Finite Group Databasebeta

← #11878730

Subgroups — #11878730

Order 1152.

915 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 915 conjugacy classes of subgroups — too many to draw. Only its 166 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10594929C211yesmaximal, characteristic
2C2×#63873C211yesmaximal, characteristic, complemented
3#10592417C211yesmaximal, characteristic, complemented
4C3×#4738C212yesmaximal, complemented
5#10591171C212yesmaximal, complemented
6#10590901C212yesmaximal, complemented
7#10590884C212yesmaximal, complemented
8#10590748C214yesmaximal, complemented
9#80034—32nomaximal
10C22×C3×Dic6C2212yes—
11C2×#3805—21no—
12C2×C4×C3⋊Dic3C2211yescharacteristic
13C3×#930C2212yes—
14C2×C3×#204C2214yes—
15#63980—21no—
16C2×#3724—21no—
17C2×#3724—21no—
18#63919C2211yescharacteristic
19#63915—21no—
20#63914C2211yescharacteristic
21#63902—21no—
22#63898C2212yes—
23#63873—21no—
24#63873—21no—
25#63873C2212yes—
26#63873C2212yes—
27C2×#3715C2214yes—
28C32×C22⋊C8C2211yescharacteristic
29#63657C2211yescharacteristic
30#63641C2212yes—
31C8×C3⋊Dic3C2212yes—
32C3×#816C2214yes—
33#63565C2214yes—
34C2×#944—32no—
35#5596—32no—
36C3×#448—32no—
37#5227—32no—
38#5175—34no—
39#5162—34no—
40#5040—32no—
41#4818—34no—
42#4792—34no—
43#4738D312yescomplemented
44#4718—32no—
45#4718—61no—
46C22×C32×C4C2311yescharacteristic
47C22×C3⋊Dic3D411yescharacteristic
48C22×C3⋊Dic3D411yescharacteristic
49C2×C3×#67—22no—
50C22×C3×Dic3—22no—
51C2×C3×D12C2312yes—
52C2×C3×Dic6—22no—
53C2×C3×Dic6—44no—
54C2×C3×Dic6C2314yes—
55C2×#627—21no—
56C2×#625—21no—
57C2×#625—21no—
58#3805—44no—
59C2×D3×Dic3—22no—
60C2×C32×C8C2312yes—
61C4×C3⋊Dic3D411yescharacteristic
62C4×C3⋊Dic3D411yescharacteristic
63C4×C3⋊Dic3C2311yescharacteristic
64C4×C3⋊Dic3C2311yescharacteristic
65C2×C3⋊(C3⋊C8)C2312yes—
66C3×(C2×D3)⋊C4—22no—
67C3×C4⋊Dic3C2312yes—
68C3×#204—44no—
69#3724—22no—
70#3724—22no—
71#3724—22no—
72#3724—22no—
73#3724—21no—
74(C2×D3)⋊Dic3—22no—
75#3715—44no—
76#3206—91noSylow
77C22×C3×Q8—32no—
78C2×#237—62no—
79C22×Dic6C2×D312yes—
80C2×C3×SD16—34no—
81C3×#121—32no—
82#944—34no—
83#944—34no—
84#944—62no—
85#944—62no—
86C2×#215—34no—
87#940—32no—
88#939—62no—
89D4×Dic3—32no—
90C2×#214—34no—
91#930C2×D312yes—
92C2×Dic3⋊C4—62no—
93C2×Dic3⋊C4—62no—
94C2×C4×Dic3—32no—
95C2×C4×Dic3—61no—
96C2×#204C2×D314yes—
97#893—32no—
98#888—32no—
99#885—62no—
100#67⋊C4—62no—
101C3×#103—34no—
102C3×C22⋊C8—21no—
103C3×C22⋊C8C2×D312yes—
104#835—34no—
105#830—32no—
106#830—61no—
107#816C2×D314yes—
108Dic3⋊C8—34no—
109Dic3⋊C8—62no—
110C8×Dic3—34no—
111C8×Dic3—62no—
112#810—34no—
113C23×C32C2×D411yescharacteristic
114C22×C3×D3—22no—
115C2×C32×C4—22no—
116C2×C32×C4C2×D411yescharacteristic
117C2×C32×C4C2411yescharacteristic
118C2×C3⋊Dic3—22no—
119C2×C3⋊Dic3SD1614yescomplemented
120C2×C3⋊Dic3C2×D411yescharacteristic
