Finite Group Databasebeta

← #27095794

Subgroups — #27095794

Order 12032.

1746 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20257253C212yesmaximal
2#20255923C218yesmaximal, complemented
3#20255848C218yesmaximal, complemented
4#20253912C211yesmaximal, characteristic, complemented
5#20253353C214yesmaximal, complemented
6#20253193C212yesmaximal, complemented
7#20253039C212yesmaximal, complemented
8#20252107C211yesmaximal, characteristic
9#20251979C212yesmaximal, complemented
10C2×C47×#441C211yesmaximal, characteristic, complemented
11C2×#13094030C2212yes—
12#15689858—22no—
13#15689851—22no—
14#15689846C2214yes—
15#15689845—24no—
16#15689833C2214yes—
17#15689814C2218yes—
18#15689720—24no—
19#15689697C2214yes—
20#15689680C2212yes—
21#15689673C2211yescharacteristic
22#15689657—24no—
23#15689604C2214yes—
24#15689532C2214yes—
25#15689532C2214yes—
26#15689463C2218yes—
27#15689462C2218yes—
28#15689457C2218yes—
29#15689452C2218yes—
30#15689430C2218yes—
31#15689418C2214yes—
32#15689408C2214yes—
33#15689378C2214yes—
34#15689378C2214yes—
35#15689361C2218yes—
36#15689196C2218yes—
37#15689163C2214yes—
38#15689162C2218yes—
39#15689161C2212yes—
40#15689161C2212yes—
41#15689158C2214yes—
42#106×D47C2214yes—
43#15689147C2211yescharacteristic
44#15689132C2214yes—
45#15689089C2212yes—
46C22×Q16×C47C2211yescharacteristic
47C2×C4×Q8×C47C2212yes—
48C47×#441C2218yes—
49C2×#106×C47C2211yescharacteristic
50C2×#103×C47C2212yes—
51C2×#97×C47C2211yescharacteristic
52C2×#10625350C2312yes—
53#13094030—28no—
54C2×#10625329C2312yes—
55C2×#10625324C2314yes—
56C2×C8×D47C2314yes—
57#13094024D411yescharacteristic
58#13094024C2312yes—
59#13094024C2312yes—
60#13094022—24no—
61#13094021D411yescharacteristic
62#13094019—28no—
63#13094018—28no—
64#13094016—24no—
65#13094016C2314yes—
66#13094014—28no—
67#13094013—24no—
68#13094012—24no—
69#13094012D412yes—
70#13094009—28no—
71#13094007—28no—
72#13094003—22no—
73#13094003—22no—
74#13094002—28no—
75#13094000C2318yes—
76#13094000C2318yes—
77#13093998D414yes—
78C4×Dic94—22no—
79C4×Dic94—22no—
80C4×Dic94—24no—
81#13093989—24no—
82C4×D188—24no—
83C4×D188C2314yes—
84#13093986—24no—
85C4×C22⋊D47—28no—
86#13093983C2314yes—
87#13093982C2318yes—
88#13093981—28no—
89C4⋊C4×D47C2318yes—
90#13093958C2314yes—
91#13093958C2314yes—
92#13093957C2318yes—
93#13093956C2312yes—
94#13093956C2312yes—
95#13093956C2312yes—
96#13093956C2312yes—
97#13093955C2318yes—
98C4⋊(C47⋊C8)C2314yes—
99#13093948C2318yes—
100#13093948C2318yes—
101(C2×D47)⋊C8C2318yes—
102C22×Q8×C47C2312yes—
103C2×Q16×C47—24no—
104C2×Q16×C47C2314yes—
105C22×C8×C47C2311yescharacteristic
106C22×C8×C47C2311yescharacteristic
107C4×Q8×C47—22no—
108C4×Q8×C47—22no—
109C4×Q8×C47C2314yes—
110C4×Q8×C47C2314yes—
111C2×C4⋊C4×C47—22no—
112C2×C4⋊C4×C47C2312yes—
113C2×C42×C47—22no—
114C2×C42×C47C2311yescharacteristic
115#106×C47C2314yes—
116#103×C47C2318yes—
117#97×C47C2314yes—
118C2×Q8×C47—22no—
119C2×Q8×C47—24no—
120C2×Q8×C47—24no—
121C2×Q8×C47C2414yes—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47C2×D412yes—
125C22×C4×C47C2411yescharacteristic
126C2×C22⋊D47C2×D414yes—
127#10625350—24no—
128#10625350—24no—
129C2×D188C2×D412yes—
130C2×D188C2×D414yes—
131C2×C4×D47—28no—
132C2×C4×D47C2414yes—
133C2×Dic94—22no—
134C2×Dic94—22no—
135C2×Dic94C2×D412yes—
