Finite Group Databasebeta

← #26186993

Subgroups — #26186993

Order 12032.

658 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253948C211yesmaximal, characteristic
2#20252101C211yesmaximal, characteristic
3#20242231C214yesmaximal, complemented
4#20242222C214yesmaximal, complemented
5#20242143C211yesmaximal, characteristic, complemented
6#20241843C211yesmaximal, characteristic, complemented
7#20241838C211yesmaximal, characteristic
8C2×#15688844C211yesmaximal, characteristic, complemented
9C47×#2960C211yesmaximal, characteristic, complemented
10C2×#13094030—21no—
11#15689673C2211yescharacteristic
12#15689519C411yescharacteristic
13#15689497C411yescharacteristic
14#15689473C411yescharacteristic
15#15689471C411yescharacteristic
16#15689354C414yes—
17C4×#10625336C2214yes—
18#15689202—21no—
19#15689202C2212yes—
20C8×D188—21no—
21C8×D188C2212yes—
22#15689147C2211yescharacteristic
23#15689089—21no—
24C2×C4×C47⋊C8C2211yescharacteristic
25C2×C4×C47⋊C8C2211yescharacteristic
26#15688873C2214yes—
27Q8⋊(C47⋊C8)C2214yes—
28#15688850C2214yes—
29#15688844C2212yes—
30#15688844C2212yes—
31C47×#533C2211yescharacteristic
32C47×#446—21no—
33C47×#446C2212yes—
34C8×Q8×C47—21no—
35C8×Q8×C47C2212yes—
36C47×#424—21no—
37C47×#423C2211yescharacteristic
38C47×#405—21no—
39C2×C4×C8×C47C2211yescharacteristic
40C2×#10625350—21no—
41C2×#10625336C2×C414yes—
42#13094030—44no—
43C2×#10625329—21no—
44C2×#10625324—22no—
45C22×C47⋊C8—21no—
46C22×C47⋊C8C2×C412yes—
47C2×C8×D47—22no—
48#13094024C2×C411yescharacteristic
49#13094021C2×C411yescharacteristic
50#13094012—21no—
51#13093998—21no—
52#13093998C2×C412yes—
53C4×D188—21no—
54C4×D188C2312yes—
55#13093955C2×C414yes—
56C4⋊(C47⋊C8)—21no—
57C4⋊(C47⋊C8)C2312yes—
58#13093948C2×C414yes—
59#13093944C2×C414yes—
60(C2×D47)⋊C8—22no—
61(C2×D47)⋊C8—22no—
62C4×C47⋊C8—21no—
63C4×C47⋊C8C2×C411yescharacteristic
64C4×C47⋊C8C2×C411yescharacteristic
65C4×C47⋊C8C2312yes—
66C4×C47⋊C8C2312yes—
67C2×M16×C47—21no—
68C22×C8×C47D411yescharacteristic
69C22×C8×C47D411yescharacteristic
70C4⋊Q8×C47C2×C411yescharacteristic
71C4⋊Q8×C47C2×C411yescharacteristic
72#125×C47—21no—
73C22⋊Q8×C47—21no—
74C22⋊Q8×C47C2×C412yes—
75C4×Q8×C47—21no—
76C4×Q8×C47C2312yes—
77#117×C47—21no—
78C2×C42×C47C2311yescharacteristic
79C4⋊C8×C47—22no—
80C4⋊C8×C47—22no—
81C4⋊C8×C47—21no—
82C4⋊C8×C47C2312yes—
83#97×C47—22no—
84C4×C8×C47—22no—
85C4×C8×C47D411yescharacteristic
86C4×C8×C47D411yescharacteristic
87C4×C8×C47C2311yescharacteristic
88C4×C8×C47C2311yescharacteristic
89C2×Q8×C47—21no—
90C2×Q8×C47C22×C412yes—
91C22×C4×C47C22×C411yescharacteristic
92C22×C4×C47C2×D411yescharacteristic
93C22×C4×C47#4111yescharacteristic
94C2×C22⋊D47—22no—
95#10625350—42no—
96#10625350—42no—
97C2×D188—21no—
98C2×D188—21no—
99C2×D188C22×C412yes—
100C2×C4×D47—22no—
101C2×Dic94—21no—
102M16×C47—42no—
103C2×C8×C47—22no—
104C2×C8×C47—22no—
105C2×C8×C47SD1614yescomplemented
106C2×C8×C47C2×D411yescharacteristic
107C2×C8×C47C2×D411yescharacteristic
108C2×C8×C47C2×D411yescharacteristic
109C2×C8×C47C2×D411yescharacteristic
110C4⋊C4×C47—21no—
111C4⋊C4×C47—21no—
112C4⋊C4×C47—21no—
113C4⋊C4×C47—22no—
114C4⋊C4×C47—22no—
115C4⋊C4×C47C22×C412yes—
116C22⋊C4×C47—22no—
