Finite Group Databasebeta

← #11791631

Subgroups — #11791631

Order 1152.

681 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
1C2×#63919C211yesmaximal, characteristic
2C4⋊C4×C3⋊Dic3C211yesmaximal, characteristic
3C2×#63572C211yesmaximal, characteristic
4C3×#4529C212yesmaximal
5#10589164C212yesmaximal
6#68800—32nomaximal
7C22×#627—21no—
8C2×C32×C4⋊C4C2211yescharacteristic
9C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
10C2×C4×C3⋊Dic3—21no—
11C2×C4×C3⋊Dic3C2211yescharacteristic
12C2×C3×C4⋊Dic3C2212yes—
13C22×C3×C3⋊C8C2212yes—
14C2×#3724—21no—
15#63919—22no—
16#63919C414yescomplemented
17#63658—21no—
18#63572—22no—
19C2×#888—32no—
20C2×Dic3⋊C8—32no—
21Dic3×C4⋊C4—32no—
22Dic3×C4⋊C4—61no—
23C3×#340—32no—
24#4551—32no—
25#4529—32no—
26#4529D312yes—
27C22×C32×C4—21no—
28C22×C32×C4C2311yescharacteristic
29C22×C3⋊Dic3—21no—
30C22×C3⋊Dic3D411yescharacteristic
31C22×C3⋊Dic3D411yescharacteristic
32C22×C3×Dic3—22no—
33C2×#627—24no—
34C2×#627—42no—
35C2×#627—42no—
36C32×C4⋊C4—21no—
37C32×C4⋊C4Q812yescomplemented
38C4⋊(C3⋊Dic3)—21no—
39C4⋊(C3⋊Dic3)Q812yescomplemented
40C4×C3⋊Dic3—42no—
41C4×C3⋊Dic3C2×C412yes—
42C4×C3⋊Dic3D412yes—
43C3×C4⋊Dic3—22no—
44C3×C4⋊Dic3C2×C414yes—
45C2×C3×C3⋊C8—22no—
46C2×C3×C3⋊C8—22no—
47C2×C3×C3⋊C8—22no—
48#3724—24no—
49#1988—91noSylow
50C22×Dic6—62no—
51C22×C3×C8—32no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—32no—
54C2×C3×C4⋊C4C2×D312yes—
55C2×C4⋊Dic3—32no—
56C2×C4⋊Dic3—61no—
57C2×C4⋊Dic3C2×D312yes—
58C2×Dic3⋊C4—62no—
59C2×C4×Dic3—32no—
60C2×C4×Dic3—61no—
61C2×C4×Dic3—62no—
62C2×C4×Dic3—62no—
63C22×C3⋊C8C2×D312yes—
64#888—38no—
65#888—64no—
66C3⋊#95—62no—
67C3⋊#95—62no—
68Dic3⋊C8—64no—
69C23×C32C2×D411yescharacteristic
70C2×C32×C4—22no—
71C2×C32×C4—41no—
72C2×C32×C4—41no—
73C2×C32×C4C22×C411yescharacteristic
74C2×C32×C4C2×D411yescharacteristic
75C2×C32×C4C2×Q811yescharacteristic
76C2×C32×C4C2×Q811yescharacteristic
77C2×C32×C4#4111yescharacteristic
78C2×C32×C4#4111yescharacteristic
79C2×C3⋊Dic3—22no—
80C2×C3⋊Dic3—21no—
81C2×C3⋊Dic3—21no—
82C2×C3⋊Dic3—41no—
83C2×C3⋊Dic3—41no—
84C2×C3⋊Dic3C22⋊C412yes—
85C2×C3⋊Dic3C22⋊C412yes—
86C2×C3⋊Dic3D812yes—
87C2×C3⋊Dic3SD1612yes—
88C2×C3×Dic3—24no—
89C2×C3×Dic3—42no—
90C2×C3×Dic3—42no—
91#627—48no—
92C3×C3⋊C8—42no—
93C2×C4⋊Q8—91no—
94C2×C4⋊C8—91no—
95C4×C4⋊C4—91no—
96#340—32no—
97#340—92no—
98C22×C3×C4—22no—
99C22×C3×C4—22no—
100C22×C3×C4—21no—
101C22×C3×C4—62no—
102C22×C3×C4C22×D312yes—
103C22×Dic3—22no—
104C22×Dic3—32no—
105C22×Dic3—32no—
106C22×Dic3—61no—
107C22×Dic3—61no—
108C22×Dic3—62no—
109C22×Dic3—62no—
110C2×Dic6—68no—
111C2×Dic6—124no—