121C2×C3⋊Dic3C2×D411yescharacteristic
122C2×C3⋊Dic3C2×D411yescharacteristic
123C2×C3⋊Dic3C2×D411yescharacteristic
124C3×#67—44no—
125C2×C3×Dic3—22no—
126C2×C3×Dic3—24no—
127C2×C3×Dic3—44no—
128C3×D12—24no—
129C3×Dic6—24no—
130C3×Dic6—24no—
131C3×Dic6—44no—
132#627—42no—
133#625—42no—
134#625—42no—
135D3×Dic3—44no—
136C32×C8—22no—
137C3⋊(C3⋊C8)—22no—
138#532—91no—
139C2×C4⋊Q8—91no—
140#492—92no—
141#487—92no—
142#461—94no—
143#448—32no—
144#448—92no—
145#431—91no—
146C2×C3×Q8—34no—
147C2×C3×Q8—62no—
148C2×C3×Q8—124no—
149C2×C3×D4—32no—
150C2×C3×D4—62no—
151C22×C3×C4—21no—
152C22×C3×C4—62no—
153C22×C3×C4C22×D312yes—
154C2×#67—22no—
155C2×#67—62no—
156C2×#67—62no—
157C22×Dic3—22no—
158C22×Dic3—32no—
159C22×Dic3—32no—
160C22×Dic3—61no—
161C22×Dic3—61no—
162C22×Dic3—62no—
163#237—128no—
164C2×D12C22×D312yes—
165C2×C4×D3—62no—
166C2×Dic6—22no—
167C2×Dic6—44no—
168C2×Dic6—62no—
169C2×Dic6C22×D314yes—
170C3×SD16—124no—
171C2×C3×C8—22no—
172C2×C3×C8C22×D314yes—
173C3×C4⋊C4—32no—
174C3×C22⋊C4—62no—
175#217—62no—
176#215—124no—
177#214—124no—
178(C2×D3)⋊C4—22no—
179(C2×D3)⋊C4—62no—
180C4⋊Dic3C22×D312yes—
181Dic3⋊C4—62no—
182Dic3⋊C4—64no—
183Dic3⋊C4—64no—
184Dic3⋊C4—64no—
185Dic3⋊C4—64no—
186C4×Dic3—32no—
187C4×Dic3—32no—
188C4×Dic3—32no—
189C4×Dic3—32no—
190C4×Dic3—61no—
191C4×Dic3—61no—
192C4×Dic3—61no—
193C4×Dic3—61no—
194C2×C3⋊C8—34no—
195C2×C3⋊C8—62no—
196#204—44no—
197C22×C32—21no—
198C22×C32—21no—
199C22×C32C2×SD1612yes—
200C22×C32C22×D411yescharacteristic
201C2×C3×D3—42no—
202C2×C3×D3—42no—
203C2×C3×D3—42no—
204C32×C4—21no—
205C32×C4H(5,𝔽2)12yes—
206C3⋊Dic3—21no—
207C3⋊Dic3C2×SD1612yes—
208C3⋊Dic3C2×SD1612yes—
209C3⋊Dic3#13712yes—
210C3×Dic3—44no—
211C3×Dic3—42no—
212C2×#41—181no—
213C22×Q8—32no—
214C2×SD16—34no—
215C2×SD16—94no—
216C4⋊Q8—91no—
217C4⋊Q8—92no—
218C4⋊Q8—92no—
219C4⋊Q8—181no—
220C4⋊Q8—181no—
221#127—91no—
222#124—181no—
223#121—32no—
224#121—181no—
225C4×D4—92no—
226C4×D4—181no—
227C2×C4⋊C4—181no—
228C2×C4⋊C4—181no—
229C2×C42—91no—
230C4⋊C8—92no—
231#103—34no—
232#103—94no—
233C22⋊C8—91no—
234C22⋊C8D3211yescharacteristic
235C4×C8—92no—
236C23×C3—21no—
237C23×C3—62no—
238C23×C3D3×D412yes—
239C22×D3—22no—
240C3×Q8—64no—
241C3×Q8—64no—
242C3×Q8—124no—
243C3×D4—64no—
244C3×D4—124no—
245C2×C3×C4—24no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—42no—
249C2×C3×C4—64no—
250C2×C3×C4—62no—
251C2×C3×C4—124no—
252C2×C3×C4D3×D412yes—
253C2×C3×C4C23×D312yes—
254#67—44no—
255#67—124no—
256#67—124no—
257C2×Dic3—24no—
258C2×Dic3—22no—
259C2×Dic3—38no—
260C2×Dic3—32no—
261C2×Dic3—32no—
262C2×Dic3—32no—
263C2×Dic3—32no—
264C2×Dic3—44no—
265C2×Dic3—64no—
266C2×Dic3—61no—
267C2×Dic3—61no—
268C2×Dic3—61no—
269C2×Dic3—61no—
270C2×Dic3—64no—
271C2×Dic3—122no—
272C2×Dic3—124no—
273D12—24no—
274C4×D3—124no—
275Dic6—24no—
276Dic6—24no—
277Dic6—44no—
278Dic6—124no—
279C3×C8—24no—
280C3×C8—42no—
