Finite Group Databasebeta

← #26200237

Subgroups — #26200237

Order 12032.

670 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253197C211yesmaximal, characteristic
2#20252871C211yesmaximal, characteristic
3C8×#10625350C211yesmaximal, characteristic
4#20242815C214yesmaximal, complemented
5#20242808C214yesmaximal, complemented
6#20242144C211yesmaximal, characteristic, complemented
7#20242073C211yesmaximal, characteristic, complemented
8#20242060C211yesmaximal, characteristic, complemented
9C2×C47×#487C211yesmaximal, characteristic, complemented
10C2×#13094055—21no—
11C4×#10625350C2211yescharacteristic
12#15689415—21no—
13#15689410—21no—
14#15689405C2212yes—
15#15689400C2212yes—
16#15689330—21no—
17#15689330C2212yes—
18#15689325—21no—
19#15689325C2212yes—
20C8×Dic94C2212yes—
21C8×D188C2212yes—
22C8×C22⋊D47—22no—
23#15689176C2211yescharacteristic
24#15689174—21no—
25#15689163—21no—
26#15689147C2211yescharacteristic
27#15689147C2211yescharacteristic
28#15689132—21no—
29#15689122—21no—
30C4×C8×D47—21no—
31#15688874C2214yes—
32#15688869C2214yes—
33#15688869C2214yes—
34C2×C4⋊Q8×C47C2211yescharacteristic
35C2×C4⋊Q8×C47C2211yescharacteristic
36C47×#487—21no—
37C47×#487—21no—
38C47×#487C2214yes—
39C2×#103×C47—22no—
40C2×C4×C8×C47C2211yescharacteristic
41C2×#10625350D411yescharacteristic
42#13094055—44no—
43C2×#10625337—22no—
44C2×#10625336—22no—
45C2×#10625329—21no—
46C2×#10625329—21no—
47C2×C8×D47—22no—
48C4×Dic94D411yescharacteristic
49C4×Dic94C2311yescharacteristic
50C4×D188D411yescharacteristic
51C4×D188C2311yescharacteristic
52C4×C22⋊D47—21no—
53C4×C22⋊D47—21no—
54#13093983D411yescharacteristic
55#13093981—21no—
56C42×D47—21no—
57#13093957—22no—
58#13093955—22no—
59C4⋊(C47⋊C8)—21no—
60C4⋊(C47⋊C8)—21no—
61C4⋊(C47⋊C8)C2312yes—
62C4⋊(C47⋊C8)C2312yes—
63C4⋊(C47⋊C8)C2312yes—
64Dic47⋊C8—22no—
65#13093948—22no—
66#13093948—22no—
67#13093947—22no—
68#13093944—22no—
69C22⋊(C47⋊C8)—21no—
70(C2×D47)⋊C8—22no—
71C4×C47⋊C8—21no—
72C8×Dic47—22no—
73C22×Q8×C47—21no—
74C22×Q8×C47—21no—
75C22×C8×C47D412yes—
76C4⋊Q8×C47—21no—
77C4⋊Q8×C47—21no—
78C4⋊Q8×C47—21no—
79C4⋊Q8×C47—21no—
80C4⋊Q8×C47—21no—
81C4⋊Q8×C47—21no—
82C4⋊Q8×C47C2312yes—
83C4⋊Q8×C47C2312yes—
84C2×C4⋊C4×C47—21no—
85C2×C4⋊C4×C47—21no—
86C2×C4⋊C4×C47—21no—
87C2×C4⋊C4×C47—21no—
88C2×C42×C47C2311yescharacteristic
89#103×C47—24no—
90#103×C47—24no—
91C4×C8×C47D411yescharacteristic
92C4×C8×C47D411yescharacteristic
93C4×C8×C47C2312yes—
94C2×Q8×C47—22no—
95C2×Q8×C47—22no—
96C2×Q8×C47—44no—
97C2×Q8×C47—41no—
98C2×Q8×C47—41no—
99C2×Q8×C47—42no—
100C2×Q8×C47—42no—
101C22×C4×C47—41no—
102C22×C4×C47—41no—
103C22×C4×C47C2×D411yescharacteristic
104C22×C4×C47C2×D411yescharacteristic
105C22×C4×C47#4111yescharacteristic
106C2×C22⋊D47—21no—
107#10625350—22no—
108#10625350SD1614yes—
109C2×D188C2×D411yescharacteristic
110C2×C4×D47—21no—
111C2×C4×D47—21no—
112C2×C4×D47—21no—
113C2×Dic94C2×D411yescharacteristic
114C2×C8×C47—42no—
115C2×C8×C47C2×D412yes—
116C2×C8×C47C2×D412yes—
117C4⋊C4×C47—22no—
118C4⋊C4×C47—22no—
119C4⋊C4×C47—22no—
120C4⋊C4×C47—22no—
121C4⋊C4×C47—41no—
122C4⋊C4×C47—41no—
123C4⋊C4×C47—41no—
124C4⋊C4×C47—41no—
