Finite Group Databasebeta

← #13156523

Subgroups — #13156523

Order 1728.

582 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 106 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#11723455C211yesmaximal, characteristic
3#11722042C212yesmaximal
4C3×#63566C212yesmaximal
5C3×#63553C211yesmaximal, characteristic
6#10591538—31nomaximal
7#10590890—32nomaximal
8C2×#6308C2211yescharacteristic
9C2×C32×Dic6C2211yescharacteristic
10C2×#6296—21no—
11C3×C4⋊(C3⋊Dic3)C2212yes—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C2×C32×C3⋊C8C2212yes—
14#88815—21no—
15#88807—21no—
16#88807—21no—
17#64153—31no—
18#63919—31no—
19#63918—32no—
20#63866—32no—
21#63866—61no—
22#63643—32no—
23C3×#835—32no—
24C3×#816—32no—
25C3×C8×Dic3—32no—
26#63567—32no—
27#63567—32no—
28#63567—62no—
29#63566D312yes—
30#63553D311yescharacteristic
31C2×C33×C4C2311yescharacteristic
32C2×C3⋊(C3⋊Dic3)—21no—
33#6308—22no—
34C2×C3×C3⋊Dic3—22no—
35C2×C3×C3⋊Dic3D411yescharacteristic
36C2×C3×C3⋊Dic3D411yescharacteristic
37C2×C32×Dic3—21no—
38C32×Dic6—22no—
39#6296—42no—
40C32×C3⋊C8—22no—
41#4792—92no—
42#4681—91no—
43C2×C32×Q8—31no—
44C2×#634—31no—
45C2×#634—32no—
46C2×#634—62no—
47C2×#634—61no—
48C2×#634—121no—
49C2×C3×Dic6—21no—
50C2×C3×Dic6C2×D312yes—
51C2×#627—62no—
52C2×#627—61no—
53C2×C32×C8—32no—
54C4⋊(C3⋊Dic3)C2×D312yes—
55#3750—61no—
56#3750—61no—
57C4×C3⋊Dic3C2×D311yescharacteristic
58C3×C4⋊Dic3—32no—
59C3×C4⋊Dic3—32no—
60C3×C4⋊Dic3—62no—
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—61no—
67Dic3⋊Dic3—62no—
68Dic3⋊Dic3—62no—
69Dic3⋊Dic3—62no—
70Dic3⋊Dic3—62no—
71Dic3⋊Dic3—62no—
72C22×C33C2×D411yescharacteristic
73C33×C4#4112yes—
74C3⋊(C3⋊Dic3)—41no—
75C3×C3⋊Dic3—42no—
76C3×C3⋊Dic3SD1612yes—
77C3×C3⋊Dic3Q1612yes—
78C32×Dic3—41no—
79#944—92no—
80#944—181no—
81#888—92no—
82#869—91no—
83#869—91no—
84C3×#103—92no—
85C3×C4×C8—91no—
86#835—32no—
87#816—32no—
88#816—92no—
89#816—92no—
90#816—182no—
91C8×Dic3—32no—
92#810—92no—
93C32×Q8—62no—
94C2×C32×C4—22no—
95C2×C32×C4—21no—
96C2×C32×C4—41no—
97C2×C32×C4—61no—
98C2×C32×C4C22×D311yescharacteristic
99C2×C32×C4C22×D312yes—
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—62no—
109#634—64no—
110#634—62no—
111#634—62no—
112#634—122no—
113#634—122no—
114#634—124no—
115C2×C3×Dic3—22no—
116C2×C3×Dic3—22no—
117C2×C3×Dic3—32no—
118C2×C3×Dic3—32no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—62no—
123C2×C3×Dic3—122no—
124C3×Dic6—22no—
125C3×Dic6—22no—
126C3×Dic6—42no—
127#627—124no—
128#627—122no—
129C32×C8—62no—
130C3×C3⋊C8—22no—
131C3×C3⋊C8—22no—
132C3×C3⋊C8—42no—
133#487—271noSylow
134C2×C33#12411yescharacteristic
135C2×C33C2×SD1611yescharacteristic
136C2×C33C2×Q1611yescharacteristic
137C2×C3×Q8—32no—
138C2×C3×Q8—61no—
139C2×Dic6—92no—
140C2×Dic6—91no—
141C2×Dic6—182no—
142C2×Dic6—181no—
143C2×Dic6—182no—
144C2×Dic6—181no—
145C2×Dic6—361no—
146C2×Dic6D3211yescharacteristic
147C2×C3×C8—32no—
148C2×C3×C8—32no—
149C2×C3×C8—62no—
150C3×C4⋊C4—92no—
151C3×C42—91no—
152C4⋊Dic3—32no—
153C4⋊Dic3—32no—
154C4⋊Dic3—62no—
155C4⋊Dic3—181no—
156C4⋊Dic3—181no—
157C4⋊Dic3—181no—
158Dic3⋊C4—182no—
159Dic3⋊C4—182no—
160Dic3⋊C4—182no—
161Dic3⋊C4—182no—
162Dic3⋊C4—182no—
163C4×Dic3—32no—
164C4×Dic3—61no—
165C2×C3⋊C8D3212yes—
166C22×C32—22no—
167C22×C32—21no—
168C22×C32—41no—
169C22×C32C2×D1211yescharacteristic
