Finite Group Databasebeta

← #13156327

Subgroups — #13156327

Order 1728.

576 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 104 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#11723406C211yesmaximal, characteristic
3#11722042C212yesmaximal
4C3×#63575C212yesmaximal
5C3×#63572C211yesmaximal, characteristic
6#10590805—32nomaximal
7#10590432—31nomaximal
8C2×#6308C2211yescharacteristic
9C2×#6296—21no—
10C3×C4⋊(C3⋊Dic3)C2212yes—
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C32×C4⋊Dic3C2211yescharacteristic
13C2×C32×C3⋊C8C2212yes—
14#88815—21no—
15#88806—21no—
16Dic3×C3⋊Dic3—21no—
17#64096—31no—
18#63919—31no—
19#63918—32no—
20#63844—32no—
21#63844—61no—
22#63643—32no—
23C3×#818—32no—
24C3×Dic3⋊C8—32no—
25C3×#807—32no—
26#63575D312yes—
27#63572D311yescharacteristic
28#63567—32no—
29#63567—32no—
30#63567—62no—
31C2×C33×C4C2311yescharacteristic
32C2×C3⋊(C3⋊Dic3)—21no—
33#6308—21no—
34#6308Q812yescomplemented
35C2×C3×C3⋊Dic3—22no—
36C2×C3×C3⋊Dic3D411yescharacteristic
37C2×C3×C3⋊Dic3D411yescharacteristic
38C2×C32×Dic3—21no—
39#6296—42no—
40C32×C3⋊C8—22no—
41#4875—92no—
42#4850—91no—
43C2×#634—31no—
44C2×#634—32no—
45C2×#634—61no—
46C2×#634—62no—
47C2×#634—121no—
48C2×#627—62no—
49C2×#627—61no—
50C2×C32×C8—32no—
51C32×C4⋊C4—31no—
52C4⋊(C3⋊Dic3)C2×D312yes—
53#3750—61no—
54C4×C3⋊Dic3—61no—
55C4×C3⋊Dic3C2×D311yescharacteristic
56C3×C4⋊Dic3—21no—
57C3×C4⋊Dic3—32no—
58C3×C4⋊Dic3—32no—
59C3×C4⋊Dic3—62no—
60C3×C4⋊Dic3C2×D312yes—
61C3×C4×Dic3—32no—
62C3×C4×Dic3—61no—
63C2×C3×C3⋊C8—22no—
64C2×C3×C3⋊C8C2×D312yes—
65C2×C3×C3⋊C8C2×D312yes—
66#3724—62no—
67#3724—62no—
68#3724—61no—
69Dic3⋊Dic3—62no—
70Dic32—62no—
71Dic32—62no—
72C22×C33C2×D411yescharacteristic
73C33×C4C2×Q811yescharacteristic
74C33×C4#4111yescharacteristic
75C3⋊(C3⋊Dic3)—22no—
76C3×C3⋊Dic3—21no—
77C3×C3⋊Dic3—42no—
78C3×C3⋊Dic3Q1612yes—
79C32×Dic3—41no—
80#888—92no—
81#887—92no—
82#887—181no—
83#869—91no—
84C4×Dic6—91no—
85C3×#107—92no—
86C3×C4⋊C8—91no—
87#818—32no—
88#816—92no—
89#816—92no—
90#816—182no—
91Dic3⋊C8—32no—
92#810—92no—
93#807—32no—
94C2×C32×C4—22no—
95C2×C32×C4—21no—
96C2×C32×C4—41no—
97C2×C32×C4—61no—
98C2×C32×C4C22×D312yes—
99C2×C32×C4C22×D311yescharacteristic
100C2×C3⋊Dic3—22no—
101C2×C3⋊Dic3—61no—
102C2×C3⋊Dic3—62no—
103C2×C3⋊Dic3—62no—
104C2×C3⋊Dic3—122no—
105C2×C3⋊Dic3—122no—
106C2×C3⋊Dic3D1211yescharacteristic
107C2×C3⋊Dic3D1211yescharacteristic
108#634—32no—
109#634—34no—
110#634—62no—
111#634—64no—
