Finite Group Databasebeta

← #11786655

Subgroups — #11786655

Order 1152.

579 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5819C211yesmaximal, characteristic
2#10592936C211yesmaximal, characteristic
3C3×#5234C211yesmaximal, characteristic
4#10591143C212yesmaximal, complemented
5#10589166C212yesmaximal, complemented
6#72027—31nomaximal
7#70001—31nomaximal
8C2×C32×#41C2211yescharacteristic
9C2×#3748C2212yes—
10C3×#948—24no—
11C2×C3×#208—22no—
12C2×C3×#208C2211yescharacteristic
13C2×C3×#208C2212yes—
14C22×C3×C3⋊C8C2211yescharacteristic
15#63663C412yes—
16#63660C412yes—
17C3×#830—22no—
18#63578—24no—
19C3×#568—31no—
20#5819D311yescharacteristic
21C3×#408—31no—
22#5237—31no—
23#5237—31no—
24#5237—61no—
25#5234D311yescharacteristic
26#5124—32no—
27#5124—32no—
28#4558—32no—
29#4552—32no—
30C32×#41—22no—
31C32×#41—22no—
32C2×C32×Q8C2×C411yescharacteristic
33C2×C32×D4C2×C411yescharacteristic
34C2×C32×D4Q812yes—
35C22×C32×C4D412yes—
36C22×C32×C4C2311yescharacteristic
37#3748—22no—
38#3748C2×C414yes—
39C2×C3⋊(C3⋊C8)—22no—
40C3×#208—24no—
41C3×#208—22no—
42C3×#208—22no—
43C3×#208D414yes—
44C2×C3×C3⋊C8—24no—
45C2×C3×C3⋊C8—22no—
46C2×C3×C3⋊C8—21no—
47C2×C3×C3⋊C8—21no—
48C2×C3×C3⋊C8—22no—
49#1858—91noSylow
50C2×C3×#41—21no—
51C2×C3×#41C2×D311yescharacteristic
52C2×C3×#41C2×D311yescharacteristic
53C3×#131—64no—
54C2×C3×M16—32no—
55C2×C3×M16—31no—
56C2×C3×M16—62no—
57C22×C3×C8—31no—
58#948—24no—
59C2×#208—22no—
60C2×#208—32no—
61C2×#208—32no—
62C2×#208—62no—
63C2×#208C2×D311yescharacteristic
64C2×#208C2×D312yes—
65C22×C3⋊C8C2×D311yescharacteristic
66C3×C22⋊C8—62no—
67#836—32no—
68#836—32no—
69#836—62no—
70#833—32no—
71#833—32no—
72#833—62no—
73#830—22no—
74#822—64no—
75#822—64no—
76C23×C32C4⋊C412yes—
77C23×C32C2×Q811yescharacteristic
78C32×Q8—22no—
79C32×D4—24no—
80C32×D4—22no—
81C2×C32×C4—22no—
82C2×C32×C4—22no—
83C2×C32×C4C4⋊C412yes—
84C2×C32×C4C22×C411yescharacteristic
85C2×C32×C4C2×D412yes—
86C2×C32×C4C2×D411yescharacteristic
87C2×C32×C4#4112yes—
88C3⋊(C3⋊C8)—24no—
89C3×C3⋊C8—24no—
90C3×C3⋊C8—41no—
91C3×C3⋊C8—41no—
92#568—31no—
93#430—92no—
94#413—91no—
95#408—31no—
96#341—92no—
97C3×#41—22no—
98C3×#41—22no—
99C3×#41—22no—
100C3×#41—22no—
101C3×#41—42no—
102C3×#41—42no—
103C2×C3×Q8—21no—
104C2×C3×Q8C4×D311yescharacteristic
105C2×C3×Q8C4×D311yescharacteristic
106C2×C3×D4—21no—
107C2×C3×D4—22no—
108C2×C3×D4Dic612yes—
109C2×C3×D4Dic612yes—
110C2×C3×D4C4×D311yescharacteristic
111C2×C3×D4C4×D311yescharacteristic
112C22×C3×C4—22no—
113C22×C3×C4—21no—
114C22×C3×C4#6712yes—
115C22×C3×C4#6712yes—
116C22×C3×C4C22×D311yescharacteristic
117C22×C3×C4C22×D311yescharacteristic
118C3×M16—34no—
119C3×M16—64no—
120C3×M16—62no—
121C3×M16—62no—
122C2×C3×C8—64no—
123C2×C3×C8—62no—
124C2×C3×C8—62no—
125C2×C3×C8—61no—
126C2×C3×C8—61no—
127#208—24no—
128#208—22no—
129#208—22no—
130#208—34no—
131#208—34no—
132#208—64no—
133#208—62no—
134#208—62no—
135#208—122no—
136#208#6714yes—
137C2×C3⋊C8—24no—
138C2×C3⋊C8—22no—
139C2×C3⋊C8—22no—
140C2×C3⋊C8—21no—
141C2×C3⋊C8—21no—
142C2×C3⋊C8—62no—
143C2×C3⋊C8—62no—
144C2×C3⋊C8—122no—
145C22×C32—22no—
146C22×C32—41no—
147C22×C32—42no—
148C22×C32C2×C4⋊C411yescharacteristic
149C22×C32C22⋊Q812yes—
150C32×C4—22no—
151C32×C4C4×D412yes—
152C32×C4#12011yescharacteristic
153C32×C4#12311yescharacteristic
154C2×#41D3211yescharacteristic
155#131—64no—
156C2×M16—32no—
157C2×M16—31no—
158C2×M16—62no—
159C2×M16—92no—
160C22×C8—31no—