112C2×Dic6—124no—
113C2×C3×C8—62no—
114C2×C3×C8—62no—
115C2×C3×C8—62no—
116C3×C4⋊C4—22no—
117C3×C4⋊C4—22no—
118C3×C4⋊C4—22no—
119C3×C4⋊C4—34no—
120C3×C4⋊C4—62no—
121C3×C4⋊C4Dic614yescomplemented
122C4⋊Dic3—22no—
123C4⋊Dic3—34no—
124C4⋊Dic3—62no—
125C4⋊Dic3—62no—
126C4⋊Dic3—62no—
127C4⋊Dic3C4×D314yes—
128Dic3⋊C4—68no—
129C4×Dic3—34no—
130C4×Dic3—34no—
131C4×Dic3—62no—
132C4×Dic3—62no—
133C4×Dic3—124no—
134C4×Dic3—124no—
135C2×C3⋊C8—22no—
136C2×C3⋊C8—22no—
137C2×C3⋊C8—22no—
138C22×C32C2×C22⋊C411yescharacteristic
139C22×C32C22⋊Q811yescharacteristic
140C22×C32C22⋊Q811yescharacteristic
141C22×C32C2×D811yescharacteristic
142C22×C32C2×SD1611yescharacteristic
143C22×C32#13711yescharacteristic
144C22×C32#13711yescharacteristic
145C32×C4—42no—
146C32×C4C4×Q811yescharacteristic
147C32×C4C4×Q811yescharacteristic
148C32×C4#12411yescharacteristic
149C32×C4C4⋊Q811yescharacteristic
150C3⋊Dic3—22no—
151C3⋊Dic3—42no—
152C3⋊Dic3#10214yes—
153C3×Dic3—44no—
154C22×Q8—181no—
155C22×C8—32no—
156C4⋊Q8—94no—
157C4⋊Q8—182no—
158C2×C4⋊C4—32no—
159C2×C4⋊C4—91no—
160C2×C4⋊C4—181no—
161C2×C4⋊C4D3211yescharacteristic
162C2×C42—91no—
163C2×C42—181no—
164C4⋊C8—182no—
165#95—181no—
166C23×C3—21no—
167C23×C3D3×D412yes—
168C2×C3×C4—24no—
169C2×C3×C4—24no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—42no—
177C2×C3×C4—42no—
178C2×C3×C4—42no—
179C2×C3×C4—42no—
180C2×C3×C4—64no—
181C2×C3×C4—122no—
182C2×C3×C4—122no—
183C2×C3×C4C2×Dic612yes—
184C2×C3×C4C2×C4×D312yes—
185C2×C3×C4#23512yes—
186C2×C3×C4D3×D412yes—
187C2×C3×C4#23712yes—
188C2×C3×C4D3×Q812yes—
189C2×Dic3—24no—
190C2×Dic3—34no—
191C2×Dic3—34no—
192C2×Dic3—34no—
193C2×Dic3—34no—
194C2×Dic3—42no—
195C2×Dic3—42no—
196C2×Dic3—62no—
197C2×Dic3—62no—
198C2×Dic3—62no—
199C2×Dic3—62no—
200C2×Dic3—62no—
201C2×Dic3—62no—
202C2×Dic3—64no—
203C2×Dic3—64no—
204C2×Dic3—121no—
205C2×Dic3—121no—
206C2×Dic3—122no—
207C2×Dic3—122no—
208C2×Dic3—122no—
209C2×Dic3—122no—
210Dic6—1216no—
211C3×C8—122no—
212C3⋊C8—42no—
213C2×C32#39111yescharacteristic
214C2×C32C2×#10211yescharacteristic
215C2×C32#41911yescharacteristic
216C2×C32#47411yescharacteristic
217C2×C32#47411yescharacteristic
218C2×C32#47511yescharacteristic
219C2×C32#47511yescharacteristic
220C2×Q8—184no—
221C2×Q8—362no—
222C2×Q8—362no—
223C22×C4—21no—
224C22×C4—62no—
225C22×C4—91no—
226C22×C4—91no—
227C22×C4—181no—
228C22×C4C2×D3211yescharacteristic
229C2×C8—62no—
230C2×C8—62no—
231C2×C8—62no—
232C4⋊C4—21no—
233C4⋊C4—34no—
234C4⋊C4—62no—
235C4⋊C4—92no—
236C4⋊C4—184no—
237C4⋊C4—181no—
238C4⋊C4#62712yescomplemented
239C42—92no—
240C42—92no—
241C42—362no—