112#634—61no—
113#634—62no—
114#634—62no—
115#634—122no—
116#634—122no—
117#634—122no—
118C2×C3×Dic3—22no—
119C2×C3×Dic3—22no—
120C2×C3×Dic3—32no—
121C2×C3×Dic3—32no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—62no—
126C2×C3×Dic3—122no—
127#627—124no—
128#627—122no—
129C32×C8—62no—
130C3×C3⋊C8—22no—
131C3×C3⋊C8—22no—
132C3×C3⋊C8—42no—
133#477—271noSylow
134C2×C33C22⋊Q811yescharacteristic
135C2×C33C2×Q1611yescharacteristic
136C2×C33AGL(1,ℤ/8)11yescharacteristic
137C2×Dic6—92no—
138C2×Dic6—91no—
139C2×Dic6—181no—
140C2×Dic6—182no—
141C2×Dic6—182no—
142C2×Dic6—181no—
143C2×Dic6—361no—
144C2×C3×C8—32no—
145C2×C3×C8—32no—
146C2×C3×C8—62no—
147C3×C4⋊C4—32no—
148C3×C4⋊C4—61no—
149C3×C4⋊C4—92no—
150C3×C42—91no—
151C4⋊Dic3—32no—
152C4⋊Dic3—32no—
153C4⋊Dic3—62no—
154C4⋊Dic3—181no—
155C4⋊Dic3D3211yescharacteristic
156Dic3⋊C4—181no—
157Dic3⋊C4—182no—
158Dic3⋊C4—182no—
159Dic3⋊C4—182no—
160C4×Dic3—32no—
161C4×Dic3—61no—
162C4×Dic3—181no—
163C4×Dic3—182no—
164C4×Dic3—182no—
165C2×C3⋊C8D3212yes—
166C22×C32—22no—
167C22×C32—21no—
168C22×C32—41no—
169C22×C32C2×D1211yescharacteristic
170C22×C32D3×D412yes—
171C32×C4—21no—
172C32×C4—22no—
173C32×C4—21no—
174C32×C4—22no—
175C32×C4—41no—
176C32×C4—41no—
177C32×C4—121no—
178C32×C4#23512yes—
179C32×C4#23711yescharacteristic
180C32×C4D3×Q812yes—
181C32×C4D3×Q811yescharacteristic
182C3⋊Dic3—21no—
183C3⋊Dic3—42no—
184C3⋊Dic3—62no—
185C3⋊Dic3—64no—
186C3⋊Dic3—64no—
187C3⋊Dic3—124no—
188C3⋊Dic3—124no—
189C3⋊Dic3Dic1212yes—
190C3×Dic3—34no—
191C3×Dic3—42no—
192C3×Dic3—42no—
193C3×Dic3—62no—
194C3×Dic3—62no—
195C3×Dic3—121no—
196C3×Dic3—122no—
197C3×Dic3—122no—
198C3×Dic3—242no—
199C4⋊Q8—271no—
200C4×Q8—271no—
201#107—92no—
202C4⋊C8—91no—
203#103—272no—
204C33#47711yescharacteristic, complemented, Sylow
205C2×C3×C4—21no—
206C2×C3×C4—22no—
207C2×C3×C4—41no—
208C2×C3×C4—62no—
209C2×C3×C4—62no—
210C2×C3×C4—91no—
211C2×C3×C4—91no—
212C2×C3×C4—182no—
213C2×C3×C4C2×D3211yescharacteristic
214C2×C3×C4C2×D3212yes—
215C2×Dic3—21no—
216C2×Dic3—32no—
217C2×Dic3—32no—
218C2×Dic3—61no—
219C2×Dic3—61no—
220C2×Dic3—62no—
221C2×Dic3—62no—
222C2×Dic3—122no—
223C2×Dic3—182no—
224C2×Dic3—182no—
225C2×Dic3—181no—
226C2×Dic3—362no—
227C2×Dic3—362no—
228Dic6—92no—
229Dic6—94no—
230Dic6—182no—
231Dic6—184no—
232Dic6—182no—
233Dic6—181no—
234Dic6—182no—
235Dic6—362no—
236Dic6—362no—
237Dic6—362no—
238Dic6—364no—
239Dic6—362no—
240C3×C8—62no—
241C3×C8—62no—
242C3×C8—122no—
