Finite Group Databasebeta

← #13156650

Subgroups — #13156650

Order 1728.

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

normal subgroups only

This group has 558 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#11723535C211yesmaximal, characteristic
2C3×#63919C211yesmaximal, characteristic
3#11722070C211yesmaximal, characteristic
4#11722054C212yesmaximal
5C3×#63574C212yesmaximal
6#10590416—31nomaximal
7#10590412—32nomaximal
8C2×#6293—21no—
9C2×C3×#627—21no—
10C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C2×C3×C3⋊(C3⋊C8)C2212yes—
13C32×C4⋊Dic3C2211yescharacteristic
14C32×C4⋊Dic3C2212yes—
15#88807—21no—
16C3×#3724—21no—
17#64097—31no—
18C3×#888—32no—
19#63919D311yescharacteristic
20#63918—32no—
21#63918—61no—
22#63635—32no—
23C3×#808—32no—
24C3×#808—32no—
25#63574D312yes—
26#63573—32no—
27#63573—32no—
28#63573—62no—
29#63553—31no—
30Dic3×C3⋊C8—32no—
31C2×C33×C4C2311yescharacteristic
32C2×C3⋊(C3⋊Dic3)—21no—
33C2×C3×C3⋊Dic3D411yescharacteristic
34C2×C3×C3⋊Dic3D411yescharacteristic
35C2×C32×Dic3—21no—
36C2×C32×Dic3—22no—
37#6293—42no—
38C3×#627—42no—
39C3×C3⋊(C3⋊C8)Q814yescomplemented
40#5080—91no—
41#4855—92no—
42C2×#634—61no—
43C2×C3×Dic6—62no—
44C2×#627—21no—
45C2×#627—62no—
46C2×#627—62no—
47C32×C4⋊C4—32no—
48C32×C4⋊C4—31no—
49C4⋊(C3⋊Dic3)—31no—
50C4⋊(C3⋊Dic3)—32no—
51C4⋊(C3⋊Dic3)—61no—
52C4⋊(C3⋊Dic3)—62no—
53C4⋊(C3⋊Dic3)—121no—
54#3750—61no—
55C4×C3⋊Dic3C2×D311yescharacteristic
56C2×C3⋊(C3⋊C8)C2×D312yes—
57C3×C4⋊Dic3—21no—
58C3×C4⋊Dic3—22no—
59C3×C4⋊Dic3C2×D312yes—
60C3×C4⋊Dic3C2×D312yes—
61C3×C4⋊Dic3C2×D312yes—
62C3×Dic3⋊C4—62no—
63C3×C4×Dic3—32no—
64C3×C4×Dic3—61no—
65C2×C3×C3⋊C8—32no—
66C2×C3×C3⋊C8—32no—
67C2×C3×C3⋊C8—62no—
68#3724—21no—
69Dic3⋊Dic3—62no—
70Dic3⋊Dic3—62no—
71C22×C33C2×D411yescharacteristic
72C33×C4C2×Q812yes—
73C3⋊(C3⋊Dic3)—41no—
74C3×C3⋊Dic3SD1612yes—
75C3×C3⋊Dic3SD1612yes—
76C32×Dic3—41no—
77C32×Dic3—42no—
78C3×C4⋊Q8—91no—
79#888—32no—
80#888—92no—
81#888—181no—
82#869—91no—
83C3×#106—92no—
84#817—92no—
85C8×Dic3—92no—
86#808—32no—
87#808—32no—
88#808—92no—
89#808—92no—
90#808—182no—
91C4×C3⋊C8—91no—
92C2×C32×C4—22no—
93C2×C32×C4—21no—
94C2×C32×C4—41no—
95C2×C32×C4—61no—
96C2×C32×C4—62no—
97C2×C32×C4C22×D312yes—
98C2×C32×C4C22×D311yescharacteristic
99C2×C3⋊Dic3—61no—
100C2×C3⋊Dic3—62no—
101C2×C3⋊Dic3—62no—
102C2×C3⋊Dic3—122no—
103C2×C3⋊Dic3—122no—
