Finite Group Databasebeta

← #26199423

Subgroups — #26199423

Order 12032.

662 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254509C211yesmaximal, characteristic
2#20254313C211yesmaximal, characteristic
3#20252101C211yesmaximal, characteristic
4#20242932C214yesmaximal, complemented
5#20242808C214yesmaximal, complemented
6#20242136C211yesmaximal, characteristic, complemented
7#20242073C211yesmaximal, characteristic, complemented
8#20242060C211yesmaximal, characteristic, complemented
9C2×C47×#487C211yesmaximal, characteristic, complemented
10C2×#13094030—21no—
11#15689673C2211yescharacteristic
12#15689526—21no—
13#15689497—21no—
14#15689473—21no—
15#15689471—21no—
16#15689436—22no—
17#15689400C2214yes—
18#15689325C2214yes—
19#15689262—22no—
20#15689202—21no—
21#15689202C2212yes—
22C8×D188—21no—
23C8×D188C2212yes—
24#15689176C2211yescharacteristic
25#15689176C2211yescharacteristic
26#15689147C2211yescharacteristic
27#15689089—21no—
28#15688876C2212yes—
29#15688874C2212yes—
30#15688869C2214yes—
31#15688869C2214yes—
32C2×C4⋊Q8×C47C2211yescharacteristic
33C2×C4⋊Q8×C47C2211yescharacteristic
34C47×#487—21no—
35C47×#487—21no—
36C47×#487C2212yes—
37C47×#487C2212yes—
38C2×#103×C47—21no—
39C2×#103×C47—21no—
40C2×C4×C8×C47C2211yescharacteristic
41C2×#10625350—21no—
42C2×#10625336—24no—
43#13094030—44no—
44C2×#10625329—21no—
45C2×#10625324—22no—
46C2×C8×D47—22no—
47#13094024D411yescharacteristic
48#13094021D411yescharacteristic
49#13094012—21no—
50#13093998—21no—
51#13093998D412yes—
52C4×D188—21no—
53C4×D188C2312yes—
54#13093957—24no—
55C4⋊(C47⋊C8)—21no—
56C4⋊(C47⋊C8)C2312yes—
57C4⋊(C47⋊C8)C2312yes—
58C4⋊(C47⋊C8)C2312yes—
59#13093948—24no—
60#13093944—24no—
61C22⋊(C47⋊C8)—21no—
62C22⋊(C47⋊C8)—21no—
63(C2×D47)⋊C8—22no—
64(C2×D47)⋊C8—22no—
65C4×C47⋊C8—21no—
66C4×C47⋊C8—21no—
67C22×Q8×C47—21no—
68C22×Q8×C47—21no—
69C22×C8×C47D411yescharacteristic
70C22×C8×C47D411yescharacteristic
71C4⋊Q8×C47—21no—
72C4⋊Q8×C47—21no—
73C4⋊Q8×C47—21no—
74C4⋊Q8×C47—21no—
75C4⋊Q8×C47—21no—
76C4⋊Q8×C47—21no—
77C4⋊Q8×C47C2312yes—
78C4⋊Q8×C47C2312yes—
79C2×C4⋊C4×C47—21no—
80C2×C4⋊C4×C47—21no—
81C2×C4⋊C4×C47—21no—
82C2×C4⋊C4×C47—21no—
83C2×C42×C47C2311yescharacteristic
84#103×C47—22no—
85#103×C47—22no—
86#103×C47—22no—
87#103×C47—22no—
88C4×C8×C47D411yescharacteristic
89C4×C8×C47D411yescharacteristic
90C4×C8×C47C2311yescharacteristic
91C4×C8×C47C2311yescharacteristic
92C2×Q8×C47—22no—
93C2×Q8×C47—22no—
94C2×Q8×C47—41no—
95C2×Q8×C47—41no—
96C2×Q8×C47—42no—
97C2×Q8×C47—42no—
98C2×Q8×C47—82no—
99C22×C4×C47—41no—
100C22×C4×C47—41no—
101C22×C4×C47C2×D411yescharacteristic
102C22×C4×C47C2×D411yescharacteristic
103C22×C4×C47#4111yescharacteristic
104C2×C22⋊D47—22no—
105#10625350—42no—
106#10625350—42no—
107C2×D188—21no—
108C2×D188—21no—
109C2×D188C2×D412yes—
110C2×C4×D47—22no—
111C2×Dic94—21no—
112C2×C8×C47—22no—
113C2×C8×C47—22no—
114C2×C8×C47C2×D411yescharacteristic
115C2×C8×C47C2×D411yescharacteristic
116C2×C8×C47C2×D411yescharacteristic
117C2×C8×C47C2×D411yescharacteristic
118C4⋊C4×C47—22no—
119C4⋊C4×C47—22no—