125C42×C47C2×D411yescharacteristic
126C42×C47C2×D411yescharacteristic
127C42×C47#4111yescharacteristic
128C42×C47C2411yescharacteristic
129C22⋊Dic47—21no—
130#10625337—44no—
131#10625336—44no—
132(C2×D47)⋊C4—21no—
133(C2×D47)⋊C4—21no—
134C4⋊Dic47C2×D411yescharacteristic
135C4⋊Dic47C2×D411yescharacteristic
136Dic47⋊C4—21no—
137Dic47⋊C4—21no—
138C4×Dic47—21no—
139C4×Dic47—21no—
140C4×Dic47—21no—
141#10625329—42no—
142#10625329—42no—
143C2×C47⋊C8—22no—
144C2×C47⋊C8—22no—
145C2×C47⋊C8—22no—
146C8×D47—44no—
147C23×C47#12111yescharacteristic
148C22×D47—21no—
149Q8×C47—42no—
150Q8×C47—42no—
151Q8×C47—44no—
152Q8×C47—82no—
153Q8×C47—82no—
154C2×C4×C47—22no—
155C2×C4×C47—22no—
156C2×C4×C47—42no—
157C2×C4×C47—42no—
158C2×C4×C47—82no—
159C2×C4×C47—82no—
160C2×C4×C47#12111yescharacteristic
161C2×C4×C47#12111yescharacteristic
162C2×C4×C47#12111yescharacteristic
163C2×C4×C47C2×SD1612yes—
164C2×C4×C47#13512yes—
165C2×C4×C47C22×D411yescharacteristic
166C2×C4×C47C22×D411yescharacteristic
167C2×C4×C47C2×#4111yescharacteristic
168C22⋊D47—24no—
169C2×Dic47—21no—
170C2×Dic47—21no—
171C2×Dic47—21no—
172D188—21no—
173D188C2×SD1612yes—
174C4×D47—24no—
175C4×D47—42no—
176C4×D47—42no—
177Dic94—21no—
178Dic94C2×SD1612yes—
179C8×C47—22no—
180C8×C47—22no—
181C47⋊C8—42no—
182C47⋊C8—42no—
183C47⋊C8—42no—
184#19275—471nomaximal, Sylow
185C22×C47—21no—
186C22×C47—21no—
187C22×C47#46512yes—
188C22×C47C2×#12111yescharacteristic
189C2×D47—22no—
190C2×D47—41no—
191C2×D47—41no—
192C4×C47—21no—
193C4×C47—21no—
194C4×C47—82no—
195C4×C47—82no—
196C4×C47#46512yes—
197C4×C47#47312yes—
198C4×C47C22×SD1611yescharacteristic
199C4×C47C2×#13511yescharacteristic
200C4×C47#57512yes—
201Dic47—22no—
202Dic47—41no—
203Dic47—41no—
204C2×#487D4711yescharacteristic, complemented
205#3079—471no—
206#3039—471no—
207C8×#41—471no—
208#1680—474no—
209#1679—474no—
210#1566—471no—
211#1549—471no—
212#1546—471no—
213C2×C47—22no—
214C2×C47#304011yescharacteristic
215C2×C47C2×#46511yescharacteristic
216C2×C47C2×#47311yescharacteristic
217D47—42no—
218C2×#137—941no—
219C2×C4⋊Q8C2×D4711yescharacteristic
220C2×C4⋊Q8C2×D4711yescharacteristic
221C4×#41—471no—
222#487—21no—
223#487—21no—
224#487C2×D4714yes—
225#477—941no—
226#476—941no—
227#475—472no—
228#474—472no—
229#462—472no—
230#462—941no—
231#461—472no—
232#461—941no—
233C8×Q8—472no—
234C8×D4—472no—
235C8×D4—942no—
236#431—471no—
237#431—941no—
238#428—941no—
239#423—471no—
240#423—471no—
241#419—941no—
242#417—941no—
243C2×#103—22no—
244C2×C4×C8—941no—
245C2×C4×C8C2×D4711yescharacteristic
246#334—474no—
247#332—474no—
248#332—474no—
249C47#1927511yescharacteristic, complemented, Sylow
250C2×#41—471no—
251C22×Q8—21no—
252C22×Q8—21no—
253#137—1884no—
254C2×Q16—942no—
255C2×SD16—942no—
256C2×M16—941no—
257C2×M16—941no—
258C22×C8—942no—
259C22×C8C22⋊D4712yes—
260C4⋊Q8—21no—
261C4⋊Q8—21no—
262C4⋊Q8—21no—
263C4⋊Q8—21no—
264C4⋊Q8—21no—
265C4⋊Q8—21no—
266C4⋊Q8C22×D4712yes—
267C4⋊Q8C22×D4712yes—
268C4×Q8—471no—
269C4×Q8—471no—
270C4×D4—471no—
271C4×D4—471no—
272C4×D4—941no—
273C4×D4—941no—
274#117—471no—
275#117—941no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C42—941no—