104C2×C3⋊Dic3#6711yescharacteristic
105C2×C3⋊Dic3#6711yescharacteristic
106#634—122no—
107C2×C3×Dic3—22no—
108C2×C3×Dic3—22no—
109C2×C3×Dic3—22no—
110C2×C3×Dic3—22no—
111C2×C3×Dic3—32no—
112C2×C3×Dic3—32no—
113C2×C3×Dic3—42no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—61no—
116C3×Dic6—124no—
117#627—42no—
118#627—124no—
119#627—124no—
120C3⋊(C3⋊C8)Dic614yescomplemented
121C3×C3⋊C8—34no—
122C3×C3⋊C8—34no—
123C3×C3⋊C8—64no—
124#498—271noSylow
125C2×C33C4⋊Q811yescharacteristic
126C2×C33C2×SD1611yescharacteristic
127C2×C33C2×SD1611yescharacteristic
128C2×C3×Q8—181no—
129C2×Dic6—62no—
130C2×Dic6—181no—
131C2×Dic6—182no—
132C2×Dic6—182no—
133C2×C3×C8—92no—
134C3×C4⋊C4—32no—
135C3×C4⋊C4—32no—
136C3×C4⋊C4—32no—
137C3×C4⋊C4—61no—
138C3×C4⋊C4—62no—
139C3×C4⋊C4—181no—
140C3×C42—91no—
141C4⋊Dic3—91no—
142C4⋊Dic3—92no—
143C4⋊Dic3—181no—
144C4⋊Dic3—182no—
145C4⋊Dic3—181no—
146C4⋊Dic3—361no—
147C4⋊Dic3D3211yescharacteristic
148C4⋊Dic3D3212yes—
149Dic3⋊C4—62no—
150Dic3⋊C4—182no—
151Dic3⋊C4—182no—
152C4×Dic3—32no—
153C4×Dic3—61no—
154C2×C3⋊C8—32no—
155C2×C3⋊C8—32no—
156C2×C3⋊C8—62no—
157C22×C32—22no—
158C22×C32—21no—
159C22×C32—41no—
160C22×C32D3×D412yes—
161C22×C32C2×#6711yescharacteristic
162C32×C4—22no—
163C32×C4—22no—
164C32×C4—22no—
165C32×C4—42no—
166C32×C4—121no—
167C32×C4—122no—
168C32×C4C2×Dic612yes—
169C32×C4D3×Q812yes—
170C32×C4D3×Q812yes—
171C3⋊Dic3—121no—
172C3⋊Dic3—122no—
173C3⋊Dic3—122no—
174C3⋊Dic3—242no—
175C3⋊Dic3—242no—
176C3⋊Dic3#21412yes—
177C3⋊Dic3#21512yes—
178C3×Dic3—34no—
179C3×Dic3—34no—
180C3×Dic3—42no—
181C3×Dic3—42no—
182C3×Dic3—42no—
183C3×Dic3—42no—
184C3×Dic3—62no—
185C3×Dic3—62no—
186C3×Dic3—82no—
187C4⋊Q8—91no—
188C4⋊Q8—271no—
189#106—92no—
190#106—272no—
191C4×C8—271no—
192C33#49811yescharacteristic, complemented, Sylow
193C3×Q8—362no—
194C2×C3×C4—22no—
195C2×C3×C4—21no—
196C2×C3×C4—41no—
197C2×C3×C4—62no—
198C2×C3×C4—62no—
199C2×C3×C4—62no—
200C2×C3×C4—62no—
201C2×C3×C4—91no—
202C2×C3×C4—91no—
203C2×C3×C4—122no—
204C2×C3×C4C2×D3211yescharacteristic
205C2×C3×C4C2×D3212yes—
206C2×Dic3—22no—
207C2×Dic3—21no—
208C2×Dic3—32no—
209C2×Dic3—32no—
210C2×Dic3—61no—
211C2×Dic3—61no—
212C2×Dic3—182no—
213C2×Dic3—182no—
214C2×Dic3—181no—
215C2×Dic3—362no—
216C2×Dic3—362no—
217Dic6—124no—
218Dic6—364no—
219Dic6—362no—
220Dic6—364no—
221C3×C8—94no—
222C3⋊C8—34no—
223C3⋊C8—34no—
224C3⋊C8—64no—
