Finite Group Databasebeta

← #13156525

Subgroups — #13156525

Order 1728.

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

normal subgroups only

This group has 570 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
1#11723678C211yesmaximal, characteristic
2#11723535C211yesmaximal, characteristic
3#11722040C212yesmaximal
4C3×#63567C212yesmaximal
5C3×#63553C211yesmaximal, characteristic
6#10590890—32nomaximal
7#10590567—31nomaximal
8C2×C3×#634C2212yes—
9C2×#6293—21no—
10C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C32×C4⋊Dic3C2211yescharacteristic
13C2×C32×C3⋊C8C2212yes—
14#88815—21no—
15#88815—21no—
16#88807—21no—
17#64097—31no—
18#63918—32no—
19#63918—61no—
20#63873—31no—
21#63866—32no—
22#63643—32no—
23C3×#816—32no—
24C3×C8×Dic3—32no—
25C3×#810—32no—
26#63567D312yes—
27#63566—32no—
28#63566—32no—
29#63566—62no—
30#63553D311yescharacteristic
31C2×C33×C4C2311yescharacteristic
32C2×C3⋊(C3⋊Dic3)—21no—
33C2×C3×C3⋊Dic3—22no—
34C2×C3×C3⋊Dic3D411yescharacteristic
35C2×C3×C3⋊Dic3D411yescharacteristic
36C3×#634—22no—
37C3×#634—22no—
38C2×C32×Dic3—21no—
39#6293—42no—
40C32×C3⋊C8—22no—
41#4792—92no—
42#4681—91no—
43C2×#634—61no—
44C2×#634C2×D312yes—
45C2×C3×Dic6—32no—
46C2×C3×Dic6—32no—
47C2×C3×Dic6—62no—
48C2×#627—62no—
49C2×#627—62no—
50C2×C32×C8—32no—
51C32×C4⋊C4—31no—
52C4⋊(C3⋊Dic3)—31no—
53C4⋊(C3⋊Dic3)—32no—
54C4⋊(C3⋊Dic3)—61no—
55C4⋊(C3⋊Dic3)—62no—
56C4⋊(C3⋊Dic3)—121no—
57#3750—61no—
58C4×C3⋊Dic3C2×D311yescharacteristic
59C3×C4⋊Dic3—21no—
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—61no—
67#3724—61no—
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#888—92no—
81#888—181no—
82#869—91no—
83#869—91no—
84C3×#103—92no—
85C3×C4×C8—91no—
86#835—92no—
87#816—32no—
88#816—92no—
89#816—92no—
90#816—182no—
91C8×Dic3—32no—
92#810—32no—
93C2×C32×C4—22no—
94C2×C32×C4—21no—
95C2×C32×C4—41no—
96C2×C32×C4—61no—
97C2×C32×C4C22×D311yescharacteristic
98C2×C32×C4C22×D312yes—
99C2×C3⋊Dic3—22no—
100C2×C3⋊Dic3—61no—
101C2×C3⋊Dic3—62no—
102C2×C3⋊Dic3—62no—
103C2×C3⋊Dic3—122no—
104C2×C3⋊Dic3—122no—
105C2×C3⋊Dic3D1211yescharacteristic
106C2×C3⋊Dic3D1211yescharacteristic
107#634—22no—
108#634—22no—
109#634—122no—
110C2×C3×Dic3—22no—
111C2×C3×Dic3—22no—
112C2×C3×Dic3—32no—
113C2×C3×Dic3—32no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—62no—
117C2×C3×Dic3—62no—
118C2×C3×Dic3—122no—
119C3×Dic6—64no—
120C3×Dic6—62no—
121C3×Dic6—62no—
122C3×Dic6—62no—
123C3×Dic6—62no—
124C3×Dic6—122no—
125#627—124no—
126#627—124no—
127C32×C8—62no—
128C3×C3⋊C8—22no—
129C3×C3⋊C8—22no—
130C3×C3⋊C8—42no—
131#487—271noSylow
132C2×C33#12411yescharacteristic
133C2×C33C2×SD1611yescharacteristic
134C2×C33C2×Q1611yescharacteristic
135C2×C3×Q8—92no—
136C2×Dic6—32no—
137C2×Dic6—32no—
138C2×Dic6—62no—
139C2×Dic6—181no—
140C2×Dic6—182no—
141C2×Dic6—182no—
142C2×C3×C8—32no—
143C2×C3×C8—32no—
144C2×C3×C8—62no—
145C3×C4⋊C4—32no—
146C3×C4⋊C4—61no—
147C3×C42—91no—
148C4⋊Dic3—91no—
149C4⋊Dic3—92no—
150C4⋊Dic3—181no—
151C4⋊Dic3—182no—
152C4⋊Dic3—181no—
153C4⋊Dic3—181no—
154C4⋊Dic3—181no—
155C4⋊Dic3—361no—
156C4⋊Dic3D3211yescharacteristic
157Dic3⋊C4—182no—
158Dic3⋊C4—182no—
159Dic3⋊C4—182no—
160Dic3⋊C4—182no—
161C4×Dic3—32no—
162C4×Dic3—61no—
163C2×C3⋊C8D3212yes—
164C22×C32—22no—
165C22×C32—21no—
166C22×C32—41no—
167C22×C32C2×D1211yescharacteristic
168C22×C32D3×D412yes—
169C32×C4—22no—
170C32×C4—22no—
171C32×C4—22no—
172C32×C4—42no—