281C2×C42C22×D4711yescharacteristic
282#107—942no—
283#106—942no—
284C4⋊C8—472no—
285C4⋊C8—472no—
286C4⋊C8—472no—
287C4⋊C8—942no—
288C4⋊C8—941no—
289C4⋊C8—941no—
290#103—24no—
291#103—24no—
292#103—942no—
293#103—942no—
294#103—942no—
295#102—942no—
296C22⋊C8—941no—
297C22⋊C8—942no—
298C4×C8—942no—
299C4×C8—941no—
300C4×C8C22⋊D4711yescharacteristic
301C4×C8C22⋊D4711yescharacteristic
302C4×C8C22×D4712yes—
303#41—474no—
304#41—942no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—44no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—471no—
313C2×D4—471no—
314C2×D4—941no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—941no—
318C22×C4—941no—
319C22×C4—941no—
320C22×C4D4×D4711yescharacteristic
321C22×C4#1062535211yescharacteristic
322C22×C4C2×C22⋊D4711yescharacteristic
323Q16—1884no—
324SD16—1884no—
325M16—1882no—
326M16—1882no—
327C2×C8—42no—
328C2×C8—942no—
329C2×C8—942no—
330C2×C8—942no—
331C2×C8—1884no—
332C2×C8C2×C22⋊D4712yes—
333C2×C8C2×C22⋊D4712yes—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—471no—
343C4⋊C4—471no—
344C4⋊C4—941no—
345C4⋊C4—941no—
346C22⋊C4—941no—
347C22⋊C4—941no—
348C22⋊C4—941no—
349C42—941no—
350C42—941no—
351C42—941no—
352C42D4×D4711yescharacteristic
353C42#1062535211yescharacteristic
354C42C2×C22⋊D4711yescharacteristic
355C42C23×D4711yescharacteristic
356C23—941no—
357C23#1309399511yescharacteristic
358Q8—42no—
359Q8—42no—
360Q8—44no—
361Q8—82no—
362Q8—82no—
363Q8—472no—
364Q8—941no—
365D4—472no—
366D4—944no—
367D4—941no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—941no—
375C2×C4—944no—
376C2×C4—941no—
377C2×C4—941no—
378C2×C4—1882no—
379C2×C4—1882no—
380C2×C4#1309399511yescharacteristic
381C2×C4#1309399511yescharacteristic
382C2×C4#1309399511yescharacteristic
383C2×C4SD16×D4712yes—
384C2×C4#1309404512yes—
385C2×C4C2×D4×D4711yescharacteristic
386C2×C4C22×C22⋊D4711yescharacteristic
387C2×C4C2×#1062535211yescharacteristic
388C8—22no—
389C8—22no—
390C8—1882no—
391C8—1882no—
392C8—1882no—
393C22—21no—
394C22—21no—
395C22—942no—
396C22—1881no—
397C22—1881no—
398C22#1568934912yes—
399C22C2×#1309399511yescharacteristic
400C4—21no—
401C4—21no—
402C4—82no—
403C4—82no—
404C4—942no—
405C4—1881no—
406C4—1881no—
407C4#1568934912yes—
408C4#1568939512yes—
409C4C2×SD16×D4711yescharacteristic
410C4C2×#1309404511yescharacteristic
411C4#1568992112yes—
412C2—22no—
413C2—1882no—
414C2#2025287811yescharacteristic
415C2C2×#1568934911yescharacteristic
416C2C2×#1568939511yescharacteristic
subgroupG/H
centerC22C2×#13093995
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC2×C47×#487C2
socleC22×C47C2×#121
prime radicalC2×#10625350D4
hypercenterC2×#487D47
hypocenterC47#19275
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. #20242815C2semidirect
5. #20242808C2semidirect
6. #20242144C2semidirect
7. #20242073C2semidirect
8. #20242060C2semidirect
9. C2×C47×#487C2semidirect
204. C2×#487D47semidirect
249. C47#19275semidirect

Maximal subgroups

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

subgroupindex
#202531972
#202528712
C8×#106253502
#202428152
#202428082
#202421442
#202420732
#202420602
C2×C47×#4872
#1927547