Finite Group Databasebeta

← #27083725

Subgroups — #27083725

Order 12032.

2578 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20258175C212yesmaximal, complemented
2#20258099C211yesmaximal, characteristic, complemented
3#20253121C2112yesmaximal, complemented
4#20253104C216yesmaximal, complemented
5C2×#15689395C216yesmaximal, complemented
6#20251999C211yesmaximal, characteristic, complemented
7#20251871C212yesmaximal, complemented
8C2×C47×#578C211yesmaximal, characteristic, complemented
9#15689959C2211yescharacteristic
10#15689938—28no—
11#15689931—28no—
12#15689921C22112yes—
13C2×#13094055C22112yes—
14#15689890C2211yescharacteristic
15#15689890C2212yes—
16#15689889C2216yes—
17#15689888—24no—
18C2×#13094030C2216yes—
19#131×D47—24no—
20#15689395C22112yescomplemented
21#15689395C22124yescomplemented
22#15689390C2214yescomplemented
23#15689383C22124yescomplemented
24#15689172C2216yescomplemented
25C2×#13093960C2216yescomplemented
26#15689107C22112yescomplemented
27C2×#13093943C22112yescomplemented
28C2×2−1+4×C47C2212yes—
29C47×#578—24no—
30C47×#578—24no—
31C2×#137×C47C2216yescomplemented
32C2×#135×C47C2213yescomplemented
33C22×Q16×C47C2213yescomplemented
34C2×#131×C47C2211yescharacteristic
35#13094072—21no—
36#13094071D414yes—
37#13094070D414yes—
38#13094069—23no—
39C2×#10625350C2316yes—
40#41×D47—24no—
41#13094055—224no—
42#13094055C23124yes—
43#13094054—212no—
44#13094050C2318yes—
45#13094049—212no—
46#13094044—224no—
47C2×#10625337—212no—
48C2×#10625336—212no—
49#13094032—212no—
50#13094031—24no—
51#13094031—28no—
52#13094030—26no—
53#13094030—26no—
54#13094030C2314yes—
55#13094030C23112yes—
56C2×#10625329C2316yes—
57C2×#10625329C23112yes—
58C2×#10625324—26no—
59M16×D47—212no—
60C2×C8×D47—26no—
61#13093960C23112yes—
62#13093960C2314yes—
63#13093943C23124yes—
64#13093941C2318yes—
652−1+4×C47—28no—
662−1+4×C47—28no—
67C2×#41×C47—26no—
68C2×#41×C47C2316yes—
69C2×#41×C47C2311yescharacteristic
70C22×Q8×C47—22no—
71C22×Q8×C47C2316yes—
72#137×C47—224no—
73#135×C47—26no—
74#135×C47—26no—
75C2×Q16×C47—26no—
76C2×Q16×C47—26no—
77C2×Q16×C47C2313yes—
78C2×Q16×C47C2316yes—
79C2×SD16×C47C2316yes—
80C2×D8×C47C2311yescharacteristic
81#131×C47D414yes—
82#131×C47D414yes—
83C2×M16×C47C2313yes—
84C22×C8×C47C2313yes—
85#41×C47—212no—
86#41×C47—212no—
87#41×C47—424no—
88#41×C47C2×D414yes—
89#41×C47C2×D414yes—
90C2×Q8×C47—26no—
91C2×Q8×C47—212no—
92C2×Q8×C47—212no—
93C2×Q8×C47—48no—
94C2×Q8×C47#4111yescharacteristic
95C2×Q8×C47C2416yes—
96C2×D4×C47—26no—
97C2×D4×C47#4113yes—
98C2×D4×C47C2412yes—
99C22×C4×C47—26no—
100C22×C4×C47—26no—
101C22×C4×C47C2413yes—
102C2×C22⋊D47—26no—
103#10625354—24no—
104Q8×D47—24no—
105#10625352—212no—
106D4×D47—212no—
107#10625350—26no—
108#10625350—26no—
109#10625350C2×D4112yes—
110#10625350C2414yes—
111C2×D188C2×D416yes—
112C2×C4×D47—26no—
113C2×Dic94C2×D416yes—
114Q16×C47—23no—
115Q16×C47—23no—
116Q16×C47—212no—
117SD16×C47—212no—
118D8×C47—21no—
119D8×C47—21no—
120M16×C47C2×D4112yes—
121C2×C8×C47C2×D416yes—
122C2×C8×C47C2×D416yes—
