Finite Group Databasebeta

← #11887599

Subgroups — #11887599

Order 1152.

1164 conjugacy classes of subgroups.

The subgroup lattice has 369 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10596201C211yesmaximal, characteristic
2Q8×C4⋊(C3⋊D3)C211yesmaximal, characteristic
3C32×#477C211yesmaximal, characteristic, complemented
4#10592792C211yesmaximal, characteristic, complemented
5C4×#3756C211yesmaximal, characteristic, complemented
6#10592586C212yesmaximal, complemented
7#10592583C212yesmaximal, complemented
8#10592548C211yesmaximal, characteristic, complemented
9#10592533C211yesmaximal, characteristic
10#10592508C212yesmaximal, complemented
11C3⋊D3×#103C212yesmaximal, complemented
12#81119—34nomaximal
13C2×Q8×C3⋊D3C2212yes—
14C32×C4⋊Q8C2211yescharacteristic
15C32×C4×Q8C2211yescharacteristic
16#64155—22no—
17C2×#3756C2211yescharacteristic
18C2×#3746C2212yes—
19C2×C8×C3⋊D3C2212yes—
20#64106—22no—
21#64106C2212yes—
22#64106C2212yes—
23#64102—21no—
24C4⋊C4×C3⋊D3C2212yes—
25#64099—21no—
26C4×C4⋊(C3⋊D3)—21no—
27C4×C4⋊(C3⋊D3)C2211yescharacteristic
28#64078—21no—
29#64078C2211yescharacteristic
30C4×#634C2211yescharacteristic
31C32×#107C2212yes—
32C32×C4⋊C8C2211yescharacteristic
33C32×#103C2212yes—
34#63662C2211yescharacteristic
35#63647C2212yes—
36#63643C2212yes—
37#63637C2212yes—
38#63637C2212yes—
39#63637C2211yescharacteristic
40#63634C2211yescharacteristic
41C4×C3⋊(C3⋊C8)C2211yescharacteristic
42#5730—34no—
43Q8×D12—34no—
44C3×#477D314yescomplemented
45#5093—34no—
46C4×#216—34no—
47#4887—38no—
48#4884—38no—
49#4849—34no—
50#4834—34no—
51#4809—38no—
52D3×#103—38no—
53C2×C32×Q8—21no—
54C2×C32×Q8C2311yescharacteristic
55Q8×C3⋊D3—24no—
56Q8×C3⋊D3—42no—
57C2×C4⋊(C3⋊D3)D411yescharacteristic
58C2×C4×C3⋊D3—22no—
59C2×C4×C3⋊D3—22no—
60C2×C4×C3⋊D3C2312yes—
61C2×#634—22no—
62C2×#634C2311yescharacteristic
63C2×#634C2312yes—
64C2×C32×C8C2312yes—
65C32×C4⋊C4—21no—
66C32×C4⋊C4—21no—
67C32×C4⋊C4C2312yes—
68C32×C4⋊C4C2311yescharacteristic
69C32×C42—21no—
70C32×C42C2311yescharacteristic
71#3756—22no—
72#3752—22no—
73#3752—22no—
74#3752D412yes—
75C4⋊(C3⋊Dic3)—21no—
76C4⋊(C3⋊Dic3)—21no—
77C4⋊(C3⋊Dic3)—22no—
78C4⋊(C3⋊Dic3)—22no—
79C4⋊(C3⋊Dic3)D411yescharacteristic
80C4⋊(C3⋊Dic3)C2311yescharacteristic
81#3750—21no—
82#3750—22no—
83C4×C3⋊Dic3—21no—
84C2×C3⋊(C3⋊C8)C2311yescharacteristic
85C2×C3⋊(C3⋊C8)C2311yescharacteristic
86#3746—42no—
87C8×C3⋊D3—42no—
88#3299—91noSylow
89C2×D3×Q8—38no—
90C3×C4⋊Q8C2×D314yes—
91C3×C4×Q8C2×D314yes—
92#946—68no—
93C2×#216—34no—
94C2×Dic12—38no—
95C2×D3×C8—38no—
96#897—38no—
97#897—38no—
98#897—68no—
99#893—64no—
100D3×C4⋊C4—38no—
101#890—64no—
102C4×D12—34no—
103C4×D12—64no—
104#869—34no—
105#869—64no—
106C4×Dic6—34no—
107C3×#107C2×D318yes—
108C3×C4⋊C8C2×D314yes—
109C3×#103C2×D318yes—
110#835—34no—
111(C2×D3)⋊C8—38no—