161#108—184no—
162#101—92no—
163#100—92no—
164C22⋊C8—62no—
165C23×C3—22no—
166C23×C3—21no—
167C23×C3Dic3⋊C412yes—
168C23×C3Dic3⋊C412yes—
169C23×C3C2×Dic611yescharacteristic
170C23×C3C2×Dic611yescharacteristic
171C3×Q8—22no—
172C3×Q8—22no—
173C3×Q8—42no—
174C3×D4—24no—
175C3×D4—24no—
176C3×D4—22no—
177C3×D4—22no—
178C3×D4—42no—
179C3×D4—44no—
180C2×C3×C4—22no—
181C2×C3×C4—22no—
182C2×C3×C4—22no—
183C2×C3×C4—22no—
184C2×C3×C4—22no—
185C2×C3×C4—22no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—22no—
189C2×C3×C4—42no—
190C2×C3×C4—42no—
191C2×C3×C4Dic3⋊C412yes—
192C2×C3×C4Dic3⋊C412yes—
193C2×C3×C4C2×C4×D311yescharacteristic
194C2×C3×C4C2×C4×D311yescharacteristic
195C2×C3×C4#23512yes—
196C2×C3×C4D3×D412yes—
197C2×C3×C4#23712yes—
198C2×C3×C4C2×#6711yescharacteristic
199C2×C3×C4C2×#6711yescharacteristic
200C2×C3×C4C2×#6712yes—
201C3×C8—64no—
202C3×C8—121no—
203C3×C8—121no—
204C3⋊C8—24no—
205C3⋊C8—41no—
206C3⋊C8—41no—
207C3⋊C8—64no—
208C3⋊C8—64no—
209C3⋊C8—124no—
210C2×C32—21no—
211C2×C32—22no—
212C2×C32—42no—
213C2×C32#38511yescharacteristic
214#41—22no—
215#41—22no—
216C2×Q8#62411yescharacteristic
217C2×D4#62411yescharacteristic
218C2×D4#62712yes—
219C22×C4#62512yes—
220C22×C4C2×D3211yescharacteristic
221M16—34no—
222M16—64no—
223M16—62no—
224M16—62no—
225M16—94no—
226M16—182no—
227C2×C8—64no—
228C2×C8—62no—
229C2×C8—62no—
230C2×C8—61no—
231C2×C8—61no—
232C2×C8—182no—
233C22×C3—22no—
234C22×C3—22no—
235C22×C3—22no—
236C22×C3—21no—
237C22×C3—42no—
238C22×C3—41no—
239C22×C3—41no—
240C22×C3—42no—
241C22×C3—42no—
242C22×C3—81no—
243C22×C3—82no—
244C22×C3#87812yes—
245C22×C3C2×Dic3⋊C411yescharacteristic
246C22×C3C2×Dic3⋊C411yescharacteristic
247C22×C3#92412yes—
248C3×C4—22no—
249C3×C4—22no—
250C3×C4—22no—
251C3×C4—21no—
252C3×C4—21no—
253C3×C4—42no—
254C3×C4#67⋊C412yes—
255C3×C4C4×#6712yes—
256C3×C4#92911yescharacteristic
257C3×C4#93511yescharacteristic
258C3×C4#93711yescharacteristic
259C3×C4#95311yescharacteristic
260C32#185811yescharacteristic, complemented, Sylow
261C23#372412yes—
262C23C2×#62711yescharacteristic
263Q8—22no—
264D4—24no—
265D4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4#372412yes—
269C2×C4#380412yes—
270C2×C4C2×#62411yescharacteristic
271C2×C4C2×#62511yescharacteristic
272C2×C4D3×#6712yes—
273C8—64no—
274C8—121no—
275C8—121no—
276C8—184no—
277C2×C3—21no—
278C2×C3—22no—
279C2×C3—21no—
280C2×C3—22no—
281C2×C3—21no—
282C2×C3—41no—
283C2×C3—42no—
284C2×C3—42no—
285C2×C3—42no—
286C2×C3—82no—
287C2×C3#494611yescharacteristic
288C2×C3#521011yescharacteristic
289C22—22no—
290C22—41no—
291C22—42no—
292C22#6396412yes—
293C22C2×#372411yescharacteristic
294C4—22no—
295C4#6395212yes—
296C4#6396111yescharacteristic
297C4#6397611yescharacteristic
298C3—21no—
299C3#7000111yescharacteristic, complemented
300C3#7202711yescharacteristic, complemented
301C2—21no—
302C2—22no—
303C2—42no—
304C2#1059169311yescharacteristic
subgroupG/H
centerC4#63976
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
Fitting, prime radicalC2×C32×#41C22
socleC2×C32#385
cosocleC22×C32×C4C23
hypercenterC2×#41D32
hypocenterC32#1858
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
4. #10591143C2semidirect
5. #10589166C2semidirect
260. C32#1858semidirect
299. C3#70001semidirect
300. C3#72027semidirect

Maximal subgroups

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

subgroupindex
C3×#58192
#105929362
C3×#52342
#105911432
#105891662
#720273
#700013