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3#87812yes—
250C22×C3D3×C22⋊C412yes—
251C22×C3#89612yes—
252C22×C3D3×D812yes—
253C22×C3D3×SD1612yes—
254C22×C3#91512yes—
255C22×C3#91812yes—
256C3×C4—21no—
257C3×C4—21no—
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—44no—
261C3×C4—44no—
262C3×C4—124no—
263C3×C4C4×Dic612yes—
264C3×C4#88512yes—
265C3×C4#88712yes—
266C3×C4#88812yes—
267Dic3—38no—
268Dic3—44no—
269Dic3—64no—
270Dic3—64no—
271Dic3—122no—
272Dic3—124no—
273Dic3—124no—
274C32#198811yescharacteristic, complemented, Sylow
275C23#381111yescharacteristic
276Q8—368no—
277C2×C4—22no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—64no—
281C2×C4—92no—
282C2×C4—92no—
283C2×C4—92no—
284C2×C4—92no—
285C2×C4—122no—
286C2×C4—122no—
287C2×C4—181no—
288C2×C4—181no—
289C2×C4—182no—
290C2×C4—361no—
291C2×C4—361no—
292C2×C4#379711yescharacteristic
293C2×C4#379811yescharacteristic
294C2×C4C4×D3211yescharacteristic
295C2×C4#380211yescharacteristic
296C2×C4#380511yescharacteristic
297C2×C4C2×#62711yescharacteristic
298C8—122no—
299C2×C3—21no—
300C2×C3—21no—
301C2×C3—21no—
302C2×C3—21no—
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3#464612yes—
307C2×C3#475912yes—
308C2×C3D3×#10212yes—
309C2×C3#479112yes—
310C2×C3#480212yes—
311C2×C3#485412yes—
312C2×C3#489012yes—
313C22#6386511yescharacteristic
314C22#6392111yescharacteristic
315C22#6392611yescharacteristic
316C22#6393111yescharacteristic
317C22#6394011yescharacteristic
318C22#6394511yescharacteristic
319C22#6398211yescharacteristic
320C4—42no—
321C4—94no—
322C4—124no—
323C4—182no—
324C4—362no—
325C4#6383811yescharacteristic
326C4#6391511yescharacteristic
327C4C4×#62711yescharacteristic
328C4#6391911yescharacteristic
329C3—21no—
330C3#6880012yescomplemented
331C2#1059043011yescharacteristic
332C2#1059043111yescharacteristic
333C2#1059059311yescharacteristic
334C2#1059060211yescharacteristic
335C2#1059139611yescharacteristic
336C2#1059149411yescharacteristic
337C2#1059163011yescharacteristic
subgroupG/H
centerC23#3811
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×C4⋊C4C22
socle, prime radicalC23×C32C2×D4
cosocleC22×C32×C4C23
hypercenterC2×C4⋊C4D32
hypocenterC32#1988
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
16. #63919C4semidirect
37. C32×C4⋊C4Q8semidirect
39. C4⋊(C3⋊Dic3)Q8semidirect
121. C3×C4⋊C4Dic6semidirect
238. C4⋊C4#627semidirect
274. C32#1988semidirect
330. C3#68800semidirect

Maximal subgroups

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

subgroupindex
C2×#639192
C4⋊C4×C3⋊Dic32
C2×#635722
C3×#45292
#105891642
#688003