123C2×C8×C47C2411yescharacteristic
124C2×C8×C47C2413yes—
125#10625337—224no—
126#10625336—224no—
127#10625335—28no—
128#10625334—28no—
129#10625329—26no—
130#10625329—212no—
131#10625329C2414yes—
132#10625329C2418yes—
133C2×C47⋊C8—26no—
134C2×C47⋊C8—212no—
135#10625324—212no—
136#10625324—24no—
137C8×D47—24no—
138C8×D47—212no—
139C23×C47—22no—
140C23×C47C2×#4113yes—
141C22×D47—26no—
142Q8×C47—26no—
143Q8×C47—26no—
144Q8×C47—412no—
145Q8×C47#12114yes—
146D4×C47—22no—
147D4×C47—22no—
148D4×C47—412no—
149D4×C47#121112yes—
150C2×C4×C47—26no—
151C2×C4×C47—412no—
152C2×C4×C47—412no—
153C2×C4×C47C22×D416yes—
154C2×C4×C47C22×D416yes—
155C2×C4×C47C2×#4113yes—
156C2×C4×C47C2511yescharacteristic
157C22⋊D47—212no—
158C22⋊D47—24no—
159C2×Dic47—26no—
160D188—26no—
161D188C22×D414yes—
162C4×D47—212no—
163C4×D47—24no—
164Dic94—26no—
165Dic94C22×D414yes—
166C8×C47C22×D411yescharacteristic
167C8×C47C22×D416yes—
168C8×C47C22×D411yescharacteristic
169C47⋊C8—24no—
170C47⋊C8—28no—
171#60472—471nomaximal, Sylow
172C22×C47—43no—
173C22×C47—42no—
174C22×C47—82no—
175C22×C47C2×#12116yes—
176C22×C47C22×#4111yescharacteristic
177C2×D47—24no—
178C2×D47—86no—
179C4×C47—46no—
180C4×C47C2×#12116yes—
181C4×C47C23×D411yescharacteristic
182C4×C47C23×D411yescharacteristic
183Dic47—24no—
184#3586—472no—
185C2×#578D4711yescharacteristic, complemented
186#3573—471no—
187#3069—4712no—
188#3067—476no—
189C2×#473—476no—
190#2916—471no—
191#2889—472no—
192C2×C47—26no—
193C2×C47—21no—
194C2×C47—82no—
195C2×C47C22×#12111yescharacteristic
196D47—44no—
197#585—471no—
198C2×2−1+4C2×D4712yes—
199#578—24no—
200#578—24no—
201#578—948no—
202#577—948no—
203#575—4712no—
204C2×#137—4712no—
205C2×#137C2×D4716yescomplemented
206C2×#135C2×D4713yescomplemented
207C22×Q16C2×D4713yescomplemented
208#568—472no—
209#568—471no—
210#568—476no—
211#568—944no—
212C2×#131—476no—
213C2×#131—944no—
214C2×#131C2×D4711yescharacteristic
215#473—4712no—
216#473—4724no—
217#472—474no—
218#471—4724no—
219#430—476no—
220C2×#108—476no—
221#413—4712no—
222C2×#101—4712no—
223C47#6047211yescharacteristic, complemented, Sylow
2242−1+4—28no—
2252−1+4—28no—
2262−1+4—474no—
2272−1+4—941no—
228H(5,𝔽2)—474no—
229H(5,𝔽2)—943no—
230C2×#41—26no—
231C2×#41—476no—
232C2×#41—944no—
233C2×#41C22×D4716yes—
234C2×#41C22×D4711yescharacteristic
235C22×Q8—22no—
236C22×Q8C22×D4716yes—
237#137—224no—
238#137—4724no—
239#137—9424no—
240#137—9412no—
241AGL(1,ℤ/8)—478no—
242AGL(1,ℤ/8)—9412no—
243#135—26no—
244#135—26no—
245#135—9424no—
246C2×Q16—26no—
247C2×Q16—26no—
248C2×Q16—9412no—
249C2×Q16C22×D4713yes—
250C2×Q16C22×D4716yes—
251C2×SD16—9412no—
252C2×SD16C22×D4716yes—
253C2×D8C22×D4711yescharacteristic
254#131—474no—
255#131—4712no—
256#131—948no—
257#131—944no—
258#131—9412no—
259#131—946no—
260#131—946no—
261#131C22⋊D4714yes—
262#131C22⋊D4714yes—
263C2×M16—476no—
264C2×M16—4712no—
265C2×M16—9412no—
266C2×M16—946no—
267C2×M16C22×D4713yes—
268C22×C8—946no—
269C22×C8C22×D4713yes—
270#108—474no—
271#108—4712no—
272#101—4724no—
273#100—478no—