243C3⋊C8—22no—
244C2×C32—22no—
245C2×C32—22no—
246C2×C32—21no—
247C2×C32—22no—
248C2×C32—21no—
249C2×C32—21no—
250C2×C32—41no—
251C2×C32—41no—
252C2×C32—41no—
253C2×C32#89612yes—
254C2×C32#89711yescharacteristic
255C2×C32C2×Dic1211yescharacteristic
256C2×C32#90811yescharacteristic
257C2×C32#91112yes—
258C2×C32D3×Q1612yes—
259C2×Q8—271no—
260C2×Q8—541no—
261C2×C8—32no—
262C4⋊C4—31no—
263C4⋊C4—92no—
264C4⋊C4—541no—
265C4⋊C4—541no—
266C42—91no—
267C42—541no—
268C22×C3—21no—
269C22×C3—22no—
270C22×C3—41no—
271C22×C3D3×D1212yes—
272C22×C3#381111yescharacteristic
273C3×C4—21no—
274C3×C4—22no—
275C3×C4—22no—
276C3×C4—21no—
277C3×C4—41no—
278C3×C4—41no—
279C3×C4—92no—
280C3×C4—122no—
281C3×C4—122no—
282C3×C4—181no—
283C3×C4—362no—
284C3×C4#379711yescharacteristic
285C3×C4#379712yes—
286C3×C4#379811yescharacteristic
287C3×C4#380412yes—
288Dic3—34no—
289Dic3—41no—
290Dic3—62no—
291Dic3—62no—
292Dic3—121no—
293Dic3—122no—
294Dic3—122no—
295Dic3—184no—
296Dic3—182no—
297Dic3—184no—
298Dic3—242no—
299Dic3—364no—
300Dic3—364no—
301C32—22no—
302C32—21no—
303C32—41no—
304C32#485011yescharacteristic, complemented
305C32#487512yescomplemented
306Q8—272no—
307Q8—541no—
308Q8—1082no—
309C2×C4—61no—
310C2×C4—91no—
311C2×C4—91no—
312C2×C4—182no—
313C2×C4—541no—
314C2×C4D3311yescharacteristic
315C8—62no—
316C2×C3—21no—
317C2×C3—22no—
318C2×C3—21no—
319C2×C3—22no—
320C2×C3—21no—
321C2×C3—22no—
322C2×C3—41no—
323C2×C3—41no—
324C2×C3—41no—
325C2×C3#6379512yes—
326C2×C3D3×Dic1212yes—
327C2×C3#6386511yescharacteristic
328C2×C3#6389512yes—
329C2×C3#6392711yescharacteristic
330C2×C3#6394411yescharacteristic
331C22#8895511yescharacteristic
332C4—92no—
333C4—121no—
334C4—181no—
335C4—362no—
336C4—542no—
337C4#8895811yescharacteristic
338C4#8895911yescharacteristic
339C3—22no—
340C3—21no—
341C3—41no—
342C3#1059043211yescharacteristic, complemented
343C3#1059080512yescomplemented
344C2#1172284911yescharacteristic
345C2#1172288011yescharacteristic
346C2#1172298211yescharacteristic
subgroupG/H
centerC22#88955
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#477
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
34. #6308Q8semidirect
204. C33#477semidirect
304. C32#4850semidirect
305. C32#4875semidirect
342. C3#10590432semidirect
343. C3#10590805semidirect

Maximal subgroups

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

subgroupindex
C4⋊#62962
#117234062
#117220422
C3×#635752
C3×#635722
#105908053
#105904323