Finite Group Databasebeta

← #13162021

Subgroups — #13162021

Order 1728.

714 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 714 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C3×#64153C211yesmaximal, characteristic
2C2×C3×#3756C212yesmaximal
3#11723678C211yesmaximal, characteristic
4C2×#88812C212yesmaximal
5#11722070C211yesmaximal, characteristic
6#10591628—31nomaximal
7#10591626—32nomaximal
8C2×C33×Q8C2211yescharacteristic
9C2×C3×#634C2212yes—
10C2×C3×#634C2211yescharacteristic
11C3×#3756—24no—
12C3×#3750—22no—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2212yes—
15#88815—22no—
16#88812—24no—
17#64153D311yescharacteristic
18C2×#3756D312yes—
19C3×#944—32no—
20C3×#944—61no—
21C2×C3×#216—32no—
22C2×C3×#216—32no—
23C2×C3×#216—62no—
24#63873—31no—
25#63866—32no—
26C2×#3719—32no—
27C2×#3719—32no—
28C2×#3718—32no—
29#63553—31no—
30Dic3×C3⋊C8—32no—
31C33×Q8—22no—
32C2×C33×C4—21no—
33C2×C33×C4C2311yescharacteristic
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3—22no—
36C2×C3×C3⋊Dic3D412yes—
37C3×#634—22no—
38C3×#634—22no—
39C3×#634—22no—
40C3×C3⋊(C3⋊C8)D414yes—
41#5183—92no—
42#5092—91no—
43C2×C32×Q8—21no—
44C2×C32×Q8—22no—
45C2×C32×Q8—41no—
46C2×C32×Q8C2×D311yescharacteristic
47C2×C32×Q8C2×D312yes—
48C2×#634C2×D311yescharacteristic
49C2×#634C2×D312yes—
50C2×C3×Dic6—32no—
51C2×C3×Dic6—32no—
52C2×C3×Dic6—32no—
53C2×C3×Dic6—62no—
54C2×C3×Dic6—61no—
55#3756—24no—
56#3750—22no—
57C4×C3⋊Dic3C2×D311yescharacteristic
58C2×C3⋊(C3⋊C8)C2×D312yes—
59C3×#216—64no—
60C3×#216—64no—
61C3×#216—68no—
62C3×Dic3⋊C4—64no—
63C3×Dic3⋊C4—64no—
64C3×C4×Dic3—32no—
65C3×C4×Dic3—61no—
66C2×C3×C3⋊C8—32no—
67C2×C3×C3⋊C8—32no—
68C2×C3×C3⋊C8—62no—
69#3724—62no—
70Dic3⋊Dic3—64no—
71#3719—64no—
72#3719—64no—
73#3718—64no—
74C22×C33C2×D411yescharacteristic
75C33×C4—41no—
76C33×C4C2×D412yes—
77C3×C3⋊Dic3—42no—
78C3×C3⋊Dic3—41no—
79C3×C3⋊Dic3Q1614yescomplemented
80C2×C3×Q16—92no—
81C3×C4⋊Q8—91no—
82#944—32no—
83#944—61no—
84#944—92no—
85C2×#216—32no—
86C2×#216—32no—
87C2×#216—62no—
88C2×#216—92no—
89C2×#216—92no—
90C2×Dic12—92no—
91#869—91no—
92C8×Dic3—92no—
93C4×C3⋊C8—91no—
94C32×Q8—22no—
95C32×Q8—24no—
96C32×Q8—22no—
97C32×Q8—22no—
98C32×Q8—42no—
99C32×Q8—42no—
100C32×Q8—44no—
101C2×C32×C4—21no—
102C2×C32×C4—22no—
103C2×C32×C4—22no—
104C2×C32×C4—22no—
105C2×C32×C4—21no—
106C2×C32×C4—41no—
107C2×C32×C4—42no—
108C2×C32×C4—42no—
109C2×C32×C4C22×D311yescharacteristic
110C2×C32×C4C22×D312yes—
111C2×C3⋊Dic3—21no—
112C2×C3⋊Dic3—22no—
113C2×C3⋊Dic3#6712yes—
114#634—22no—
115#634—22no—
116#634—22no—
117C2×C3×Dic3—34no—
118C2×C3×Dic3—62no—
119C2×C3×Dic3—62no—
120C2×C3×Dic3—62no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—62no—
123C2×C3×Dic3—122no—
124C3×Dic6—64no—
125C3×Dic6—62no—
126C3×Dic6—62no—
127C3×Dic6—62no—
128C3×Dic6—62no—
129C3×Dic6—62no—
130C3×Dic6—62no—
131C3×Dic6—122no—
132C3×Dic6—122no—
133C3⋊(C3⋊C8)#6714yes—
134C3×C3⋊C8—34no—
135C3×C3⋊C8—34no—
136C3×C3⋊C8—64no—
137#494—271noSylow
138C2×C33#12711yescharacteristic
139C2×C33C2×Q1612yes—
140C2×C3×Q8—21no—
141C2×C3×Q8—22no—
142C2×C3×Q8—41no—
143C2×C3×Q8—91no—
144C2×C3×Q8—92no—
145C2×C3×Q8D3211yescharacteristic
146C2×C3×Q8D3212yes—
147C2×Dic6—32no—
148C2×Dic6—32no—
149C2×Dic6—32no—
150C2×Dic6—62no—
151C2×Dic6—61no—
152C3×Q16—184no—
153C2×C3×C8—92no—
154C3×C4⋊C4—182no—
155C3×C42—91no—
156#216—64no—
157#216—64no—
158#216—68no—
159#216—184no—
160#216—184no—
161C4⋊Dic3—182no—
162Dic3⋊C4—64no—
163Dic3⋊C4—64no—
164Dic3⋊C4—184no—
165C4×Dic3—32no—
