Finite Group Databasebeta

← #11895460

Subgroups — #11895460

Order 1152.

1008 conjugacy classes of subgroups.

The subgroup lattice has 339 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10596038C211yesmaximal, characteristic
2Q8×#634C211yesmaximal, characteristic
3C32×C4×Q16C211yesmaximal, characteristic
4#10592881C212yesmaximal
5#10592730C211yesmaximal, characteristic
6#10592720C211yesmaximal, characteristic
7#10592574C212yesmaximal
8#10592491C212yesmaximal
9#10592489C212yesmaximal
10#10592380C211yesmaximal, characteristic
11C8×#634C211yesmaximal, characteristic
12#79699—34nomaximal
13C2×C32×Q16C2211yescharacteristic
14C32×C4×Q8C2211yescharacteristic
15C32×C4×Q8C2211yescharacteristic
16Q8×C3⋊Dic3C2212yes—
17#64153C2212yes—
18C2×#3756C2212yes—
19#64098—22no—
20#64097—22no—
21#64097C2212yes—
22#64096C2212yes—
23#64079—21no—
24#64078—21no—
25#64078C2211yescharacteristic
26#64078C2211yescharacteristic
27C4×#634—21no—
28C4×#634—21no—
29C4×#634C2211yescharacteristic
30C32×#107C2211yescharacteristic
31C32×#103C2211yescharacteristic
32C32×#103C2211yescharacteristic
33C32×C4×C8C2211yescharacteristic
34#63662C2212yes—
35#63643C2212yes—
36#63643C2212yes—
37#63641C2212yes—
38C8×C3⋊Dic3C2212yes—
39#63637C2212yes—
40#63634C2212yes—
41C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
42#5567—34no—
43Q8×Dic6—34no—
44C3×C4×Q16D314yes—
45#5182—38no—
46#5031—34no—
47#5021—34no—
48#4875—38no—
49#4792—38no—
50#4790—38no—
51#4681—34no—
52C8×Dic6—34no—
53C2×C32×Q8C2311yescharacteristic
54C2×C32×Q8C2311yescharacteristic
55C2×#634—21no—
56C2×#634—22no—
57C2×#634D411yescharacteristic
58C2×#634C2312yes—
59C32×Q16—22no—
60C2×C32×C8C2311yescharacteristic
61C2×C32×C8C2311yescharacteristic
62C32×C4⋊C4—21no—
63C32×C4⋊C4—21no—
64C32×C4⋊C4C2311yescharacteristic
65C32×C4⋊C4C2311yescharacteristic
66C32×C42—21no—
67C32×C42—21no—
68C32×C42C2311yescharacteristic
69#3756—24no—
70C4⋊(C3⋊Dic3)—21no—
71C4⋊(C3⋊Dic3)—21no—
72C4⋊(C3⋊Dic3)—21no—
73C4⋊(C3⋊Dic3)—22no—
74C4⋊(C3⋊Dic3)D411yescharacteristic
75C4⋊(C3⋊Dic3)C2312yes—
76#3750—22no—
77#3750—22no—
78#3750—22no—
79#3750—22no—
80#3750D412yes—
81C4×C3⋊Dic3—22no—
82C4×C3⋊Dic3—22no—
83C4×C3⋊Dic3C2312yes—
84C2×C3⋊(C3⋊C8)C2312yes—
85#3363—91noSylow
86C2×C3×Q16C2×D314yes—
87C3×C4×Q8C2×D314yes—
88C3×C4×Q8C2×D314yes—
89Q8×Dic3—38no—
90#944—38no—
91C2×#216—38no—
92#889—68no—
93#888—38no—
94#888—68no—
95#887—38no—
96#870—64no—
97#869—34no—
98#869—34no—
99#869—64no—
100C4×Dic6—34no—
101C4×Dic6—64no—
102C4×Dic6—64no—
103C3×#107C2×D314yes—