112#816—38no—
113#810—38no—
114#810—34no—
115#810—38no—
116#807—34no—
117C4×C3⋊C8—34no—
118C22×C3⋊D3C2×D412yes—
119C32×Q8—21no—
120C32×Q8—81no—
121C32×Q8#4112yes—
122C2×C32×C4—21no—
123C2×C32×C4—22no—
124C2×C32×C4—21no—
125C2×C32×C4C2×D411yescharacteristic
126C2×C32×C4C2×D411yescharacteristic
127C2×C32×C4C2411yescharacteristic
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3—21no—
130C2×C3⋊Dic3—22no—
131C2×C3⋊Dic3C2×D412yes—
132C4⋊(C3⋊D3)Q1614yescomplemented
133C4×C3⋊D3—22no—
134C4×C3⋊D3—22no—
135C4×C3⋊D3—44no—
136C4×C3⋊D3—82no—
137#634—21no—
138#634—22no—
139#634—21no—
140#634—22no—
141#634—44no—
142C32×C8—22no—
143C3⋊(C3⋊C8)—21no—
144C3⋊(C3⋊C8)—21no—
145#554—91no—
146D4×Q8—91no—
147#487—91no—
148#484—92no—
149#477C3⋊D311yescharacteristic, complemented
150#467—92no—
151#462—91no—
152#451—92no—
153C4×Q16—91no—
154C8×D4—91no—
155C2×#103—92no—
156C2×C3×Q8—24no—
157C2×C3×Q8C22×D314yes—
158D3×Q8—616no—
159D3×Q8—128no—
160C2×D12—34no—
161C2×C4×D3—38no—
162C2×C4×D3—68no—
163C2×C4×D3—68no—
164C2×Dic6—38no—
165C2×Dic6—34no—
166C2×Dic6—68no—
167C2×C3×C8C22×D318yes—
168C3×C4⋊C4—24no—
169C3×C4⋊C4—24no—
170C3×C4⋊C4C22×D314yes—
171C3×C4⋊C4C22×D318yes—
172C3×C42—24no—
173C3×C42C22×D314yes—
174#216—68no—
175(C2×D3)⋊C4—38no—
176(C2×D3)⋊C4—68no—
177(C2×D3)⋊C4—68no—
178C4⋊Dic3—34no—
179C4⋊Dic3—34no—
180C4⋊Dic3—68no—
181C4⋊Dic3—68no—
182C4⋊Dic3—64no—
183C4⋊Dic3—64no—
184Dic3⋊C4—68no—
185Dic3⋊C4—64no—
186C4×Dic3—64no—
187C2×C3⋊C8—34no—
188C2×C3⋊C8—34no—
189Dic12—128no—
190D3×C8—128no—
191C22×C32C22×D411yescharacteristic
192C2×C3⋊D3—22no—
193C2×C3⋊D3—22no—
194C2×C3⋊D3C2×Q1614yes—
195C32×C4—22no—
196C32×C4—21no—
197C32×C4—42no—
198C32×C4—41no—
199C32×C4C2×Q1612yes—
200C32×C4C2×#4111yescharacteristic
201C32×C4H(5,𝔽2)11yescharacteristic
202C3⋊Dic3—22no—
203C3⋊Dic3—42no—
204C3⋊Dic3—41no—
205C3⋊Dic3—41no—
206C22×Q8—92no—
207C2×Q16—91no—
208C2×Q16—92no—
209C22×C8—92no—
210C4⋊Q8—91no—
211C4⋊Q8—181no—
212C4⋊Q8C2×C3⋊D311yescharacteristic
213#125—181no—
214C22⋊Q8—92no—
215C22⋊Q8—92no—
216C22⋊Q8—182no—
217C22⋊Q8—182no—
218C4×Q8—91no—
219C4×Q8C2×C3⋊D311yescharacteristic
220C4×D4—91no—
221C4×D4—181no—
222C4×D4—181no—
223C2×C4⋊C4—92no—
224#107—91no—
225#107C2×C3⋊D312yes—
226C4⋊C8C2×C3⋊D311yescharacteristic
227#103—91no—
228#103—92no—
229#103—92no—
230#103—91no—
231#103—92no—
232#103C2×C3⋊D312yes—
233C22⋊C8—92no—
234C4×C8—91no—
235C22×D3—38no—
236C3×Q8—24no—
237C3×Q8—84no—
238C3×Q8#23918yes—
239C2×C3×C4—24no—
240C2×C3×C4—28no—
241C2×C3×C4—24no—
242C2×C3×C4D3×D414yes—
243C2×C3×C4D3×D414yes—
244C2×C3×C4C23×D314yes—
245C2×Dic3—38no—
246C2×Dic3—64no—
247C2×Dic3—64no—
248C2×Dic3—68no—
249D12—316no—
250C4×D3—68no—
251C4×D3—68no—
252C4×D3—1216no—
253C4×D3—248no—
254Dic6—64no—
255Dic6—68no—
256Dic6—64no—
257Dic6—68no—
258Dic6—1216no—
259C3×C8—28no—
260C3⋊C8—64no—
261C3⋊C8—64no—
