Finite Group Databasebeta

← #13156652

Subgroups — #13156652

Order 1728.

558 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 122 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
1C4⋊#6296C211yesmaximal, characteristic
2#11723535C211yesmaximal, characteristic
3#11722052C212yesmaximal
4C3×#63573C212yesmaximal
5C3×#63553C211yesmaximal, characteristic
6#10590889—32nomaximal
7#10590416—31nomaximal
8C2×#6296—21no—
9C2×#6293—21no—
10C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
11C3×C4⋊(C3⋊Dic3)C2212yes—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C32×C4⋊Dic3C2211yescharacteristic
14C2×C32×C3⋊C8C2212yes—
15#88815—21no—
16#88807—21no—
17#64097—31no—
18#63919—31no—
19#63918—32no—
20#63918—32no—
21#63918—61no—
22#63644—32no—
23C3×#817—32no—
24C3×C8×Dic3—32no—
25C3×#808—32no—
26#63573—32no—
27#63573—32no—
28#63573—62no—
29#63573D312yes—
30#63553D311yescharacteristic
31C2×C33×C4C2311yescharacteristic
32C2×C3⋊(C3⋊Dic3)—21no—
33C2×C3×C3⋊Dic3—22no—
34C2×C3×C3⋊Dic3D411yescharacteristic
35C2×C3×C3⋊Dic3D411yescharacteristic
36C2×C32×Dic3—21no—
37#6296—42no—
38#6293—42no—
39C32×C3⋊C8Q814yescomplemented
40#4855—92no—
41#4680—91no—
42C2×#634—61no—
43C2×#627—62no—
44C2×#627—62no—
45C2×#627—62no—
46C2×#627—61no—
47C2×C32×C8—32no—
48C32×C4⋊C4—31no—
49C4⋊(C3⋊Dic3)—31no—
50C4⋊(C3⋊Dic3)—32no—
51C4⋊(C3⋊Dic3)—61no—
52C4⋊(C3⋊Dic3)—62no—
53C4⋊(C3⋊Dic3)—121no—
54C4⋊(C3⋊Dic3)C2×D312yes—
55#3750—61no—
56C4×C3⋊Dic3C2×D311yescharacteristic
57C3×C4⋊Dic3—21no—
58C3×C4⋊Dic3—32no—
59C3×C4⋊Dic3—32no—
60C3×C4⋊Dic3—62no—
61C3×C4⋊Dic3C2×D312yes—
62C3×C4×Dic3—32no—
63C3×C4×Dic3—61no—
64C2×C3×C3⋊C8—22no—
65C2×C3×C3⋊C8C2×D312yes—
66C2×C3×C3⋊C8C2×D312yes—
67#3724—61no—
68Dic3⋊Dic3—62no—
69Dic3⋊Dic3—62no—
70Dic3⋊Dic3—62no—
71C22×C33C2×D411yescharacteristic
72C33×C4C2×Q812yes—
73C3⋊(C3⋊Dic3)—41no—
74C3×C3⋊Dic3—42no—
75C3×C3⋊Dic3SD1612yes—
76C3×C3⋊Dic3SD1612yes—
77C32×Dic3—41no—
78#888—92no—
79#888—92no—
80#888—181no—
81#869—91no—
82#869—91no—
83C3×#106—92no—
84C3×C4×C8—91no—
85#817—32no—
86#817—92no—
87#817—92no—
88#817—182no—
89C8×Dic3—32no—
90#808—32no—
91#808—92no—
92C2×C32×C4—22no—
93C2×C32×C4—21no—
94C2×C32×C4—41no—
95C2×C32×C4—61no—
96C2×C32×C4C22×D311yescharacteristic
97C2×C32×C4C22×D312yes—
98C2×C3⋊Dic3—22no—
99C2×C3⋊Dic3—61no—
100C2×C3⋊Dic3—62no—
101C2×C3⋊Dic3—62no—
102C2×C3⋊Dic3—122no—
103C2×C3⋊Dic3—122no—
104C2×C3⋊Dic3D1211yescharacteristic