104C3×#103C2×D314yes—
105C3×#103C2×D314yes—
106C3×C4×C8C2×D314yes—
107#835—38no—
108#816—38no—
109#816—38no—
110Dic3⋊C8—38no—
111C8×Dic3—38no—
112#810—38no—
113#807—38no—
114C4⋊(C3⋊C8)—34no—
115C32×Q8—21no—
116C32×Q8—21no—
117C32×Q8—21no—
118C32×Q8#4112yes—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4C2×D411yescharacteristic
124C2×C32×C4C2×D411yescharacteristic
125C2×C32×C4C2411yescharacteristic
126C2×C3⋊Dic3—22no—
127C2×C3⋊Dic3—22no—
128C2×C3⋊Dic3C2×D412yes—
129C2×C3⋊Dic3C2×D412yes—
130#634—22no—
131#634—22no—
132#634—82no—
133#634—81no—
134#634Q1614yescomplemented
135C32×C8—21no—
136C32×C8—21no—
137C3⋊(C3⋊C8)—22no—
138Q82—91no—
139#557—91no—
140#487—92no—
141#487—91no—
142#477—91no—
143#477—92no—
144#462—92no—
145#462—91no—
146C8×Q8—91no—
147C4×Q16—92no—
148C4×Q16C3⋊D311yescharacteristic
149C2×C3×Q8C22×D314yes—
150C2×C3×Q8C22×D314yes—
151C2×Dic6—34no—
152C2×Dic6—38no—
153C2×Dic6—68no—
154C2×Dic6—64no—
155C3×Q16—28no—
156C2×C3×C8C22×D314yes—
157C2×C3×C8C22×D314yes—
158C3×C4⋊C4—24no—
159C3×C4⋊C4—24no—
160C3×C4⋊C4C22×D314yes—
161C3×C4⋊C4C22×D314yes—
162C3×C42—24no—
163C3×C42—24no—
164C3×C42C22×D314yes—
165#216—616no—
166C4⋊Dic3—34no—
167C4⋊Dic3—38no—
168C4⋊Dic3—68no—
169C4⋊Dic3—64no—
170C4⋊Dic3—64no—
171C4⋊Dic3—64no—
172Dic3⋊C4—38no—
173Dic3⋊C4—68no—
174Dic3⋊C4—68no—
175Dic3⋊C4—68no—
176Dic3⋊C4—68no—
177C4×Dic3—38no—
178C4×Dic3—68no—
179C4×Dic3—68no—
180C2×C3⋊C8—38no—
181C22×C32C22×D411yescharacteristic
182C32×C4—22no—
183C32×C4—21no—
184C32×C4—41no—
185C32×C4—41no—
186C32×C4—41no—
187C32×C4C2×Q1612yes—
188C32×C4C2×#4111yescharacteristic
189C32×C42−1+411yescharacteristic
190C3⋊Dic3—22no—
191C3⋊Dic3—42no—
192C3⋊Dic3—42no—
193C3⋊Dic3C2×Q1614yes—
194C2×Q16—92no—
195C2×Q16C2×C3⋊D311yescharacteristic
196C4⋊Q8—92no—
197C4⋊Q8—92no—
198C4⋊Q8—91no—
199C4⋊Q8—91no—
200C4⋊Q8—181no—
201C4⋊Q8—182no—
202#125—181no—
203#125—182no—
204C4×Q8—91no—
205C4×Q8—92no—
206C4×Q8—92no—
207C4×Q8—181no—
208C4×Q8—181no—
209C4×Q8C2×C3⋊D311yescharacteristic
210C4×Q8C2×C3⋊D311yescharacteristic
211#107—92no—
212#107C2×C3⋊D311yescharacteristic
213C4⋊C8—92no—
214C4⋊C8—91no—
215#103—92no—
216#103—92no—
217#103—92no—
218#103—92no—
219#103C2×C3⋊D311yescharacteristic
220#103C2×C3⋊D311yescharacteristic
221C4×C8—92no—
222C4×C8C2×C3⋊D311yescharacteristic
223C3×Q8—24no—
224C3×Q8—24no—
225C3×Q8—24no—
226C3×Q8#23518yes—
227C2×C3×C4—24no—
228C2×C3×C4—24no—
229C2×C3×C4—24no—
230C2×C3×C4—24no—
231C2×C3×C4D3×D414yes—
232C2×C3×C4D3×D414yes—
233C2×C3×C4C23×D314yes—
234C2×Dic3—38no—
235C2×Dic3—38no—