274#41—212no—
275#41—212no—
276#41—424no—
277#41—4712no—
278#41—474no—
279#41—944no—
280#41—9412no—
281#41—946no—
282#41—946no—
283#41D4×D4714yes—
284#41C2×C22⋊D4714yes—
285C2×Q8—26no—
286C2×Q8—212no—
287C2×Q8—212no—
288C2×Q8—48no—
289C2×Q8—476no—
290C2×Q8—944no—
291C2×Q8#1062535211yescharacteristic
292C2×Q8C23×D4716yes—
293C2×D4—26no—
294C2×D4—476no—
295C2×D4—946no—
296C2×D4—9412no—
297C2×D4#1062535213yes—
298C2×D4C23×D4712yes—
299C22×C4—26no—
300C22×C4—26no—
301C22×C4—946no—
302C22×C4C23×D4713yes—
303Q16—23no—
304Q16—23no—
305Q16—212no—
306Q16—9424no—
307SD16—212no—
308SD16—948no—
309SD16—9424no—
310D8—21no—
311D8—21no—
312D8—948no—
313M16—474no—
314M16—478no—
315M16—9412no—
316M16—946no—
317M16—9412no—
318M16—944no—
319M16C2×C22⋊D47112yes—
320C2×C8—9412no—
321C2×C8—946no—
322C2×C8—944no—
323C2×C8—9412no—
324C2×C8C2×C22⋊D4716yes—
325C2×C8C2×C22⋊D4716yes—
326C2×C8C23×D4711yescharacteristic
327C2×C8C23×D4713yes—
328C23—22no—
329C23—946no—
330C23C2×#1062535213yes—
331Q8—26no—
332Q8—26no—
333Q8—412no—
334Q8—474no—
335Q8—946no—
336Q8#1309399514yes—
337D4—22no—
338D4—22no—
339D4—412no—
340D4—474no—
341D4—9412no—
342D4—944no—
343D4—946no—
344D4#13093995112yes—
345C2×C4—26no—
346C2×C4—412no—
347C2×C4—412no—
348C2×C4—946no—
349C2×C4—9412no—
350C2×C4—944no—
351C2×C4C2×D4×D4716yes—
352C2×C4C22×C22⋊D4716yes—
353C2×C4C2×#1062535213yes—
354C2×C4C24×D4711yescharacteristic
355C8—948no—
356C8—944no—
357C8C22×C22⋊D4711yescharacteristic
358C8C22×C22⋊D4716yes—
359C8C22×C22⋊D4711yescharacteristic
360C22—43no—
361C22—42no—
362C22—82no—
363C22—944no—
364C22—3766no—
365C22C2×#1309399516yes—
366C22C22×#1062535211yescharacteristic
367C4—46no—
368C4—944no—
369C4C2×#1309399516yes—
370C4C22×D4×D4711yescharacteristic
371C4C23×C22⋊D4711yescharacteristic
372C2—21no—
373C2—26no—
374C2—82no—
375C2—1884no—
376C2C22×#1309399511yescharacteristic
subgroupG/H
centerC2C22×#13093995
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#578C2
socleC2×C47C22×#121
hypercenterC2×#578D47
hypocenterC47#60472
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×#41 ◁ C2×#578

Complements

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

NH ≅ G/Ntype
1. #20258175C2semidirect
2. #20258099C2semidirect
3. #20253121C2semidirect
4. #20253104C2semidirect
5. C2×#15689395C2semidirect
6. #20251999C2semidirect
7. #20251871C2semidirect
8. C2×C47×#578C2semidirect
20. #15689395C22semidirect
21. #15689395C22semidirect
22. #15689390C22semidirect
23. #15689383C22semidirect
24. #15689172C22semidirect
25. C2×#13093960C22semidirect
26. #15689107C22semidirect
27. C2×#13093943C22semidirect
31. C2×#137×C47C22semidirect
32. C2×#135×C47C22semidirect
33. C22×Q16×C47C22semidirect
185. C2×#578D47semidirect
205. C2×#137C2×D47semidirect
206. C2×#135C2×D47semidirect
207. C22×Q16C2×D47semidirect
223. C47#60472semidirect

Maximal subgroups

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

subgroupindex
#202581752
#202580992
#202531212
#202531042
C2×#156893952
#202519992
#202518712
C2×C47×#5782
#6047247