105C2×C3⋊Dic3D1211yescharacteristic
106#634—122no—
107C2×C3×Dic3—22no—
108C2×C3×Dic3—22no—
109C2×C3×Dic3—32no—
110C2×C3×Dic3—32no—
111C2×C3×Dic3—61no—
112C2×C3×Dic3—61no—
113C2×C3×Dic3—62no—
114C2×C3×Dic3—62no—
115C2×C3×Dic3—122no—
116#627—124no—
117#627—124no—
118#627—124no—
119#627—122no—
120C32×C8—34no—
121C3×C3⋊C8—24no—
122C3×C3⋊C8Dic614yescomplemented
123C3×C3⋊C8Dic614yescomplemented
124#498—271noSylow
125C2×C33C4⋊Q811yescharacteristic
126C2×C33C2×SD1611yescharacteristic
127C2×C33C2×SD1611yescharacteristic
128C2×Dic6—181no—
129C2×Dic6—182no—
130C2×Dic6—182no—
131C2×Dic6—181no—
132C2×Dic6—182no—
133C2×C3×C8—32no—
134C2×C3×C8—32no—
135C2×C3×C8—62no—
136C3×C4⋊C4—32no—
137C3×C4⋊C4—61no—
138C3×C4⋊C4—92no—
139C3×C42—91no—
140C4⋊Dic3—32no—
141C4⋊Dic3—32no—
142C4⋊Dic3—62no—
143C4⋊Dic3—92no—
144C4⋊Dic3—91no—
145C4⋊Dic3—181no—
146C4⋊Dic3—182no—
147C4⋊Dic3—181no—
148C4⋊Dic3—181no—
149C4⋊Dic3—361no—
150C4⋊Dic3D3211yescharacteristic
151Dic3⋊C4—182no—
152Dic3⋊C4—182no—
153Dic3⋊C4—182no—
154C4×Dic3—32no—
155C4×Dic3—61no—
156C2×C3⋊C8D3212yes—
157C22×C32—22no—
158C22×C32—21no—
159C22×C32—41no—
160C22×C32C2×D1211yescharacteristic
161C22×C32D3×D412yes—
162C32×C4—22no—
163C32×C4—22no—
164C32×C4—22no—
165C32×C4—42no—
166C32×C4—121no—
167C32×C4C2×Dic612yes—
168C32×C4C2×Dic612yes—
169C32×C4D3×Q812yes—
170C3⋊Dic3—42no—
171C3⋊Dic3—121no—
172C3⋊Dic3—122no—
173C3⋊Dic3—122no—
174C3⋊Dic3—242no—
175C3⋊Dic3—242no—
176C3⋊Dic3#20412yes—
177C3⋊Dic3#20412yes—
178C3×Dic3—34no—
179C3×Dic3—34no—
180C3×Dic3—42no—
181C3×Dic3—42no—
182C3×Dic3—62no—
183C3×Dic3—62no—
184C3×Dic3—122no—
185C3×Dic3—122no—
186C3×Dic3—242no—
187C4⋊Q8—271no—
188C4⋊Q8—271no—
189#106—92no—
190#106—272no—
191C4×C8—91no—
192C33#49811yescharacteristic, complemented, Sylow
193C2×C3×C4—21no—
194C2×C3×C4—22no—
195C2×C3×C4—41no—
196C2×C3×C4—62no—
197C2×C3×C4—62no—
198C2×C3×C4—91no—
199C2×C3×C4—91no—
200C2×C3×C4—182no—
201C2×C3×C4C2×D3211yescharacteristic
202C2×C3×C4C2×D3212yes—
203C2×Dic3—21no—
204C2×Dic3—32no—
205C2×Dic3—32no—
206C2×Dic3—61no—
207C2×Dic3—61no—
208C2×Dic3—62no—
209C2×Dic3—62no—
210C2×Dic3—122no—
211C2×Dic3—182no—
212C2×Dic3—182no—
213C2×Dic3—181no—
214C2×Dic3—362no—
215C2×Dic3—362no—
216Dic6—364no—
217Dic6—364no—
218Dic6—362no—
219Dic6—364no—
220Dic6—362no—
221C3×C8—34no—
222C3×C8—34no—
223C3×C8—64no—
224C3⋊C8#62714yescomplemented
225C2×C32—22no—