236C2×Dic3—68no—
237C2×Dic3—68no—
238Dic6—316no—
239Dic6—68no—
240Dic6—68no—
241Dic6—248no—
242Dic6—244no—
243C3×C8—24no—
244C3×C8—24no—
245C3⋊C8—68no—
246C2×C32#54811yescharacteristic
247C2×C32C22×Q1611yescharacteristic
248C2×C32#57711yescharacteristic
249C2×Q8—91no—
250C2×Q8—92no—
251C2×Q8—182no—
252C2×Q8—181no—
253C2×Q8C22×C3⋊D311yescharacteristic
254C2×Q8C22×C3⋊D311yescharacteristic
255Q16—22no—
256Q16—184no—
257C2×C8—92no—
258C2×C8C22×C3⋊D311yescharacteristic
259C2×C8C22×C3⋊D311yescharacteristic
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—92no—
263C4⋊C4—91no—
264C4⋊C4—92no—
265C4⋊C4—182no—
266C4⋊C4—182no—
267C4⋊C4—182no—
268C4⋊C4—182no—
269C4⋊C4—182no—
270C4⋊C4—181no—
271C4⋊C4—181no—
272C4⋊C4—181no—
273C4⋊C4C22×C3⋊D311yescharacteristic
274C4⋊C4C22×C3⋊D311yescharacteristic
275C42—21no—
276C42—21no—
277C42—92no—
278C42—182no—
279C42—182no—
280C42C22×C3⋊D311yescharacteristic
281C22×C3C2×D3×D414yes—
282C3×C4—24no—
283C3×C4—28no—
284C3×C4—44no—
285C3×C4—44no—
286C3×C4—44no—
287C3×C4D3×Q1618yes—
288C3×C4C2×#23514yes—
289C3×C4#101014yes—
290Dic3—316no—
291Dic3—68no—
292Dic3—128no—
293Dic3—128no—
294C32#336311yescharacteristic, complemented, Sylow
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—94no—
299Q8—182no—
300Q8—182no—
301Q8—721no—
302Q8—722no—
303Q8#382812yes—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—92no—
309C2×C4—92no—
310C2×C4—182no—
311C2×C4—182no—
312C2×C4D4×C3⋊D311yescharacteristic
313C2×C4D4×C3⋊D311yescharacteristic
314C2×C4C23×C3⋊D311yescharacteristic
315C8—21no—
316C8—21no—
317C8—182no—
318C2×C3#555214yes—
319C2×C3C2×D3×Q1614yes—
320C2×C3#577014yes—
321C22C2×D4×C3⋊D311yescharacteristic
322C4—21no—
323C4—22no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—94no—
328C4—182no—
329C4—362no—
330C4—362no—
331C4Q16×C3⋊D312yes—
332C4C2×#382811yescharacteristic
333C4#6436711yescharacteristic
334C3#7969914yescomplemented
335C2#1059602311yescharacteristic
336C2C2×Q16×C3⋊D311yescharacteristic
337C2#1059624111yescharacteristic
subgroupG/H
centerC22C2×D4×C3⋊D3
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×C4×Q16C2
socle, prime radicalC22×C32C22×D4
hypercenterC4×Q16C3⋊D3
hypocenterC32#3363
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
134. #634Q16semidirect
294. C32#3363semidirect
334. C3#79699semidirect

Maximal subgroups

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

subgroupindex
#105960382
Q8×#6342
C32×C4×Q162
#105928812
#105927302
#105927202
#105925742
#105924912
#105924892
#105923802
C8×#6342
#796993