281C3⋊C8—64no—
282C3⋊C8—122no—
283C2×C32—22no—
284C2×C32#53511yescharacteristic
285C2×C32C22×SD1611yescharacteristic
286C2×C32C2×#13711yescharacteristic
287C3×D3—82no—
288#41—364no—
289C2×Q8—34no—
290C2×Q8—62no—
291C2×Q8—124no—
292C2×Q8—181no—
293C2×D4—32no—
294C2×D4—62no—
295C2×D4—181no—
296C2×D4—181no—
297C22×C4—62no—
298C22×C4—91no—
299C22×C4—91no—
300C22×C4—182no—
301C22×C4C2×D3211yescharacteristic
302SD16—124no—
303SD16—364no—
304C2×C8—92no—
305C2×C8C2×D3212yes—
306C4⋊C4—32no—
307C4⋊C4—182no—
308C4⋊C4—182no—
309C4⋊C4—182no—
310C4⋊C4—182no—
311C4⋊C4—181no—
312C22⋊C4—62no—
313C22⋊C4—182no—
314C42—91no—
315C42—91no—
316C42—91no—
317C42—91no—
318C22×C3—22no—
319C22×C3—22no—
320C22×C3—21no—
321C22×C3—22no—
322C22×C3—41no—
323C22×C3—41no—
324C22×C3—122no—
325C22×C3—122no—
326C22×C3—122no—
327C22×C3D3×SD1614yes—
328C22×C3C2×D3×D412yes—
329C2×D3—42no—
330C2×D3—42no—
331C2×D3—42no—
332C3×C4—22no—
333C3×C4—22no—
334C3×C4—41no—
335C3×C4—122no—
336C3×C4—124no—
337C3×C4#100414yes—
338Dic3—34no—
339Dic3—34no—
340Dic3—34no—
341Dic3—42no—
342Dic3—44no—
343Dic3—62no—
344Dic3—62no—
345Dic3—62no—
346Dic3—62no—
347Dic3—121no—
348C32#320611yescharacteristic, complemented, Sylow
349C23—62no—
350C23#380211yescharacteristic
351Q8—64no—
352Q8—64no—
353Q8—124no—
354Q8—362no—
355D4—64no—
356D4—124no—
357D4—362no—
358D4—362no—
359C2×C4—22no—
360C2×C4—62no—
361C2×C4—64no—
362C2×C4—91no—
363C2×C4—91no—
364C2×C4—91no—
365C2×C4—94no—
366C2×C4—91no—
367C2×C4—124no—
368C2×C4—182no—
369C2×C4—364no—
370C2×C4#380211yescharacteristic
371C2×C4C22×D3211yescharacteristic
372C8—22no—
373C8—182no—
374C2×C3—24no—
375C2×C3—21no—
376C2×C3—21no—
377C2×C3—21no—
378C2×C3—42no—
379C2×C3—242no—
380C2×C3#559212yes—
381C2×C3C2×D3×SD1612yes—
382C2×C3D3×#13712yes—
383D3—82no—
384C22—21no—
385C22—21no—
386C22—122no—
387C22—122no—
388C22—122no—
389C22#6379612yes—
390C22C2×#380211yescharacteristic
391C4—21no—
392C4—92no—
393C4—92no—
394C4—92no—
395C4—122no—
396C4—124no—
397C4—181no—
398C4#6431312yes—
399C3—21no—
400C3#8003412yescomplemented
401C2—22no—
402C2—242no—
403C2#1059419211yescharacteristic
404C2C2×#6379611yescharacteristic
405C2#1059471711yescharacteristic
subgroupG/H
centerC22C2×#3802
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C22⋊C8C22
socleC22×C32C22×D4
prime radical#10594929C2
hypercenterC22⋊C8D32
hypocenterC32#3206
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#63873C2semidirect
3. #10592417C2semidirect
4. C3×#4738C2semidirect
5. #10591171C2semidirect
6. #10590901C2semidirect
7. #10590884C2semidirect
8. #10590748C2semidirect
43. #4738D3semidirect
119. C2×C3⋊Dic3SD16semidirect
348. C32#3206semidirect
400. C3#80034semidirect

Maximal subgroups

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

subgroupindex
#105949292
C2×#638732
#105924172
C3×#47382
#105911712
#105909012
#105908842
#105907482
#800343