225C2×C32—22no—
226C2×C32—21no—
227C2×C32—22no—
228C2×C32—21no—
229C2×C32—22no—
230C2×C32—21no—
231C2×C32—41no—
232C2×C32—41no—
233C2×C32—41no—
234C2×C32#88812yes—
235C2×C32D3×SD1612yes—
236C2×C32D3×SD1612yes—
237C2×C32C2×#21411yescharacteristic
238C2×C32C2×#21511yescharacteristic
239C2×C32#94411yescharacteristic
240C2×Q8—181no—
241C2×Q8—541no—
242C2×C8—92no—
243C4⋊C4—31no—
244C4⋊C4—32no—
245C4⋊C4—181no—
246C4⋊C4—271no—
247C4⋊C4—541no—
248C42—91no—
249C22×C3—21no—
250C22×C3—22no—
251C22×C3—41no—
252C22×C3D3×#6712yes—
253C22×C3#381111yescharacteristic
254C3×C4—22no—
255C3×C4—22no—
256C3×C4—22no—
257C3×C4—42no—
258C3×C4—92no—
259C3×C4—92no—
260C3×C4—122no—
261C3×C4—122no—
262C3×C4—122no—
263C3×C4—122no—
264C3×C4—242no—
265C3×C4D3×Dic612yes—
266C3×C4D3×Dic612yes—
267C3×C4#379712yes—
268Dic3—34no—
269Dic3—34no—
270Dic3—42no—
271Dic3—41no—
272Dic3—62no—
273Dic3—62no—
274Dic3—362no—
275Dic3—361no—
276Dic3—362no—
277Dic3—722no—
278Dic3—722no—
279C32—22no—
280C32—21no—
281C32—41no—
282C32#485512yescomplemented
283C32#508011yescharacteristic, complemented
284Q8—362no—
285Q8—1082no—
286C2×C4—62no—
287C2×C4—61no—
288C2×C4—91no—
289C2×C4—91no—
290C2×C4—541no—
291C2×C4D3311yescharacteristic
292C8—94no—
293C2×C3—22no—
294C2×C3—21no—
295C2×C3—21no—
296C2×C3—22no—
297C2×C3—21no—
298C2×C3—22no—
299C2×C3—41no—
300C2×C3—41no—
301C2×C3—41no—
302C2×C3#6383912yes—
303C2×C3#6384011yescharacteristic
304C2×C3D3×#21412yes—
305C2×C3#6393011yescharacteristic
306C2×C3D3×#21512yes—
307C2×C3#6394011yescharacteristic
308C22#8895211yescharacteristic
309C4—92no—
310C4—92no—
311C4—122no—
312C4—121no—
313C4—1081no—
314C4#8895712yes—
315C3—21no—
316C3—22no—
317C3—41no—
318C3#1059041212yescomplemented
319C3#1059041611yescharacteristic, complemented
320C2#1172283711yescharacteristic
321C2#1172283811yescharacteristic
322C2#1172303411yescharacteristic
subgroupG/H
centerC22#88952
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#498
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
39. C3×C3⋊(C3⋊C8)Q8semidirect
120. C3⋊(C3⋊C8)Dic6semidirect
192. C33#498semidirect
282. C32#4855semidirect
283. C32#5080semidirect
318. C3#10590412semidirect
319. C3#10590416semidirect

Maximal subgroups

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

subgroupindex
#117235352
C3×#639192
#117220702
#117220542
C3×#635742
#105904163
#105904123