—
181#1767—471no—
182C2×#332—471no—
183C2×#332D4711yescharacteristic, complemented
184C2×#331—471no—
185C2×#331—471no—
186C2×C47#155312yes—
187C2×C47#155712yes—
188C2×C47#196111yescharacteristic
189C2×C47#196711yescharacteristic
190C2×C47#198311yescharacteristic
191D47—162no—
192C2×C4⋊Q8C2×D4711yescharacteristic
193C2×#127—471no—
194C2×#124—942no—
195C2×C4⋊C8—471no—
196C2×C4⋊C8—471no—
197C2×C4⋊C8C2×D4711yescharacteristic
198C2×C4⋊C8C2×D4711yescharacteristic
199C2×#97—941no—
200#390—474no—
201#390—941no—
202#390—941no—
203C43—471no—
204#336—942no—
205#336—941no—
206#332—22no—
207#332—942no—
208#331—942no—
209#331—942no—
210C47#971011yescharacteristic, complemented, Sylow
211C22×Q8—21no—
212C22×D4—472no—
213C22×D4—941no—
214C22×D4—941no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—941no—
218C22×C8—941no—
219C4⋊Q8—21no—
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223#127—941no—
224#127—941no—
225#127—941no—
226#127—941no—
227#124—1884no—
228#124—1884no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4C4×D4712yes—
231C2×C22⋊C4—944no—
232C2×C22⋊C4—944no—
233C2×C42—472no—
234C2×C42—471no—
235C2×C42—471no—
236C2×C42—472no—
237C2×C42C22×D4711yescharacteristic
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—472no—
242C4⋊C8—941no—
243C4⋊C8—941no—
244C4⋊C8—941no—
245C4⋊C8C22⋊D4712yes—
246#97—1882no—
247C24—942no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×D4—942no—
253C2×D4—942no—
254C2×D4—1884no—
255C2×D4—1881no—
256C2×D4—1881no—
257C2×D4—1881no—
258C2×D4—1881no—
259C2×D4—1881no—
260C2×D4—1881no—
261C2×D4—1882no—
262C2×D4—1882no—
263C2×D4—1881no—
264C2×D4—1881no—
265C22×C4—22no—
266C22×C4—472no—
267C22×C4—472no—
268C22×C4C2×C4×D4711yescharacteristic
269C22×C4D4×D4711yescharacteristic
270C22×C4D4×D4711yescharacteristic
271C2×C8—22no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—42no—
276C2×C8—942no—
277C2×C8—1881no—
278C2×C8—1882no—
279C2×C8—1881no—
280C2×C8—1881no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C22⋊C4—1888no—
286C22⋊C4—1888no—
287C42—941no—
288C42—941no—
289C42—942no—
290C42—941no—
291C42—942no—
292C42—941no—
293C42—1882no—
294C42#1062535011yescharacteristic
295C42D4×D4711yescharacteristic
296C42#1062535211yescharacteristic
297C42C2×C22⋊D4711yescharacteristic
298C23—1882no—
299C23—1884no—
300C23—1882no—
301C23—1882no—
302C23—1884no—
303C23C22⋊C4×D4711yescharacteristic
304Q8—82no—
305Q8—81no—
306Q8—81no—
307D4—3762no—
308D4—3762no—
309D4—3762no—
310D4—3761no—
311D4—3761no—
312D4—3761no—
313D4—3761no—
314D4—3761no—
315D4—3761no—
316C2×C4—22no—
317C2×C4—42no—
318C2×C4—44no—
319C2×C4—942no—
320C2×C4—942no—
321C2×C4—942no—
322C2×C4—942no—
323C2×C4—942no—
324C2×C4—942no—
325C2×C4(C22⋊D47)⋊C411yescharacteristic
326C2×C4C4×C22⋊D4711yescharacteristic
327C2×C4#1309399611yescharacteristic
328C2×C4#1309399711yescharacteristic
329C2×C4#1309401011yescharacteristic
330C2×C4#1309402011yescharacteristic
331C2×C4#1309404511yescharacteristic
332C2×C4#1309404711yescharacteristic
333C2×C4#1309404812yes—
334C2×C4#1309405112yes—
335C2×C4#1309405111yescharacteristic
336C2×C4#1309405611yescharacteristic
337C8—42no—
338C8—81no—
339C8—3761no—
340C8—1882no—
341C22—3762no—
342C22—3762no—
343C22—3764no—
344C22—3764no—
345C22—3762no—
346C22#1568902311yescharacteristic
347C22#1568910612yes—
348C22#1568911811yescharacteristic
349C22#1568912311yescharacteristic
350C22#1568912612yes—
351C4—21no—
352C4—21no—
353C4—82no—
354C4—1882no—
355C4—1882no—
356C4#1568934011yescharacteristic
357C4#1568934111yescharacteristic
358C4#1568934711yescharacteristic
359C4#1568934811yescharacteristic
360C4#1568948612yes—
361C4#1568950911yescharacteristic
362C4#1568951111yescharacteristic
363C2—7522no—
364C2#2024209212yes—
365C2#2024210912yes—
366C2#2024421011yescharacteristic
367C2#2024423611yescharacteristic
368C2#2024431711yescharacteristic
subgroupG/H
centerC23C22⋊C4×D47
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC2×C47×#332C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#13094021C22
cosocleC2×C42×C47C23
hypercenterC2×#332D47
hypocenterC47#9710
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
2. #20243504C2semidirect
3. #20243243C2semidirect
4. C2×#15688869C2semidirect
7. C2×C47×#332C2semidirect
183. C2×#332D47semidirect
210. C47#9710semidirect

Maximal subgroups

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

subgroupindex
#202517462
#202435042
#202432432
C2×#156888692
C2×#156888662
C2×C47×#3322
#971047