136Q16×C47—28no—
137C2×C8×C47—22no—
138C2×C8×C47#4114yes—
139C2×C8×C47C2412yes—
140C2×C8×C47C2412yes—
141C4⋊C4×C47—24no—
142C4⋊C4×C47—24no—
143C4⋊C4×C47—22no—
144C4⋊C4×C47C2414yes—
145C42×C47—24no—
146C42×C47—24no—
147C42×C47C2×D411yescharacteristic
148C42×C47C2×D411yescharacteristic
149C42×C47C2412yes—
150C22⋊Dic47—24no—
151(C2×D47)⋊C4—28no—
152(C2×D47)⋊C4—28no—
153(C2×D47)⋊C4—28no—
154(C2×D47)⋊C4C2×D418yes—
155C4⋊Dic47—22no—
156C4⋊Dic47—22no—
157C4⋊Dic47C2×D414yes—
158C4⋊Dic47C2414yes—
159C4⋊Dic47C2414yes—
160Dic47⋊C4—24no—
161Dic47⋊C4—24no—
162Dic47⋊C4—28no—
163Dic47⋊C4—28no—
164Dic47⋊C4—28no—
165C4×Dic47—24no—
166C4×Dic47—28no—
167#10625329—24no—
168C2×C47⋊C8C2414yes—
169#10625324—28no—
170C8×D47—28no—
171C23×C47C22×D411yescharacteristic
172C22×D47C22×D414yes—
173Q8×C47—24no—
174Q8×C47—24no—
175Q8×C47—44no—
176C2×C4×C47—24no—
177C2×C4×C47—24no—
178C2×C4×C47—42no—
179C2×C4×C47—44no—
180C2×C4×C47—82no—
181C2×C4×C47C22×D412yes—
182C2×C4×C47C22×D412yes—
183C2×C4×C47C22×D411yescharacteristic
184C2×C4×C47C2×#4112yes—
185C2×C4×C472−1+412yes—
186C2×C4×C47C2511yescharacteristic
187C22⋊D47—28no—
188C2×Dic47—24no—
189C2×Dic47—24no—
190C2×Dic47C22×D414yes—
191D188—24no—
192D188—44no—
193C4×D47—24no—
194C4×D47—24no—
195C4×D47—88no—
196Dic94—24no—
197Dic94—44no—
198Dic94—82no—
199C8×C47—22no—
200C8×C47C2×#4114yes—
201C47⋊C8—24no—
202#61292—471nomaximal, Sylow
203C22×C47—22no—
204C22×C47#54712yes—
205C22×C47C23×D411yescharacteristic
206C2×D47—28no—
207C2×D47—24no—
208C4×C47—22no—
209C4×C47—22no—
210C4×C47—44no—
211C4×C47—44no—
212C4×C47#54712yes—
213C4×C47C22×#4111yescharacteristic
214C4×C47C2×2−1+411yescharacteristic
215Dic47—24no—
216Dic47—44no—
217Dic47—44no—
218#3507—472no—
219#3349—478no—
220#3344—478no—
221#3162—471no—
222#3097—474no—
223#3078—472no—
224#3059—472no—
225C2×#441D4711yescharacteristic, complemented
226#2932—471no—
227#2912—472no—
228C2×C47—22no—
229C2×C47C2×#54711yescharacteristic
230C2×C47#358612yes—
231D47—44no—
232C22×Q16C2×D4711yescharacteristic
233C2×#131—472no—
234#559—942no—
235#557—942no—
236#556—474no—
237#556—944no—
238#554—474no—
239#552—478no—
240#539—944no—
241#536—474no—
242#533—472no—
243#532—471no—
244#529—944no—
245#520—474no—
246C2×C4×Q8C2×D4712yes—
247#501—474no—
248#501—474no—
249#485—478no—
250#485—478no—
251#484—478no—
252#483—478no—
253#479—478no—
254#477—474no—
255#475—474no—
256#470—474no—
257#470—474no—
258#467—478no—
259#441C2×D4718yes—
260#435—478no—
261#428—474no—
262#428—478no—
263#427—472no—
264#427—472no—
265#427—474no—
266C2×#106—474no—
267C2×#106C2×D4711yescharacteristic
268#423—471no—
269#419—474no—
270C2×#103C2×D4712yes—
271#408—472no—
272C2×#97C2×D4711yescharacteristic
273C47#6129211yescharacteristic, complemented, Sylow
274C2×#41—472no—
275C22×Q8C22×D4712yes—
276C2×Q16—24no—
277C2×Q16C22×D4714yes—
278#131—948no—
279C2×M16—472no—
280C2×M16—474no—
281C22×C8—474no—
282C22×C8C22×D4711yescharacteristic
283C22×C8C22×D4711yescharacteristic
284C4⋊Q8—471no—
285C4⋊Q8—472no—
286C4⋊Q8—472no—
287C4⋊Q8—944no—
288#127—471no—
289#126—948no—
290#126—948no—
291#125—474no—
292#125—944no—
293#125—948no—
294#124—472no—
295#124—944no—
296#124—944no—
297#123—948no—
298#123—948no—
299C22⋊Q8—478no—
300C22⋊Q8—478no—
301C22⋊Q8—948no—
302C22⋊Q8—942no—
303C22⋊Q8—942no—
304#121—474no—
