Finite Group Databasebeta

← #13178711

Subgroups — #13178711

Order 1728.

1398 conjugacy classes of subgroups; only the 327 normal subgroups are shown.

normal subgroups only

This group has 1398 conjugacy classes of subgroups — too many to draw. Only its 327 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
1C2×C3×Q8×C3⋊D3C211yesmaximal, characteristic
2C3×Q8×C3⋊Dic3C212yesmaximal, complemented
3#11723673C213yesmaximal
4#11723668C216yesmaximal, complemented
5#11723667C213yesmaximal
6#10595347—32nomaximal
7#10595339—31nomaximal
8C2×C33×Q8C2211yescharacteristic
9C3×Q8×C3⋊D3C418yescomplemented
10C2×C3×C4×C3⋊D3C2213yes—
11C2×C3×#634C2213yes—
12C3×C4⋊(C3⋊Dic3)C2216yes—
13C3×C4×C3⋊Dic3C2216yes—
14#88815C2216yes—
15#88814C2213yes—
16#88814C2211yescharacteristic
17#88813C2216yes—
18C2×Q8×C3⋊D3D311yescharacteristic
19C2×C3×D3×Q8—32no—
20C2×C3×D3×Q8—61no—
21Q8×C3⋊Dic3D312yescomplemented
22C3×Q8×Dic3—32no—
23C3×Q8×Dic3—32no—
24C3×Q8×Dic3—62no—
25#63874—33no—
26#63855—36no—
27#63849—33no—
28#63844—36no—
29#63843—36no—
30Dic3×Dic6—36no—
31#63841—36no—
32C22×C3×C3⋊D3C2311yescharacteristic
33C33×Q8C2×C414yes—
34C2×C33×C4C2313yes—
35C2×C3×C3⋊Dic3C2312yes—
36C2×C3×C3⋊Dic3C2316yes—
37C2×C3×C3⋊Dic3C2313yes—
38C3×C4×C3⋊D3C2×C4112yes—
39C3×#634C2×C4112yes—
40#5812—91no—
41#5572—92no—
42C2×C32×Q8—22no—
43C2×C32×Q8—21no—
44C2×C32×Q8—41no—
45C2×C32×Q8C2×D312yes—
46C2×C32×Q8C2×D311yescharacteristic
47Q8×C3⋊D3Dic318yescomplemented
48C2×C4×C3⋊D3C2×D313yes—
49C2×#634C2×D313yes—
50C3×D3×Q8—316no—
51C3×D3×Q8—68no—
52C2×C3×C4×D3—36no—
53C2×C3×C4×D3—63no—
54C2×C3×Dic6—36no—
55C2×C3×Dic6—63no—
56C4⋊(C3⋊Dic3)C2×D316yes—
57C4×C3⋊Dic3C2×D316yes—
58C3×C4⋊Dic3—36no—
59C3×C4⋊Dic3—36no—
60C3×C4⋊Dic3—66no—
61C3×C4×Dic3—36no—
62C3×C4×Dic3—36no—
63C3×C4×Dic3—66no—
64#3724—36no—
65Dic3⋊Dic3—36no—
66Dic3⋊Dic3—36no—
67#3722—33no—
68#3722—31no—
69(C2×D3)⋊Dic3—36no—
70(C2×D3)⋊Dic3—32no—
71Dic32—312no—
72Dic32—36no—
73C22×C33C2411yescharacteristic
74C2×C3×C3⋊D3—22no—
75C2×C3×C3⋊D3C22×C412yes—
76C33×C4C22×C416yes—
77C3×C3⋊Dic3—22no—
78C3×C3⋊Dic3—26no—
79C3×C3⋊Dic3C22×C416yes—
80C22×C3×Q8—91no—
81C2×D3×Q8—32no—
82C2×D3×Q8—61no—
83C3×C4×Q8—92no—
84Q8×Dic3—32no—
85Q8×Dic3—32no—
86Q8×Dic3—62no—
87Q8×Dic3—96no—
88#926—93no—
89#926—93no—
90#892—96no—
91#887—96no—
92#872—96no—
93C4×Dic6—96no—
94C22×C3⋊D3C22×D311yescharacteristic
95C22×C3×D3—32no—
96C22×C3×D3—61no—
