Finite Group Databasebeta

← #13156643

Subgroups — #13156643

Order 1728.

762 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 114 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, complemented
2#11723440C211yesmaximal, characteristic
3C2×#88797C212yesmaximal
4C2×C3×#3719C212yesmaximal, complemented
5C3×#63555C211yesmaximal, characteristic, complemented
6#10591629—31nomaximal
7#10590899—32nomaximal
8C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
9C2×C3×#634C2212yes—
10C2×C32×Dic6C2211yescharacteristic
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C2×C32×C3⋊C8C2212yes—
13#88815—22no—
14#88804—22no—
15#88797—24no—
16C3×#3719—24no—
17#64156—31no—
18C2×#3744—32no—
19C2×C3×#216—32no—
20C2×C3×Dic12—32no—
21#63873—31no—
22#63866—32no—
23#63864—32no—
24#63864—61no—
25C2×#3719D312yescomplemented
26C2×#3717—32no—
27C2×#3717—32no—
28C2×#3717—62no—
29C3×#815—32no—
30#63555D311yescharacteristic, complemented
31C22×C3⋊(C3⋊D3)—21no—
32C2×C33×C4C2311yescharacteristic
33C4⋊(C3⋊(C3⋊D3))—22no—
34C2×C3×C3⋊Dic3—22no—
35C2×C3×C3⋊Dic3D412yes—
36C3×#634—22no—
37C3×#634—22no—
38C2×C32×Dic3—21no—
39C32×Dic6—22no—
40C32×C3⋊C8D414yes—
41#5190—92no—
42#4715—91no—
43C2×C32×Q8—31no—
44C2×C4⋊(C3⋊D3)—31no—
45C2×C4⋊(C3⋊D3)—32no—
46C2×C4⋊(C3⋊D3)—62no—
47C2×C4⋊(C3⋊D3)—61no—
48C2×C4⋊(C3⋊D3)—121no—
49C2×#634C2×D312yes—
50C2×C3×Dic6—21no—
51C2×C3×Dic6—32no—
52C2×C3×Dic6—32no—
53C2×C3×Dic6—62no—
54C2×C3×Dic6C2×D312yes—
55C2×C32×C8—32no—
56#3752—62no—
57C4×C3⋊Dic3C2×D311yescharacteristic
58#3744—64no—
59C3×#216—64no—
60C3×C4×Dic3—32no—
61C3×C4×Dic3—61no—
62C2×C3×C3⋊C8—22no—
63C2×C3×C3⋊C8C2×D312yes—
64C2×C3×C3⋊C8C2×D312yes—
65C3×Dic12—64no—
66#3724—62no—
67Dic3⋊Dic3—64no—
68#3722—64no—
69#3722—64no—
70#3719—24no—
71#3717—68no—
72#3717—64no—
73#3717—64no—
74C22×C33C2×D411yescharacteristic
75C2×C3⋊(C3⋊D3)—41no—
76C2×C3⋊(C3⋊D3)—42no—
77C33×C4C2×D412yes—
78C3×C3⋊Dic3—22no—
79C3×C3⋊Dic3—42no—
80C32×Dic3—41no—
81C2×C3×Q16—92no—
82#947—92no—
83#947—181no—
84#944—92no—
85C2×#216—32no—
86C2×#215—92no—
87C2×Dic12—32no—
88C2×#204—92no—
89C2×#204—92no—
90C2×#204—182no—
91#875—91no—
92#869—91no—
93C3×#97—91no—
94#815—32no—
95C22×C3⋊D3—61no—
96C22×C3⋊D3—62no—
97C22×C3⋊D3—62no—
98C22×C3⋊D3—122no—
99C22×C3⋊D3—122no—
100C32×Q8—62no—
101C2×C32×C4—22no—
102C2×C32×C4—21no—
103C2×C32×C4—41no—
104C2×C32×C4—61no—
105C2×C32×C4C22×D311yescharacteristic
106C2×C32×C4C22×D312yes—
107C2×C3⋊Dic3—22no—
108C2×C3⋊Dic3D1212yes—
109C4⋊(C3⋊D3)—62no—
110C4⋊(C3⋊D3)—64no—