226C2×C32—22no—
227C2×C32—21no—
228C2×C32—22no—
229C2×C32—21no—
230C2×C32—21no—
231C2×C32—41no—
232C2×C32—41no—
233C2×C32—41no—
234C2×C32#86911yescharacteristic
235C2×C32#88812yes—
236C2×C32C2×#20411yescharacteristic
237C2×C32C2×#20411yescharacteristic
238C2×C32D3×SD1612yes—
239C2×C32D3×SD1612yes—
240C2×Q8—541no—
241C2×Q8—541no—
242C2×C8—32no—
243C4⋊C4—31no—
244C4⋊C4—92no—
245C4⋊C4—271no—
246C4⋊C4—541no—
247C4⋊C4—541no—
248C42—91no—
249C22×C3—21no—
250C22×C3—22no—
251C22×C3—41no—
252C22×C3D3×D1212yes—
253C22×C3#381111yescharacteristic
254C3×C4—22no—
255C3×C4—22no—
256C3×C4—22no—
257C3×C4—42no—
258C3×C4—92no—
259C3×C4—92no—
260C3×C4—122no—
261C3×C4—122no—
262C3×C4—362no—
263C3×C4D3×Dic612yes—
264C3×C4D3×Dic612yes—
265C3×C4C2×#62712yes—
266Dic3—34no—
267Dic3—34no—
268Dic3—41no—
269Dic3—62no—
270Dic3—62no—
271Dic3—122no—
272Dic3—122no—
273Dic3—242no—
274Dic3—362no—
275Dic3—362no—
276Dic3—361no—
277Dic3—722no—
278Dic3—722no—
279C32—22no—
280C32—21no—
281C32—41no—
282C32#468011yescharacteristic, complemented
283C32#485512yescomplemented
284Q8—1082no—
285Q8—1082no—
286C2×C4—61no—
287C2×C4—91no—
288C2×C4—91no—
289C2×C4—182no—
290C2×C4—541no—
291C2×C4D3311yescharacteristic
292C8—34no—
293C2×C3—21no—
294C2×C3—22no—
295C2×C3—21no—
296C2×C3—22no—
297C2×C3—21no—
298C2×C3—22no—
299C2×C3—41no—
300C2×C3—41no—
301C2×C3—41no—
302C2×C3D3×#20412yes—
303C2×C3D3×#20412yes—
304C2×C3#6384011yescharacteristic
305C2×C3#6391812yes—
306C2×C3#6393011yescharacteristic
307C2×C3#6394011yescharacteristic
308C22#8895511yescharacteristic
309C4—92no—
310C4—92no—
311C4—121no—
312C4—362no—
313C4—1081no—
314C4D3×#62712yes—
315C3—21no—
316C3—22no—
317C3—41no—
318C3#1059041611yescharacteristic, complemented
319C3#1059088912yescomplemented
320C2#1172284211yescharacteristic
321C2#1172284311yescharacteristic
322C2#1172303611yescharacteristic
subgroupG/H
centerC22#88955
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#498
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
39. C32×C3⋊C8Q8semidirect
122. C3×C3⋊C8Dic6semidirect
123. C3×C3⋊C8Dic6semidirect
192. C33#498semidirect
224. C3⋊C8#627semidirect
282. C32#4680semidirect
283. C32#4855semidirect
318. C3#10590416semidirect
319. C3#10590889semidirect

Maximal subgroups

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

subgroupindex
C4⋊#62962
#117235352
#117220522
C3×#635732
C3×#635532
#105908893
#105904163