97C32×Q8—28no—
98C32×Q8—24no—
99C32×Q8—44no—
100C32×Q8C4×D318yes—
101C32×Q8C2×Dic314yes—
102C2×C32×C4—26no—
103C2×C32×C4—23no—
104C2×C32×C4—43no—
105C2×C32×C4C22×D316yes—
106C2×C32×C4C22×D313yes—
107C2×C3⋊Dic3C22×D316yes—
108C2×C3⋊Dic3C22×D312yes—
109C2×C3⋊Dic3C22×D313yes—
110C4×C3⋊D3C2×Dic3112yes—
111#634C2×Dic3112yes—
112C2×C3×Dic3—36no—
113C2×C3×Dic3—36no—
114C2×C3×Dic3—36no—
115C2×C3×Dic3—32no—
116C2×C3×Dic3—32no—
117C2×C3×Dic3—62no—
118C2×C3×Dic3—63no—
119C2×C3×Dic3—66no—
120C3×C4×D3—324no—
121C3×C4×D3—612no—
122C3×Dic6—324no—
123C3×Dic6—612no—
124#519—271noSylow
125C2×C33C23×C411yescharacteristic
126C2×C332−1+412yes—
127C3×C3⋊D3—22no—
128C2×C3×Q8—22no—
129C2×C3×Q8—21no—
130C2×C3×Q8—41no—
131C2×C3×Q8—98no—
132C2×C3×Q8—93no—
133C2×C3×Q8D3212yes—
134C2×C3×Q8D3211yescharacteristic
135C22×C3×C4—93no—
136D3×Q8—316no—
137D3×Q8—68no—
138C2×C4×D3—36no—
139C2×C4×D3—63no—
140C2×Dic6—36no—
141C2×Dic6—63no—
142C3×C4⋊C4—96no—
143C3×C42—96no—
144#217—91no—
145#217—93no—
146(C2×D3)⋊C4—92no—
147(C2×D3)⋊C4—96no—
148C4⋊Dic3—36no—
149C4⋊Dic3—36no—
150C4⋊Dic3—66no—
151C4⋊Dic3—96no—
152Dic3⋊C4—96no—
153Dic3⋊C4—96no—
154C4×Dic3—36no—
155C4×Dic3—36no—
156C4×Dic3—66no—
157C4×Dic3—912no—
158C4×Dic3—96no—
159C22×C32—22no—
160C22×C32—21no—
161C22×C32—41no—
162C22×C32C23×D312yes—
163C22×C32C23×D311yescharacteristic
164C2×C3⋊D3—22no—
165C2×C3⋊D3C22×Dic312yes—
166C2×C3×D3—34no—
167C2×C3×D3—62no—
168C2×C3×D3—64no—
169C2×C3×D3—62no—
170C2×C3×D3—62no—
171C32×C4—212no—
172C32×C4—26no—
173C32×C4—46no—
174C32×C4C2×C4×D3112yes—
175C32×C4C22×Dic316yes—
176C3⋊Dic3—22no—
177C3⋊Dic3—26no—
178C3⋊Dic3C22×Dic316yes—
179C3×Dic3—312no—
180C3×Dic3—66no—
181C3×Dic3—64no—
182C3×Dic3—66no—
183C3×Dic3—66no—
184C3×Dic3—62no—
185C3×Dic3—62no—
186C3×Dic3—126no—
187C22×Q8—91no—
188C4×Q8—92no—
189C4×Q8—276no—
190#117—273no—
191#117—273no—
192C33#51911yescharacteristic, complemented, Sylow
193C23×C3—91no—
194C22×D3—32no—
195C22×D3—61no—
196C3×Q8—28no—
197C3×Q8—24no—
198C3×Q8—44no—
199C3×Q8—912no—
200C3×Q8D3×Dic318yes—
201C3×Q8#62414yes—
202C2×C3×C4—26no—
203C2×C3×C4—23no—
204C2×C3×C4—43no—
205C2×C3×C4—93no—
206C2×C3×C4—912no—
207C2×C3×C4—96no—
208C2×C3×C4—92no—
209C2×C3×C4C2×D3216yes—
210C2×C3×C4C2×D3213yes—
211C2×Dic3—36no—
212C2×Dic3—32no—
213C2×Dic3—32no—
214C2×Dic3—36no—
215C2×Dic3—36no—
216C2×Dic3—62no—
217C2×Dic3—63no—
218C2×Dic3—66no—
219C4×D3—324no—
220C4×D3—612no—