111C4⋊(C3⋊D3)—62no—
112C4⋊(C3⋊D3)—62no—
113C4⋊(C3⋊D3)—122no—
114C4⋊(C3⋊D3)—122no—
115C4⋊(C3⋊D3)—124no—
116#634—22no—
117#634—22no—
118C2×C3×Dic3—22no—
119C2×C3×Dic3—22no—
120C2×C3×Dic3—34no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—62no—
123C2×C3×Dic3—62no—
124C2×C3×Dic3—122no—
125C3×Dic6—22no—
126C3×Dic6—22no—
127C3×Dic6—42no—
128C3×Dic6—64no—
129C3×Dic6—62no—
130C3×Dic6—62no—
131C3×Dic6—62no—
132C3×Dic6—62no—
133C3×Dic6—122no—
134C32×C8—34no—
135C3×C3⋊C8—24no—
136C3×C3⋊C8D1214yes—
137C3×C3⋊C8#6714yes—
138#497—271noSylow
139C2×C33#12711yescharacteristic
140C2×C33#13712yes—
141C3⋊(C3⋊D3)—81no—
142C2×C3×Q8—32no—
143C2×C3×Q8—61no—
144C2×C3×Q8—92no—
145C2×D12—91no—
146C2×D12—92no—
147C2×D12—182no—
148C2×D12—181no—
149C2×D12—361no—
150C2×Dic6—32no—
151C2×Dic6—32no—
152C2×Dic6—62no—
153C2×Dic6D3211yescharacteristic
154C3×Q16—184no—
155C2×C3×C8—32no—
156C2×C3×C8—32no—
157C2×C3×C8—62no—
158C3×C42—91no—
159#216—64no—
160#215—184no—
161(C2×D3)⋊C4—182no—
162(C2×D3)⋊C4—184no—
163(C2×D3)⋊C4—184no—
164C4⋊Dic3—182no—
165Dic3⋊C4—184no—
166C4×Dic3—32no—
167C4×Dic3—61no—
168C2×C3⋊C8D3212yes—
169Dic12—64no—
170#204—184no—
171#204—184no—
172#204—188no—
173C22×C32—22no—
174C22×C32—21no—
175C22×C32—41no—
176C22×C32C2×D1211yescharacteristic
177C22×C32D3×D412yes—
178C2×C3⋊D3—124no—
179C2×C3⋊D3—122no—
180C2×C3⋊D3—124no—
181C2×C3⋊D3—121no—
182C2×C3⋊D3—122no—
183C2×C3⋊D3—122no—
184C2×C3⋊D3—124no—
185C2×C3⋊D3—124no—
186C2×C3⋊D3—244no—
187C2×C3⋊D3—244no—
188C32×C4—22no—
189C32×C4—22no—
190C32×C4—22no—
191C32×C4—42no—
192C32×C4—121no—
193C32×C4C2×D1212yes—
194C32×C4D3×D412yes—
195C32×C4C2×#6712yes—
196C3⋊Dic3—22no—
197C3⋊Dic3—42no—
198C3×Dic3—42no—
199C3×Dic3—42no—
200C3×Dic3—64no—
201C3×Dic3—122no—
202C3×Dic3—124no—
203C3×Dic3—122no—
204C3×Dic3—122no—
205C2×Q16—92no—
206C2×SD16—272no—
207C4⋊Q8—271no—
208#124—271no—
209#97—91no—
210C33#49711yescharacteristic, complemented, Sylow
211C22×D3—182no—
212C22×D3—182no—
213C22×D3—181no—
214C22×D3—362no—
215C22×D3—362no—
216C3×Q8—62no—
217C3×Q8—62no—
218C3×Q8—122no—
219C3×Q8—182no—
220C3×Q8—182no—
221C2×C3×C4—21no—
222C2×C3×C4—22no—
223C2×C3×C4—41no—
224C2×C3×C4—62no—
225C2×C3×C4—62no—
226C2×C3×C4—92no—
227C2×C3×C4—182no—
228C2×C3×C4C2×D3212yes—
229C2×C3×C4C2×D3211yescharacteristic
230C2×Dic3—21no—
231C2×Dic3—34no—
232C2×Dic3—62no—
233C2×Dic3—62no—
234C2×Dic3—62no—
235C2×Dic3—122no—
236D12—184no—
237D12—182no—
238D12—182no—
239D12—182no—
240D12—362no—
241D12—362no—
242D12—364no—
243Dic6—22no—
244Dic6—64no—