170C22×C32D3×D412yes—
171C32×C4—22no—
172C32×C4—22no—
173C32×C4—22no—
174C32×C4—42no—
175C32×C4—121no—
176C32×C4#23512yes—
177C32×C4#23512yes—
178C32×C4#23712yes—
179C3⋊Dic3—42no—
180C3⋊Dic3—124no—
181C3⋊Dic3—122no—
182C3⋊Dic3—124no—
183C3⋊Dic3—121no—
184C3⋊Dic3—122no—
185C3⋊Dic3#20412yes—
186C3⋊Dic3Dic1212yes—
187C3×Dic3—34no—
188C3×Dic3—34no—
189C3×Dic3—42no—
190C3×Dic3—42no—
191C3×Dic3—62no—
192C3×Dic3—62no—
193C3×Dic3—122no—
194C3×Dic3—122no—
195C3×Dic3—242no—
196C4⋊Q8—271no—
197C4⋊Q8—271no—
198#103—92no—
199#103—272no—
200C4×C8—91no—
201C33#48711yescharacteristic, complemented, Sylow
202C3×Q8—62no—
203C3×Q8—62no—
204C3×Q8—122no—
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×D3212yes—
214C2×C3×C4C2×D3211yescharacteristic
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—22no—
229Dic6—184no—
230Dic6—182no—
231Dic6—182no—
232Dic6—182no—
233Dic6—362no—
234Dic6—362no—
235Dic6—364no—
236Dic6—364no—
237Dic6—362no—
238C3×C8—62no—
239C3×C8—62no—
240C3×C8—122no—
241C3⋊C8—22no—
242C2×C32—22no—
243C2×C32—22no—
244C2×C32—22no—
245C2×C32—21no—
246C2×C32—21no—
247C2×C32—21no—
248C2×C32—41no—
249C2×C32—41no—
250C2×C32—41no—
251C2×C32#87511yescharacteristic
252C2×C32#88512yes—
253C2×C32C2×#20411yescharacteristic
254C2×C32C2×Dic1211yescharacteristic
255C2×C32D3×SD1612yes—
256C2×C32D3×Q1612yes—
257C2×Q8—31no—
258C2×Q8—271no—
259C2×Q8—541no—
260C2×C8—32no—
261C4⋊C4—92no—
262C4⋊C4—541no—
263C4⋊C4—541no—
264C4⋊C4—541no—
265C42—91no—
266C22×C3—21no—
267C22×C3—22no—
268C22×C3—41no—
269C22×C3D3×D1212yes—
270C22×C3#381111yescharacteristic
271C3×C4—22no—
272C3×C4—22no—
273C3×C4—22no—
274C3×C4—42no—
275C3×C4—92no—
276C3×C4—92no—
277C3×C4—122no—
278C3×C4—122no—
279C3×C4—362no—
280C3×C4#379812yes—
281C3×C4#380412yes—
282C3×C4#380412yes—
283Dic3—34no—
284Dic3—34no—
285Dic3—41no—
286Dic3—62no—
287Dic3—62no—
288Dic3—122no—
289Dic3—122no—
290Dic3—242no—
291Dic3—364no—
292Dic3—362no—
293Dic3—364no—
294Dic3—362no—
295Dic3—361no—
296C32—21no—
297C32—22no—
298C32—41no—
299C32#468111yescharacteristic, complemented
300C32#479212yescomplemented
301Q8—62no—
302Q8—542no—
303Q8—1082no—
304C2×C4—61no—
305C2×C4—91no—
306C2×C4—91no—
307C2×C4—182no—
308C2×C4—541no—
309C2×C4D3311yescharacteristic
310C8—62no—
311C2×C3—21no—
312C2×C3—22no—
313C2×C3—21no—
314C2×C3—22no—
315C2×C3—21no—
316C2×C3—22no—
317C2×C3—41no—
318C2×C3—41no—
319C2×C3—41no—
320C2×C3D3×#20412yes—
321C2×C3D3×Dic1212yes—
322C2×C3#6385212yes—
323C2×C3#6394011yescharacteristic
324C2×C3#6394411yescharacteristic
325C2×C3#6395311yescharacteristic
326C22#8895511yescharacteristic
327C4—92no—
328C4—92no—
329C4—121no—
330C4—362no—
331C4—1081no—
332C4#8895912yes—
333C3—21no—
334C3—22no—
335C3—41no—
336C3#1059089012yescomplemented
337C3#1059153811yescharacteristic, complemented
338C2#1172284311yescharacteristic
339C2#1172284911yescharacteristic
340C2#1172300811yescharacteristic
subgroupG/H
centerC22#88955
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#487
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
201. C33#487semidirect
299. C32#4681semidirect
300. C32#4792semidirect
336. C3#10590890semidirect
337. C3#10591538semidirect

Maximal subgroups

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

subgroupindex
C4⋊#62962
#117234552
#117220422
C3×#635662
C3×#635532
#105915383
#105908903