120C4⋊C4×C47—22no—
121C4⋊C4×C47—22no—
122C4⋊C4×C47—41no—
123C4⋊C4×C47—41no—
124C4⋊C4×C47—41no—
125C4⋊C4×C47—41no—
126C42×C47C2×D411yescharacteristic
127C42×C47C2×D411yescharacteristic
128C42×C47#4111yescharacteristic
129C42×C47C2411yescharacteristic
130#10625336—84no—
131(C2×D47)⋊C4—22no—
132(C2×D47)⋊C4—22no—
133C4⋊Dic47—21no—
134C4⋊Dic47C2×D412yes—
135#10625329—42no—
136C2×C47⋊C8—22no—
137C2×C47⋊C8—22no—
138C2×C47⋊C8—22no—
139#10625324—44no—
140C8×D47—44no—
141C23×C47#12111yescharacteristic
142C22×D47—22no—
143Q8×C47—42no—
144Q8×C47—42no—
145Q8×C47—44no—
146Q8×C47—82no—
147Q8×C47—82no—
148C2×C4×C47—22no—
149C2×C4×C47—22no—
150C2×C4×C47—42no—
151C2×C4×C47—42no—
152C2×C4×C47—82no—
153C2×C4×C47—82no—
154C2×C4×C47#12111yescharacteristic
155C2×C4×C47#12111yescharacteristic
156C2×C4×C47#12111yescharacteristic
157C2×C4×C47#13512yes—
158C2×C4×C47AGL(1,ℤ/8)12yes—
159C2×C4×C47C22×D411yescharacteristic
160C2×C4×C47C22×D411yescharacteristic
161C2×C4×C47C2×#4111yescharacteristic
162C22⋊D47—44no—
163C2×Dic47—22no—
164D188—24no—
165D188—42no—
166D188—42no—
167C4×D47—42no—
168C4×D47—42no—
169Dic94—42no—
170C8×C47—21no—
171C8×C47—21no—
172C8×C47—21no—
173C8×C47—21no—
174C47⋊C8—42no—
175C47⋊C8—42no—
176C47⋊C8—42no—
177#19197—471nomaximal, Sylow
178C22×C47—21no—
179C22×C47—21no—
180C22×C47#46312yes—
181C22×C47C2×#12111yescharacteristic
182C2×D47—42no—
183C2×D47—42no—
184C2×D47—42no—
185C4×C47—21no—
186C4×C47—21no—
187C4×C47—82no—
188C4×C47—82no—
189C4×C47#46312yes—
190C4×C47#47312yes—
191C4×C47C2×#13511yescharacteristic
192C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
193C4×C47#57712yes—
194Dic47—42no—
195#3226—471no—
196#3206—471no—
197C2×#487D4711yescharacteristic, complemented
198#2931—471no—
199#1699—474no—
200#1679—474no—
201#1564—471no—
202#1549—471no—
203#1546—471no—
204C2×C47—22no—
205C2×C47C2×#46311yescharacteristic
206C2×C47#305711yescharacteristic
207C2×C47C2×#47311yescharacteristic
208D47—82no—
209C2×#131—941no—
210#532—471no—
211C2×C4⋊Q8C2×D4711yescharacteristic
212C2×C4⋊Q8C2×D4711yescharacteristic
213#500—941no—
214#492—941no—
215#487—21no—
216#487—21no—
217#487—941no—
218#487C2×D4712yes—
219#487C2×D4712yes—
220#486—941no—
221#480—942no—
222#474—474no—
223#461—474no—
224#448—942no—
225#436—472no—
226#436—941no—
227C8×D4—472no—
228C8×D4—941no—
229#431—471no—
230#431—471no—
231#423—471no—
232C2×#103—21no—
233C2×#103—21no—
234#408—941no—
235C2×C4×C8C2×D4711yescharacteristic
236#335—472no—
237#334—472no—
238#332—474no—
239#332—474no—
240C47#1919711yescharacteristic, complemented, Sylow
241C2×#41—941no—
242C22×Q8—21no—
243C22×Q8—21no—
244C2×SD16—944no—
245#131—1884no—
246C2×M16—941no—
247C2×M16—942no—
248C22×C8—942no—
249C22×C8C22⋊D4711yescharacteristic
250C22×C8C22⋊D4711yescharacteristic
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8—21no—
257C4⋊Q8—471no—
258C4⋊Q8C22×D4712yes—
259C4⋊Q8C22×D4712yes—
260#127—471no—
261#124—941no—
262#121—472no—
263#121—941no—
264C4×D4—472no—
265C4×D4—941no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C42C22×D4711yescharacteristic