117C42×C47—22no—
118C42×C47C22×C411yescharacteristic
119C42×C47C2×D411yescharacteristic
120C42×C47#4111yescharacteristic
121C42×C47C2411yescharacteristic
122#10625336—44no—
123(C2×D47)⋊C4—22no—
124(C2×D47)⋊C4—22no—
125C4⋊Dic47—21no—
126C4⋊Dic47C22×C412yes—
127#10625329—42no—
128C2×C47⋊C8—22no—
129C2×C47⋊C8—22no—
130C2×C47⋊C8—42no—
131C2×C47⋊C8—42no—
132C2×C47⋊C8C22×C412yes—
133C2×C47⋊C8C22×C412yes—
134#10625324—44no—
135C8×D47—44no—
136C23×C47C4×D411yescharacteristic
137C22×D47—22no—
138Q8×C47—24no—
139Q8×C47—42no—
140C2×C4×C47—22no—
141C2×C4×C47—22no—
142C2×C4×C47—22no—
143C2×C4×C47—22no—
144C2×C4×C47C4×D411yescharacteristic
145C2×C4×C47C4×D411yescharacteristic
146C2×C4×C47C4×D411yescharacteristic
147C2×C4×C47C2×SD1612yes—
148C2×C4×C47#13512yes—
149C2×C4×C47C23×C411yescharacteristic
150C2×C4×C47C22×D411yescharacteristic
151C2×C4×C47C2×#4111yescharacteristic
152C22⋊D47—44no—
153C2×Dic47—22no—
154D188—24no—
155D188—42no—
156D188—42no—
157C4×D47—42no—
158C4×D47—42no—
159Dic94—42no—
160C8×C47—21no—
161C8×C47—21no—
162C8×C47—42no—
163C8×C47C2×SD1612yes—
164C8×C47C2×SD1612yes—
165C47⋊C8—22no—
166C47⋊C8—22no—
167C47⋊C8—42no—
168C47⋊C8—42no—
169#17976—471nomaximal, Sylow
170C22×C47—21no—
171C22×C47—21no—
172C22×C47C4×SD1612yes—
173C22×C47C2×C4×D411yescharacteristic
174C2×D47—42no—
175C2×D47—42no—
176C2×D47—42no—
177C4×C47—21no—
178C4×C47—21no—
179C4×C47—42no—
180C4×C47—42no—
181C4×C47C4×SD1612yes—
182C4×C47#44412yes—
183C4×C47#56812yes—
184C4×C47C22×SD1611yescharacteristic
185C4×C47C2×#13511yescharacteristic
186Dic47—42no—
187#3167—471no—
188#2960D4711yescharacteristic, complemented
189#2931—471no—
190#1583—474no—
191#1582—474no—
192#1566—471no—
193#1493—471no—
194#1492—471no—
195C2×#322—471no—
196C2×C47—22no—
197C2×C47#293211yescharacteristic
198C2×C47C2×C4×SD1611yescharacteristic
199C2×C47C2×#44411yescharacteristic
200D47—82no—
201C2×#131—941no—
202#533C2×D4711yescharacteristic
203#532—471no—
204#498—471no—
205#492—471no—
206#487—471no—
207#486—471no—
208#465—474no—
209#446—21no—
210#446C2×D4712yes—
211C8×Q8—21no—
212C8×Q8C2×D4712yes—
213C4×SD16—474no—
214#436—472no—
215#436—941no—
216C8×D4—472no—
217C8×D4—941no—
218#424—21no—
219#423—471no—
220#423C2×D4711yescharacteristic
221#408—941no—
222#405—21no—
223C2×C4×C8—471no—
224C2×C4×C8—471no—
225C2×C4×C8C2×D4711yescharacteristic
226#334—474no—
227Q8⋊C8—474no—
228D4⋊C8—474no—
229#322—472no—
230#322—472no—
231C47#1797611yescharacteristic, complemented, Sylow
232C2×#41—941no—
233C2×SD16—474no—
234#131—1884no—
235C2×M16—21no—
236C2×M16—941no—
237C2×M16—942no—
238C22×C8—472no—
239C22×C8—941no—
240C22×C8—942no—
241C22×C8C22⋊D4711yescharacteristic
242C22×C8C22⋊D4711yescharacteristic
243C4⋊Q8—471no—
244C4⋊Q8C4×D4711yescharacteristic
245C4⋊Q8C4×D4711yescharacteristic
246#127—471no—
247#125—21no—
248#124—941no—
249C22⋊Q8—21no—
250C22⋊Q8C4×D4712yes—
251#121—472no—
252#121—941no—
253C4×Q8—21no—
254C4×Q8C22×D4712yes—
255C4×D4—472no—
256C4×D4—941no—
257#117—21no—
258C2×C42C22×D4711yescharacteristic
259#106—474no—
260C4⋊C8—22no—
261C4⋊C8—22no—
262C4⋊C8—21no—
263C4⋊C8—472no—
264C4⋊C8—941no—
265C4⋊C8C22×D4712yes—
266#103—474no—
267#102—474no—
268C22⋊C8—942no—