262C2×C32#54611yescharacteristic
263C2×C32C22×Q1611yescharacteristic
264C2×C32#57711yescharacteristic
265C3⋊D3—24no—
266C2×Q8—21no—
267C2×Q8—92no—
268C2×Q8—91no—
269C2×Q8—182no—
270C2×Q8—184no—
271C2×Q8—362no—
272C2×Q8C22×C3⋊D311yescharacteristic
273C2×D4—91no—
274C22×C4—92no—
275C22×C4—182no—
276C22×C4—182no—
277Q16—182no—
278Q16—362no—
279C2×C8—91no—
280C2×C8—91no—
281C2×C8—362no—
282C2×C8C22×C3⋊D312yes—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—91no—
286C4⋊C4—91no—
287C4⋊C4—182no—
288C4⋊C4—182no—
289C4⋊C4—182no—
290C4⋊C4—181no—
291C4⋊C4—181no—
292C4⋊C4—181no—
293C4⋊C4C22×C3⋊D312yes—
294C4⋊C4C22×C3⋊D311yescharacteristic
295C22⋊C4—92no—
296C22⋊C4—182no—
297C22⋊C4—182no—
298C42—21no—
299C42—181no—
300C42C22×C3⋊D311yescharacteristic
301C22×C3C2×D3×D414yes—
302C2×D3—316no—
303C2×D3—68no—
304C2×D3—68no—
305C3×C4—24no—
306C3×C4—28no—
307C3×C4—48no—
308C3×C4—44no—
309C3×C4D3×Q1618yes—
310C3×C4C2×#23914yes—
311C3×C4#100914yes—
312Dic3—68no—
313Dic3—124no—
314Dic3—128no—
315Dic3—124no—
316C32#329911yescharacteristic, complemented, Sylow
317C23—92no—
318Q8—21no—
319Q8—81no—
320Q8—181no—
321Q8—182no—
322Q8—181no—
323Q8—182no—
324Q8—364no—
325Q8#383212yes—
326D4—94no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—92no—
331C2×C4—182no—
332C2×C4—182no—
333C2×C4—181no—
334C2×C4—181no—
335C2×C4—182no—
336C2×C4—364no—
337C2×C4—722no—
338C2×C4D4×C3⋊D311yescharacteristic
339C2×C4D4×C3⋊D311yescharacteristic
340C2×C4C23×C3⋊D311yescharacteristic
341C8—22no—
342C8—181no—
343C8—181no—
344C2×C3#563014yes—
345C2×C3C2×D3×Q1614yes—
346C2×C3#577514yes—
347D3—616no—
348C22—94no—
349C22—182no—
350C22—182no—
351C22C2×D4×C3⋊D311yescharacteristic
352C4—22no—
353C4—21no—
354C4—42no—
355C4—41no—
356C4—182no—
357C4—362no—
358C4—361no—
359C4—361no—
360C4Q16×C3⋊D312yes—
361C4C2×#383211yescharacteristic
362C4#6436611yescharacteristic
363C3#8111914yescomplemented
364C2—184no—
365C2#1059610111yescharacteristic
366C2C2×Q16×C3⋊D311yescharacteristic
367C2#1059624611yescharacteristic
subgroupG/H
centerC22C2×D4×C3⋊D3
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×#477C2
socleC22×C32C22×D4
prime radicalC2×C4⋊(C3⋊D3)D4
hypercenter#477C3⋊D3
hypocenterC32#3299
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
3. C32×#477C2semidirect
4. #10592792C2semidirect
5. C4×#3756C2semidirect
6. #10592586C2semidirect
7. #10592583C2semidirect
8. #10592548C2semidirect
10. #10592508C2semidirect
11. C3⋊D3×#103C2semidirect
44. C3×#477D3semidirect
132. C4⋊(C3⋊D3)Q16semidirect
149. #477C3⋊D3semidirect
316. C32#3299semidirect
363. C3#81119semidirect

Maximal subgroups

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

subgroupindex
#105962012
Q8×C4⋊(C3⋊D3)2
C32×#4772
#105927922
C4×#37562
#105925862
#105925832
#105925482
#105925332
#105925082
C3⋊D3×#1032
#811193