271#107—944no—
272C4⋊C8—472no—
273C4⋊C8—472no—
274C4⋊C8—472no—
275C4⋊C8—941no—
276#103—22no—
277#103—22no—
278#103—22no—
279#103—22no—
280#103—944no—
281#102—944no—
282C22⋊C8—941no—
283C22⋊C8—941no—
284C22⋊C8—942no—
285C22⋊C8—942no—
286C4×C8—941no—
287C4×C8—941no—
288C4×C8C22⋊D4711yescharacteristic
289C4×C8C22⋊D4711yescharacteristic
290C4×C8C22×D4711yescharacteristic
291C4×C8C22×D4711yescharacteristic
292#41—1882no—
293#41—1882no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—82no—
301C2×Q8—941no—
302C2×D4—472no—
303C2×D4—942no—
304C2×D4—941no—
305C2×D4—941no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—942no—
309C22×C4D4×D4711yescharacteristic
310C22×C4#1062535211yescharacteristic
311C22×C4C2×C22⋊D4711yescharacteristic
312SD16—3764no—
313M16—1884no—
314M16—1882no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—942no—
318C2×C8—942no—
319C2×C8—942no—
320C2×C8—1884no—
321C2×C8C2×C22⋊D4711yescharacteristic
322C2×C8C2×C22⋊D4711yescharacteristic
323C2×C8C2×C22⋊D4711yescharacteristic
324C2×C8C2×C22⋊D4711yescharacteristic
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—472no—
334C4⋊C4—941no—
335C22⋊C4—942no—
336C22⋊C4—942no—
337C42D4×D4711yescharacteristic
338C42#1062535211yescharacteristic
339C42C2×C22⋊D4711yescharacteristic
340C42C23×D4711yescharacteristic
341C23—942no—
342C23#1309399511yescharacteristic
343Q8—42no—
344Q8—42no—
345Q8—44no—
346Q8—82no—
347Q8—82no—
348Q8—1882no—
349D4—944no—
350D4—1884no—
351D4—1882no—
352D4—1882no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—942no—
360C2×C4—1882no—
361C2×C4—1882no—
362C2×C4#1309399511yescharacteristic
363C2×C4#1309399511yescharacteristic
364C2×C4#1309399511yescharacteristic
365C2×C4#1309404412yes—
366C2×C4#1309404912yes—
367C2×C4C2×D4×D4711yescharacteristic
368C2×C4C22×C22⋊D4711yescharacteristic
369C2×C4C2×#1062535211yescharacteristic
370C8—21no—
371C8—21no—
372C8—21no—
373C8—21no—
374C8—1882no—
375C8—1882no—
376C8—1882no—
377C22—21no—
378C22—21no—
379C22—1882no—
380C22—1882no—
381C22—1882no—
382C22#1568933512yes—
383C22C2×#1309399511yescharacteristic
384C4—21no—
385C4—21no—
386C4—82no—
387C4—82no—
388C4—1882no—
389C4#1568933512yes—
390C4#1568939512yes—
391C4C2×#1309404411yescharacteristic
392C4C2×#1309404911yescharacteristic
393C4#1568992912yes—
394C2—22no—
395C2—3762no—
396C2C2×#1568933511yescharacteristic
397C2#2025302311yescharacteristic
398C2C2×#1568939511yescharacteristic
subgroupG/H
centerC22C2×#13093995
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC2×C47×#487C2
socleC22×C47C2×#121
prime radical#15689673C22
hypercenterC2×#487D47
hypocenterC47#19197
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
4. #20242932C2semidirect
5. #20242808C2semidirect
6. #20242136C2semidirect
7. #20242073C2semidirect
8. #20242060C2semidirect
9. C2×C47×#487C2semidirect
197. C2×#487D47semidirect
240. C47#19197semidirect

Maximal subgroups

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

subgroupindex
#202545092
#202543132
#202521012
#202429322
#202428082
#202421362
#202420732
#202420602
C2×C47×#4872
#1919747