Finite Group Databasebeta

← #11794284

Subgroups — #11794284

Order 1152.

537 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 537 conjugacy classes of subgroups — too many to draw. Only its 126 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
1C3×#5500C211yesmaximal, characteristic
2#10592676C211yesmaximal, characteristic
3#10591131C212yesmaximal
4C3×#4530C211yesmaximal, characteristic
5#10589160C212yesmaximal
6#71901—31nomaximal
7#68979—31nomaximal
8C2×C32×C4⋊C4C2211yescharacteristic
9C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
10C2×C4×C3⋊Dic3C2211yescharacteristic
11C3×#926C2212yes—
12C3×#924—22no—
13C2×C3×#208C2212yes—
14C3×#890—22no—
15C3×#888—22no—
16#63658—22no—
17#63575—22no—
18#63573—22no—
19C3×#544—31no—
20#5500D311yescharacteristic
21#5121—32no—
22#4995—32no—
23#4977—31no—
24#4977—31no—
25#4977—61no—
26C3×#343—31no—
27#4555—32no—
28#4532—32no—
29#4530D311yescharacteristic
30C22×C32×C4D412yes—
31C22×C32×C4C2311yescharacteristic
32C22×C3⋊Dic3—21no—
33C22×C3⋊Dic3—21no—
34C2×C3×Dic6—22no—
35C32×C4⋊C4—22no—
36C4⋊(C3⋊Dic3)—22no—
37C4×C3⋊Dic3—21no—
38C4×C3⋊Dic3Q812yescomplemented
39C3×#217—22no—
40C3×C4⋊Dic3—22no—
41C3×C4⋊Dic3D412yes—
42C3×C4⋊Dic3D412yes—
43C3×Dic3⋊C4—24no—
44C3×C4×Dic3—22no—
45C3×#208—22no—
46C3×#208Q814yescomplemented
47C2×C3×C3⋊C8—22no—
48#2061—91noSylow
49C2×C3×M16—32no—
50C3×C4⋊Q8—62no—
51C3×#125—62no—
52C3×C22⋊Q8—62no—
53C3×#117—32no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4C2×D311yescharacteristic
56C2×C3×C4⋊C4C2×D311yescharacteristic
57#926C2×D312yes—
58C2×C4⋊Dic3—31no—
59C2×C4⋊Dic3—31no—
60C2×C4⋊Dic3—61no—
61#924—22no—
62C2×C4×Dic3—31no—
63C2×C4×Dic3—31no—
64C2×C4×Dic3—61no—
65C2×#208C2×D312yes—
66#890—22no—
67#888—22no—
68C3⋊#95—64no—
69C3⋊#95—62no—
70C3⋊#95—62no—
71#818—62no—
72#817—62no—
73#808—62no—
74#807—62no—
75C23×C32C2×D411yescharacteristic
76C2×C32×C4—22no—
77C2×C32×C4—42no—
78C2×C32×C4C2×D412yes—
79C2×C32×C4C2×Q812yes—
80C2×C32×C4C2×Q811yescharacteristic
81C2×C32×C4#4111yescharacteristic
82C2×C3⋊Dic3—22no—
83C2×C3⋊Dic3—41no—
84C2×C3⋊Dic3—41no—
85C2×C3⋊Dic3—42no—
86C2×C3⋊Dic3—41no—
87C2×C3×Dic3—22no—
88C2×C3×Dic3—22no—
89C3×Dic6—82no—
90C3×C3⋊C8—24no—
91#544—31no—
92#428—92no—
93#382—91no—
94#343—31no—
95#339—92no—
96C2×C3×Q8—62no—
97C22×C3×C4—22no—
98C22×C3×C4—21no—
99C22×C3×C4D1212yes—
100C22×C3×C4#6712yes—
101C22×C3×C4C22×D311yescharacteristic
102C22×C3×C4C22×D311yescharacteristic
103C22×Dic3—62no—
104C22×Dic3—61no—
105C22×Dic3—61no—
106C22×Dic3—62no—
107C22×Dic3—61no—
108C22×Dic3—61no—
109C2×Dic6—22no—
110C3×M16—34no—
111C3×M16—62no—
112C2×C3×C8—62no—
113C3×C4⋊C4—22no—
114C3×C4⋊C4—22no—
115C3×C4⋊C4—32no—
116C3×C4⋊C4—32no—
117C3×C4⋊C4—42no—
118C3×C4⋊C4—64no—
119C3×C4⋊C4—62no—
120C3×C22⋊C4—62no—
121C3×C42—62no—
122#217—22no—
123C4⋊Dic3—22no—
124C4⋊Dic3—62no—
125C4⋊Dic3—62no—
126C4⋊Dic3—122no—
127C4⋊Dic3#6712yes—
128C4⋊Dic3#6712yes—
129Dic3⋊C4—24no—
130C4×Dic3—22no—
131C4×Dic3—32no—
132C4×Dic3—32no—
133C4×Dic3—62no—
134C4×Dic3—62no—
135C4×Dic3—61no—
136C4×Dic3—61no—
137#208—22no—
138#208Dic614yescomplemented
139C2×C3⋊C8—22no—
140C22×C32—21no—
141C22×C32—21no—
142C22×C32#12011yescharacteristic
143C22×C32C22⋊Q811yescharacteristic
144C22×C32C4⋊Q811yescharacteristic
145C32×C4—42no—
146C32×C4C22⋊Q812yes—
147C32×C4C4⋊Q812yes—
148C3⋊Dic3—42no—
149C3⋊Dic3—81no—
150C3×Dic3—42no—
151C3×Dic3—42no—
152C2×M16—32no—
153C4⋊Q8—62no—
154#125—62no—
155C22⋊Q8—62no—
156#117—32no—
157C2×C4⋊C4—91no—
158C2×C4⋊C4D3211yescharacteristic
159C2×C42—91no—
160#107—182no—
161#106—182no—
162#95—182no—
163C23×C3—21no—
164C23×C3C2×D1211yescharacteristic
165C23×C3C2×#6711yescharacteristic