245Dic6—62no—
246Dic6—62no—
247Dic6—62no—
248Dic6—62no—
249Dic6—122no—
250C3×C8—34no—
251C3×C8—34no—
252C3×C8—64no—
253C3⋊C8#62614yes—
254C2×C32—24no—
255C2×C32—22no—
256C2×C32—22no—
257C2×C32—21no—
258C2×C32—41no—
259C2×C32—42no—
260C2×C32#87411yescharacteristic
261C2×C32#90912yes—
262C2×C32#91814yes—
263C2×C32#94012yes—
264C3⋊D3—242no—
265C3⋊D3—244no—
266C3⋊D3—244no—
267C3⋊D3—242no—
268C3⋊D3—241no—
269C2×Q8—31no—
270C2×Q8—92no—
271C2×D4—271no—
272Q16—184no—
273SD16—544no—
274C2×C8—32no—
275C4⋊C4—542no—
276C22⋊C4—542no—
277C42—91no—
278C22×C3—21no—
279C22×C3—22no—
280C22×C3—41no—
281C22×C3D3×D1212yes—
282C22×C3#381111yescharacteristic
283C2×D3—364no—
284C2×D3—362no—
285C2×D3—364no—
286C2×D3—364no—
287C2×D3—364no—
288C2×D3—362no—
289C2×D3—362no—
290C2×D3—361no—
291C2×D3—724no—
292C2×D3—724no—
293C3×C4—22no—
294C3×C4—22no—
295C3×C4—22no—
296C3×C4—42no—
297C3×C4—122no—
298C3×C4—122no—
299C3×C4—182no—
300C3×C4—362no—
301C3×C4D3×D1212yes—
302C3×C4C2×#62612yes—
303C3×C4D3×#6712yes—
304Dic3—41no—
305Dic3—64no—
306Dic3—122no—
307Dic3—124no—
308Dic3—122no—
309Dic3—122no—
310C32—21no—
311C32—22no—
312C32—41no—
313C32#471511yescharacteristic, complemented
314C32#519012yescomplemented
315C23—541no—
316Q8—62no—
317Q8—182no—
318Q8—182no—
319D4—542no—
320C2×C4—61no—
321C2×C4—92no—
322C2×C4—182no—
323C2×C4D3311yescharacteristic
324C8—34no—
325C2×C3—22no—
326C2×C3—21no—
327C2×C3—22no—
328C2×C3—24no—
329C2×C3—41no—
330C2×C3—42no—
331C2×C3#6380114yes—
332C2×C3#6391112yes—
333C2×C3#6394512yes—
334C2×C3#6397911yescharacteristic
335D3—722no—
336D3—724no—
337D3—724no—
338D3—721no—
339D3—722no—
340C22—1081no—
341C22—1082no—
342C22#8895511yescharacteristic
343C4—121no—
344C4—182no—
345C4—362no—
346C4D3×#62612yes—
347C3—22no—
348C3—21no—
349C3—41no—
350C3#1059089912yescomplemented
351C3#1059162911yescharacteristic, complemented
352C2—2161no—
353C2#1172289712yes—
354C2#1172303011yescharacteristic
subgroupG/H
centerC22#88955
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypocenterC33#497
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
1. #11723678C2semidirect
4. C2×C3×#3719C2semidirect
5. C3×#63555C2semidirect
25. C2×#3719D3semidirect
30. #63555D3semidirect
210. C33#497semidirect
313. C32#4715semidirect
314. C32#5190semidirect
350. C3#10590899semidirect
351. C3#10591629semidirect

Maximal subgroups

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

subgroupindex
#117236782
#117234402
C2×#887972
C2×C3×#37192
C3×#635552
#105916293
#105908993