269C22⋊C8—942no—
270#97—22no—
271C4×C8—22no—
272C4×C8—471no—
273C4×C8—471no—
274C4×C8—472no—
275C4×C8—472no—
276C4×C8—941no—
277C4×C8C22⋊D4711yescharacteristic
278C4×C8C22⋊D4711yescharacteristic
279C4×C8C22×D4711yescharacteristic
280C4×C8C22×D4711yescharacteristic
281#41—1882no—
282#41—1882no—
283C2×Q8—21no—
284C2×Q8—941no—
285C2×Q8C2×C4×D4712yes—
286C2×D4—472no—
287C2×D4—942no—
288C2×D4—941no—
289C2×D4—941no—
290C22×C4—942no—
291C22×C4C2×C4×D4711yescharacteristic
292C22×C4#1062535011yescharacteristic
293C22×C4C2×C22⋊D4711yescharacteristic
294SD16—1884no—
295M16—42no—
296M16—1882no—
297M16—1884no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—472no—
301C2×C8—472no—
302C2×C8—942no—
303C2×C8—942no—
304C2×C8—1882no—
305C2×C8—1882no—
306C2×C8—1884no—
307C2×C8#1062533614yescomplemented
308C2×C8C2×C22⋊D4711yescharacteristic
309C2×C8C2×C22⋊D4711yescharacteristic
310C2×C8C2×C22⋊D4711yescharacteristic
311C2×C8C2×C22⋊D4711yescharacteristic
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—472no—
318C4⋊C4—941no—
319C4⋊C4C2×C4×D4712yes—
320C22⋊C4—22no—
321C22⋊C4—942no—
322C22⋊C4—942no—
323C42—22no—
324C42C2×C4×D4711yescharacteristic
325C42#1062535011yescharacteristic
326C42C2×C22⋊D4711yescharacteristic
327C42C23×D4711yescharacteristic
328C23—942no—
329C23C4×C22⋊D4711yescharacteristic
330Q8—24no—
331Q8—42no—
332Q8—1882no—
333D4—944no—
334D4—1884no—
335D4—1882no—
336D4—1882no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—942no—
342C2×C4—1882no—
343C2×C4—1882no—
344C2×C4C4×C22⋊D4711yescharacteristic
345C2×C4C4×C22⋊D4711yescharacteristic
346C2×C4C4×C22⋊D4711yescharacteristic
347C2×C4C2×#1062533612yes—
348C2×C4#1309404412yes—
349C2×C4C22×C4×D4711yescharacteristic
350C2×C4C22×C22⋊D4711yescharacteristic
351C2×C4C2×#1062535011yescharacteristic
352C8—21no—
353C8—21no—
354C8—42no—
355C8—942no—
356C8—942no—
357C8—1882no—
358C8—1882no—
359C8C2×#1062533612yes—
360C8C2×#1062533612yes—
361C22—21no—
362C22—21no—
363C22—1882no—
364C22—1882no—
365C22—1882no—
366C22C4×#1062533612yes—
367C22C2×C4×C22⋊D4711yescharacteristic
368C4—21no—
369C4—21no—
370C4—42no—
371C4—42no—
372C4—1882no—
373C4C4×#1062533612yes—
374C4#1568924312yes—
375C4#1568988812yes—
376C4C22×#1062533611yescharacteristic
377C4C2×#1309404411yescharacteristic
378C2—22no—
379C2—3762no—
380C2#2025210411yescharacteristic
381C2C2×C4×#1062533611yescharacteristic
382C2C2×#1568924311yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#2960C2
socleC22×C47C2×C4×D4
prime radical#15689673C22
cosocleC42×C47C24
hypercenter#2960D47
hypocenterC47#17976
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ #2960

Complements

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

NH ≅ G/Ntype
3. #20242231C2semidirect
4. #20242222C2semidirect
5. #20242143C2semidirect
6. #20241843C2semidirect
8. C2×#15688844C2semidirect
9. C47×#2960C2semidirect
105. C2×C8×C47SD16semidirect
188. #2960D47semidirect
231. C47#17976semidirect
307. C2×C8#10625336semidirect

Maximal subgroups

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

subgroupindex
#202539482
#202521012
#202422312
#202422222
#202421432
#202418432
#202418382
C2×#156888442
C47×#29602
#1797647