221Dic6—324no—
222Dic6—612no—
223C2×C32—22no—
224C2×C32—22no—
225C2×C32—22no—
226C2×C32—22no—
227C2×C32—21no—
228C2×C32—41no—
229C2×C32—42no—
230C2×C32C22×C4×D312yes—
231C2×C32#100712yes—
232C2×C32#100712yes—
233C2×C32#101012yes—
234C2×C32C23×Dic311yescharacteristic
235C3⋊D3—22no—
236C3×D3—64no—
237C3×D3—64no—
238C2×Q8—98no—
239C2×Q8—93no—
240C2×Q8#111311yescharacteristic
241C22×C4—93no—
242C4⋊C4—96no—
243C4⋊C4—276no—
244C22⋊C4—271no—
245C22⋊C4—273no—
246C42—96no—
247C42—276no—
248C22×C3—22no—
249C22×C3—21no—
250C22×C3—41no—
251C22×C3—92no—
252C22×C3—182no—
253C22×C3C22×D3211yescharacteristic
254C22×C3C22×D3212yes—
255C2×D3—34no—
256C2×D3—62no—
257C2×D3—64no—
258C2×D3—62no—
259C2×D3—62no—
260C3×C4—26no—
261C3×C4—212no—
262C3×C4—46no—
263C3×C4—96no—
264C3×C4—182no—
265C3×C4—186no—
266C3×C4C2×D3×Dic3112yes—
267C3×C4C2×#62416yes—
268Dic3—312no—
269Dic3—66no—
270Dic3—64no—
271Dic3—62no—
272Dic3—66no—
273Dic3—66no—
274Dic3—62no—
275Dic3—126no—
276C32—21no—
277C32—22no—
278C32—41no—
279C32#557212yescomplemented
280C32#581211yescharacteristic, complemented
281C23—91no—
282Q8—912no—
283Q8#629414yes—
284C2×C4—93no—
285C2×C4—912no—
286C2×C4—92no—
287C2×C4—96no—
288C2×C4C2×#111313yes—
289C2×C3—22no—
290C2×C3—22no—
291C2×C3—22no—
292C2×C3—21no—
293C2×C3—22no—
294C2×C3—42no—
295C2×C3—41no—
296C2×C3—182no—
297C2×C3#6431512yes—
298C2×C3#6431512yes—
299C2×C3#6431612yes—
300C2×C3C22×D3×Dic312yes—
301C2×C3C22×#62411yescharacteristic
302D3—64no—
303D3—64no—
304C22—92no—
305C22—182no—
306C22C22×#111311yescharacteristic
307C4—96no—
308C4—182no—
309C4—186no—
310C4C2×#629416yes—
311C3—21no—
312C3—22no—
313C3—41no—
314C3#1059533911yescharacteristic, complemented
315C3#1059534712yescomplemented
316C2—182no—
317C2#1172462412yes—
318C2C22×#629411yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutatorC2×C33C23×C4
FittingC2×C33×Q8C22
socle, cosocleC22×C33C24
prime radicalC22×C3×C3⋊D3C23
hypercenterC2×Q8#1113
hypocenterC33#519
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C3×Q8×C3⋊Dic3C2semidirect
4. #11723668C2semidirect
9. C3×Q8×C3⋊D3C4semidirect
21. Q8×C3⋊Dic3D3semidirect
47. Q8×C3⋊D3Dic3semidirect
192. C33#519semidirect
279. C32#5572semidirect
280. C32#5812semidirect
314. C3#10595339semidirect
315. C3#10595347semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×C3⋊D32
C3×Q8×C3⋊Dic32
#117236732
#117236682
#117236672
#105953473
#105953393