305C4×Q8—22no—
306C4×Q8—22no—
307C4×Q8—944no—
308C4×Q8—944no—
309C4×Q8—942no—
310C4×Q8—942no—
311C4×Q8C22×D4714yes—
312C4×Q8C22×D4714yes—
313C4×D4—474no—
314C4×D4—944no—
315C4×D4—944no—
316C4×D4—948no—
317#117—474no—
318#117—478no—
319#117—948no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—478no—
322C2×C4⋊C4C22×D4712yes—
323C2×C42—22no—
324C2×C42C22×D4711yescharacteristic
325#107—474no—
326#107—474no—
327#107—478no—
328#106—472no—
329#106—472no—
330#106—472no—
331#106—472no—
332#106—478no—
333#106C22×D4714yes—
334C4⋊C8—474no—
335#103—478no—
336#103—478no—
337#103C22×D4718yes—
338C22⋊C8—478no—
339#97C22×D4714yes—
340#41—944no—
341#41—944no—
342C2×Q8—22no—
343C2×Q8—24no—
344C2×Q8—24no—
345C2×Q8—472no—
346C2×Q8—942no—
347C2×Q8—942no—
348C2×Q8C23×D4714yes—
349C2×D4—474no—
350C2×D4—472no—
351C2×D4—474no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—474no—
355C22×C4—948no—
356C22×C4D4×D4712yes—
357C22×C4C23×D4711yescharacteristic
358Q16—28no—
359M16—944no—
360M16—948no—
361C2×C8—22no—
362C2×C8—474no—
363C2×C8—948no—
364C2×C8#1062535014yes—
365C2×C8C23×D4712yes—
366C2×C8C23×D4712yes—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—22no—
370C4⋊C4—474no—
371C4⋊C4—474no—
372C4⋊C4—474no—
373C4⋊C4—948no—
374C4⋊C4—948no—
375C4⋊C4—948no—
376C4⋊C4—944no—
377C4⋊C4—944no—
378C4⋊C4—942no—
379C4⋊C4—942no—
380C4⋊C4C23×D4714yes—
381C22⋊C4—478no—
382C22⋊C4—948no—
383C22⋊C4—948no—
384C22⋊C4—948no—
385C22⋊C4—944no—
386C42—24no—
387C42—24no—
388C42—948no—
389C42—944no—
390C42D4×D4711yescharacteristic
391C42D4×D4711yescharacteristic
392C42C23×D4712yes—
393C23—474no—
394C23C2×D4×D4711yescharacteristic
395Q8—24no—
396Q8—24no—
397Q8—44no—
398Q8—944no—
399Q8—3762no—
400Q8—1884no—
401D4—948no—
402D4—944no—
403D4—1884no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—82no—
409C2×C4—474no—
410C2×C4—944no—
411C2×C4—944no—
412C2×C4—944no—
413C2×C4—944no—
414C2×C4—3768no—
415C2×C4C2×D4×D4712yes—
416C2×C4C2×D4×D4712yes—
417C2×C4C2×D4×D4711yescharacteristic
418C2×C4C2×#1062535012yes—
419C2×C4#1309407112yes—
420C2×C4C24×D4711yescharacteristic
421C8—22no—
422C8—944no—
423C8C2×#1062535014yes—
424C22—22no—
425C22—944no—
426C22—948no—
427C22#1568977812yes—
428C22C22×D4×D4711yescharacteristic
429C4—22no—
430C4—22no—
431C4—44no—
432C4—44no—
433C4—944no—
434C4—1884no—
435C4—1884no—
436C4#1568977812yes—
437C4C22×#1062535011yescharacteristic
438C4C2×#1309407111yescharacteristic
439C2—22no—
440C2—1884no—
441C2C2×#1568977811yescharacteristic
442C2#2025817112yes—
subgroupG/H
centerC22C22×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#441C2
socleC22×C47C23×D4
prime radical#15689673C22
hypercenterC2×#441D47
hypocenterC47#61292
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#441

Complements

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

NH ≅ G/Ntype
2. #20255923C2semidirect
3. #20255848C2semidirect
4. #20253912C2semidirect
5. #20253353C2semidirect
6. #20253193C2semidirect
7. #20253039C2semidirect
9. #20251979C2semidirect
10. C2×C47×#441C2semidirect
225. C2×#441D47semidirect
273. C47#61292semidirect

Maximal subgroups

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

subgroupindex
#202572532
#202559232
#202558482
#202539122
#202533532
#202531932
#202530392
#202521072
#202519792
C2×C47×#4412
#6129247