173C32×C4—121no—
174C32×C4#23512yes—
175C32×C4#23512yes—
176C32×C4#23712yes—
177C3⋊Dic3—42no—
178C3⋊Dic3—121no—
179C3⋊Dic3—122no—
180C3⋊Dic3—122no—
181C3⋊Dic3—242no—
182C3⋊Dic3—242no—
183C3⋊Dic3#20412yes—
184C3⋊Dic3Dic1212yes—
185C3×Dic3—34no—
186C3×Dic3—34no—
187C3×Dic3—42no—
188C3×Dic3—42no—
189C3×Dic3—62no—
190C3×Dic3—62no—
191C3×Dic3—124no—
192C3×Dic3—122no—
193C3×Dic3—122no—
194C4⋊Q8—271no—
195C4⋊Q8—271no—
196#103—92no—
197#103—272no—
198C4×C8—91no—
199C33#48711yescharacteristic, complemented, Sylow
200C3×Q8—182no—
201C3×Q8—182no—
202C2×C3×C4—21no—
203C2×C3×C4—22no—
204C2×C3×C4—41no—
205C2×C3×C4—62no—
206C2×C3×C4—62no—
207C2×C3×C4—91no—
208C2×C3×C4—91no—
209C2×C3×C4—182no—
210C2×C3×C4C2×D3211yescharacteristic
211C2×C3×C4C2×D3212yes—
212C2×Dic3—21no—
213C2×Dic3—32no—
214C2×Dic3—32no—
215C2×Dic3—61no—
216C2×Dic3—61no—
217C2×Dic3—62no—
218C2×Dic3—62no—
219C2×Dic3—122no—
220C2×Dic3—182no—
221C2×Dic3—182no—
222C2×Dic3—181no—
223C2×Dic3—362no—
224C2×Dic3—362no—
225Dic6—64no—
226Dic6—62no—
227Dic6—62no—
228Dic6—62no—
229Dic6—62no—
230Dic6—122no—
231Dic6—364no—
232Dic6—362no—
233Dic6—364no—
234C3×C8—62no—
235C3×C8—62no—
236C3×C8—122no—
237C3⋊C8—22no—
238C2×C32—22no—
239C2×C32—22no—
240C2×C32—22no—
241C2×C32—21no—
242C2×C32—21no—
243C2×C32—21no—
244C2×C32—41no—
245C2×C32—41no—
246C2×C32—41no—
247C2×C32#87511yescharacteristic
248C2×C32#88512yes—
249C2×C32C2×#20411yescharacteristic
250C2×C32C2×Dic1211yescharacteristic
251C2×C32D3×SD1612yes—
252C2×C32D3×Q1612yes—
253C2×Q8—92no—
254C2×Q8—541no—
255C2×C8—32no—
256C4⋊C4—31no—
257C4⋊C4—271no—
258C4⋊C4—541no—
259C4⋊C4—541no—
260C4⋊C4—541no—
261C42—91no—
262C22×C3—21no—
263C22×C3—22no—
264C22×C3—41no—
265C22×C3D3×D1212yes—
266C22×C3#381111yescharacteristic
267C3×C4—22no—
268C3×C4—22no—
269C3×C4—22no—
270C3×C4—42no—
271C3×C4—92no—
272C3×C4—92no—
273C3×C4—122no—
274C3×C4—122no—
275C3×C4—362no—
276C3×C4#379512yes—
277C3×C4#379812yes—
278C3×C4#380412yes—
279Dic3—34no—
280Dic3—34no—
281Dic3—41no—
282Dic3—62no—
283Dic3—62no—
284Dic3—124no—
285Dic3—122no—
286Dic3—122no—
287Dic3—362no—
288Dic3—362no—
289Dic3—361no—
290Dic3—722no—
291Dic3—722no—
292C32—21no—
293C32—22no—
294C32—41no—
295C32#468111yescharacteristic, complemented
296C32#479212yescomplemented
297Q8—182no—
298Q8—182no—
299Q8—1082no—
300C2×C4—61no—
301C2×C4—91no—
302C2×C4—91no—
303C2×C4—182no—
304C2×C4—541no—
305C2×C4D3311yescharacteristic
306C8—62no—
307C2×C3—22no—
308C2×C3—21no—
309C2×C3—21no—
310C2×C3—22no—
311C2×C3—21no—
312C2×C3—22no—
313C2×C3—41no—
314C2×C3—41no—
315C2×C3—41no—
316C2×C3D3×#20412yes—
317C2×C3D3×Dic1212yes—
318C2×C3#6385212yes—
319C2×C3#6391311yescharacteristic
320C2×C3#6393011yescharacteristic
321C2×C3#6394411yescharacteristic
322C22#8895511yescharacteristic
323C4—92no—
324C4—92no—
325C4—121no—
326C4—362no—
327C4—1081no—
328C4#8896012yes—
329C3—21no—
330C3—22no—
331C3—41no—
332C3#1059056711yescharacteristic, complemented
333C3#1059089012yescomplemented
334C2#1172284211yescharacteristic
335C2#1172284911yescharacteristic
336C2#1172301011yescharacteristic
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
199. C33#487semidirect
295. C32#4681semidirect
296. C32#4792semidirect
332. C3#10590567semidirect
333. C3#10590890semidirect

Maximal subgroups

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

subgroupindex
#117236782
#117235352
#117220402
C3×#635672
C3×#635532
#105908903
#105905673