166C3×Q8—242no—
167C2×C3×C4—22no—
168C2×C3×C4—22no—
169C2×C3×C4—22no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—22no—
173C2×C3×C4—42no—
174C2×C3×C4—42no—
175C2×C3×C4—42no—
176C2×C3×C4—44no—
177C2×C3×C4—62no—
178C2×C3×C4—62no—
179C2×C3×C4C2×Dic612yes—
180C2×C3×C4D3×D412yes—
181C2×C3×C4#23711yescharacteristic
182C2×C3×C4D3×Q812yes—
183C2×C3×C4D3×Q811yescharacteristic
184C2×C3×C4D3×Q811yescharacteristic
185C2×C3×C4#23911yescharacteristic
186C2×C3×C4C2×#6712yes—
187C2×Dic3—22no—
188C2×Dic3—22no—
189C2×Dic3—64no—
190C2×Dic3—62no—
191C2×Dic3—62no—
192C2×Dic3—122no—
193C2×Dic3—122no—
194C2×Dic3—121no—
195C2×Dic3—121no—
196C2×Dic3—121no—
197C2×Dic3—121no—
198C2×Dic3—122no—
199C2×Dic3—124no—
200C2×Dic3—122no—
201C2×Dic3—122no—
202C2×Dic3—121no—
203C2×Dic3—121no—
204Dic6—82no—
205C3×C8—64no—
206C3⋊C8—24no—
207C2×C32—21no—
208C2×C32—21no—
209C2×C32#39511yescharacteristic
210C2×C32#45411yescharacteristic
211C2×C32#45511yescharacteristic
212C2×Q8—62no—
213C22×C4—181no—
214C22×C4—181no—
215C22×C4#62612yes—
216C22×C4C2×D3211yescharacteristic
217M16—34no—
218M16—62no—
219C2×C8—62no—
220C4⋊C4—22no—
221C4⋊C4—32no—
222C4⋊C4—32no—
223C4⋊C4—62no—
224C4⋊C4—64no—
225C4⋊C4—182no—
226C22⋊C4—62no—
227C42—62no—
228C42—92no—
229C42—181no—
230C22×C3—22no—
231C22×C3—22no—
232C22×C3—21no—
233C22×C3—21no—
234C22×C3—21no—
235C22×C3—21no—
236C22×C3—21no—
237C22×C3—21no—
238C22×C3—21no—
239C22×C3#86911yescharacteristic
240C22×C3#88211yescharacteristic
241C22×C3#89711yescharacteristic
242C22×C3#94411yescharacteristic
243C22×C3#94611yescharacteristic
244C22×C3#95311yescharacteristic
245C3×C4—22no—
246C3×C4—22no—
247C3×C4—42no—
248C3×C4—42no—
249C3×C4—82no—
250C3×C4—122no—
251C3×C4—122no—
252C3×C4#87812yes—
253C3×C4#88812yes—
254C3×C4#94412yes—
255C3×C4#94612yes—
256Dic3—42no—
257Dic3—42no—
258Dic3—124no—
259Dic3—122no—
260Dic3—122no—
261Dic3—242no—
262Dic3—241no—
263Dic3—241no—
264C32#206111yescharacteristic, complemented, Sylow
265C23C2×#62611yescharacteristic
266Q8—242no—
267C2×C4—22no—
268C2×C4—42no—
269C2×C4—62no—
270C2×C4—62no—
271C2×C4—182no—
272C2×C4—361no—
273C2×C4—361no—
274C2×C4—362no—
275C2×C4—361no—
276C2×C4D3×Dic612yes—
277C2×C4#379611yescharacteristic
278C2×C4#379711yescharacteristic
279C2×C4D3×#6712yes—
280C8—64no—
281C2×C3—22no—
282C2×C3—22no—
283C2×C3—21no—
284C2×C3—21no—
285C2×C3—21no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—21no—
289C2×C3—21no—
290C2×C3#465611yescharacteristic
291C2×C3#474611yescharacteristic
292C2×C3#475311yescharacteristic
293C2×C3#523011yescharacteristic
294C2×C3#524211yescharacteristic
295C2×C3#524511yescharacteristic
296C22—21no—
297C22—21no—
298C22#6386611yescharacteristic
299C22#6387111yescharacteristic
300C22#6397711yescharacteristic
301C4—42no—
302C4—122no—
303C4—122no—
304C4—362no—
305C4—721no—
306C4#6383912yes—
307C4#6388712yes—
308C3—21no—
309C3#6897911yescharacteristic, complemented
310C3#7190111yescharacteristic, complemented
311C2—21no—
312C2—21no—
313C2#1059134611yescharacteristic
314C2#1059151311yescharacteristic
315C2#1059164611yescharacteristic
subgroupG/H
centerC22#63977
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC2×C32×C4⋊C4C22
socleC22×C32#120
prime radicalC23×C32C2×D4
hypercenterC2×C4⋊C4D32
hypocenterC32#2061
LCSG ▷ C22×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
38. C4×C3⋊Dic3Q8semidirect
46. C3×#208Q8semidirect
138. #208Dic6semidirect
264. C32#2061semidirect
309. C3#68979semidirect
310. C3#71901semidirect

Maximal subgroups

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

subgroupindex
C3×#55002
#105926762
#105911312
C3×#45302
#105891602
#719013
#689793