166C4×Dic3—61no—
167C2×C3⋊C8—32no—
168C2×C3⋊C8—32no—
169C2×C3⋊C8—62no—
170Dic12—184no—
171C22×C32—22no—
172C22×C32—21no—
173C22×C32—41no—
174C22×C32D3×D412yes—
175C22×C32C2×#6711yescharacteristic
176C32×C4—22no—
177C32×C4—22no—
178C32×C4—22no—
179C32×C4—42no—
180C32×C4—42no—
181C32×C4—44no—
182C32×C4—44no—
183C32×C4—41no—
184C32×C4—42no—
185C32×C4D3×D412yes—
186C32×C4D3×D412yes—
187C32×C4C2×#6712yes—
188C3⋊Dic3—42no—
189C3⋊Dic3—41no—
190C3⋊Dic3#21614yescomplemented
191C3×Dic3—38no—
192C3×Dic3—64no—
193C3×Dic3—124no—
194C3×Dic3—122no—
195C3×Dic3—122no—
196C3×Dic3—122no—
197C3×Dic3—122no—
198C2×Q16—92no—
199C2×Q16—272no—
200C4⋊Q8—91no—
201C4⋊Q8—271no—
202C4×C8—271no—
203C33#49411yescharacteristic, complemented, Sylow
204C3×Q8—24no—
205C3×Q8—22no—
206C3×Q8—22no—
207C3×Q8—22no—
208C3×Q8—42no—
209C3×Q8—42no—
210C3×Q8—44no—
211C3×Q8—182no—
212C3×Q8—182no—
213C3×Q8—182no—
214C2×C3×C4—22no—
215C2×C3×C4—22no—
216C2×C3×C4—21no—
217C2×C3×C4—22no—
218C2×C3×C4—21no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—41no—
222C2×C3×C4—92no—
223C2×C3×C4—182no—
224C2×C3×C4—181no—
225C2×C3×C4C2×D3211yescharacteristic
226C2×C3×C4C2×D3212yes—
227C2×Dic3—34no—
228C2×Dic3—62no—
229C2×Dic3—62no—
230C2×Dic3—62no—
231C2×Dic3—62no—
232C2×Dic3—62no—
233C2×Dic3—122no—
234Dic6—64no—
235Dic6—62no—
236Dic6—62no—
237Dic6—62no—
238Dic6—62no—
239Dic6—62no—
240Dic6—62no—
241Dic6—122no—
242Dic6—122no—
243C3×C8—94no—
244C3⋊C8—34no—
245C3⋊C8—34no—
246C3⋊C8—64no—
247C2×C32—24no—
248C2×C32—22no—
249C2×C32—22no—
250C2×C32—21no—
251C2×C32—41no—
252C2×C32—42no—
253C2×C32D3×Q1614yes—
254C2×C32#94012yes—
255C2×C32#94011yescharacteristic
256C2×C32C2×#21612yes—
257C2×Q8—91no—
258C2×Q8—92no—
259C2×Q8#111311yescharacteristic
260Q16—184no—
261Q16—544no—
262C2×C8—92no—
263C4⋊C4—182no—
264C4⋊C4—542no—
265C42—91no—
266C22×C3—21no—
267C22×C3—22no—
268C22×C3—41no—
269C22×C3D3×#6712yes—
270C22×C3#381111yescharacteristic
271C3×C4—22no—
272C3×C4—22no—
273C3×C4—22no—
274C3×C4—42no—
275C3×C4—44no—
276C3×C4—44no—
277C3×C4—42no—
278C3×C4—41no—
279C3×C4—42no—
280C3×C4—94no—
281C3×C4—362no—
282C3×C4—361no—
283C3×C4D3×#6712yes—
284C3×C4D3×#6712yes—
285C3×C4#381112yes—
286Dic3—38no—
287Dic3—64no—
288Dic3—124no—
289Dic3—122no—
290Dic3—122no—
291Dic3—122no—
292Dic3—122no—
293C32—22no—
294C32—21no—
295C32—41no—
296C32#509211yescharacteristic, complemented
297C32#518312yescomplemented
298Q8—22no—
299Q8—182no—
300Q8—182no—
301Q8—182no—
302C2×C4—21no—
303C2×C4—92no—
304C2×C4—181no—
305C2×C4—182no—
306C2×C4C2×#111311yescharacteristic
307C8—94no—
308C2×C3—22no—
309C2×C3—24no—
310C2×C3—22no—
311C2×C3—21no—
312C2×C3—42no—
313C2×C3—41no—
314C2×C3D3×#21614yes—
315C2×C3#6394412yes—
316C2×C3#6397912yes—
317C2×C3#6397911yescharacteristic
318C22#8904911yescharacteristic
319C4—41no—
320C4—94no—
321C4—361no—
322C4—362no—
323C4#8904912yes—
324C3—22no—
325C3—21no—
326C3—41no—
327C3#1059162612yescomplemented
328C3#1059162811yescharacteristic, complemented
329C2#1172371012yes—
330C2#1172372811yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×Q8#1113
hypocenterC33#494
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
79. C3×C3⋊Dic3Q16semidirect
190. C3⋊Dic3#216semidirect
203. C33#494semidirect
296. C32#5092semidirect
297. C32#5183semidirect
327. C3#10591626semidirect
328. C3#10591628semidirect

Maximal subgroups

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

subgroupindex
C3×#641532
C2×C3×#37562
#117236782
C2×#888122
#117220702
#105916283
#105916263