Finite Group Databasebeta

← #13156453

Subgroups — #13156453

Order 1728.

612 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 102 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#11723431C211yesmaximal, characteristic
2C3×#63900C211yesmaximal, characteristic
3#11722072C211yesmaximal, characteristic
4#11722041C212yesmaximal, complemented
5C3×#63574C212yesmaximal, complemented
6#10590443—31nomaximal
7#10590426—32nomaximal
8C2×#6308C2211yescharacteristic
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C3×#624—21no—
11C2×C3×C3⋊(C3⋊C8)C2212yes—
12C32×C4⋊Dic3C2211yescharacteristic
13C32×C4⋊Dic3C2212yes—
14Dic3⋊(C3⋊Dic3)—21no—
15#88803—21no—
16C3×#3724—21no—
17#64106—31no—
18C3×D3×C4⋊C4—32no—
19#63900D311yescharacteristic
20#63857—32no—
21#63857—61no—
22#63637—32no—
23C3×#808—32no—
24C3×#808—32no—
25#63574D312yescomplemented
26#63567—32no—
27#63567—32no—
28#63567—62no—
29#63557—31no—
30(C2×D3)⋊(C3⋊C8)—32no—
31C22×C3×C3⋊D3D411yescharacteristic
32C2×C33×C4C2311yescharacteristic
33C2×C3⋊(C3⋊Dic3)—21no—
34#6308—22no—
35C2×C3×C3⋊Dic3D411yescharacteristic
36C3×C4×C3⋊D3—22no—
37C2×C32×Dic3—21no—
38C2×C32×Dic3—22no—
39C3×#624—42no—
40C3×C3⋊(C3⋊C8)—22no—
41#4971—91no—
42#4862—92no—
43C2×C4×C3⋊D3C2×D311yescharacteristic
44C2×#634—31no—
45C2×#634—32no—
46C2×#634—62no—
47C2×#634—61no—
48C2×#634—121no—
49C2×C3×C4×D3—32no—
50C2×C3×C4×D3—62no—
51C2×C3×C4×D3—61no—
52C2×#624—21no—
53C32×C4⋊C4—32no—
54C32×C4⋊C4—31no—
55#3752—61no—
56C4⋊(C3⋊Dic3)—61no—
57C2×C3⋊(C3⋊C8)C2×D312yes—
58C3×C4⋊Dic3—21no—
59C3×C4⋊Dic3—22no—
60C3×C4⋊Dic3C2×D312yes—
61C3×C4⋊Dic3C2×D312yes—
62C3×C4⋊Dic3C2×D312yes—
63C3×Dic3⋊C4—62no—
64C2×C3×C3⋊C8—32no—
65C2×C3×C3⋊C8—32no—
66C2×C3×C3⋊C8—62no—
67#3724—21no—
68Dic3⋊Dic3—62no—
69Dic3⋊Dic3—62no—
70(C2×D3)⋊Dic3—62no—
71(C2×D3)⋊Dic3—62no—
72C22×C33C2×D411yescharacteristic
73C2×C3×C3⋊D3—21no—
74C2×C3×C3⋊D3—21no—
75C2×C3×C3⋊D3SD1612yes—
76C33×C4#4112yes—
77C3⋊(C3⋊Dic3)—41no—
78C3×C3⋊Dic3—21no—
79C32×Dic3—42no—
80C32×Dic3—41no—
81C2×C3×C4⋊C4—91no—
82#924—91no—
83#897—92no—
84#897—181no—
85D3×C4⋊C4—32no—
86C3×#106—92no—
87#830—91no—
88(C2×D3)⋊C8—92no—
89#816—92no—
90#810—92no—
91#810—92no—
92#810—182no—
93#808—32no—
94#808—32no—
95C22×C3⋊D3#6711yescharacteristic
96C22×C3×D3—32no—
97C22×C3×D3—61no—
98C2×C32×C4—22no—
99C2×C32×C4—21no—
100C2×C32×C4—41no—
101C2×C32×C4—61no—
102C2×C32×C4—62no—
103C2×C32×C4C22×D311yescharacteristic
104C2×C32×C4C22×D312yes—
105C2×C3⋊Dic3—61no—
106C2×C3⋊Dic3—62no—
107C2×C3⋊Dic3—62no—
108C2×C3⋊Dic3—122no—
109C2×C3⋊Dic3—122no—
110C2×C3⋊Dic3#6711yescharacteristic
111C4×C3⋊D3—22no—
112#634—62no—
113#634—64no—
114#634—62no—
115#634—62no—
116#634—122no—
117#634—122no—
118#634—124no—
119C2×C3×Dic3—22no—
120C2×C3×Dic3—22no—
121C2×C3×Dic3—22no—
122C2×C3×Dic3—22no—
123C2×C3×Dic3—32no—
124C2×C3×Dic3—42no—
125C2×C3×Dic3—61no—
126C3×C4×D3—62no—
127C3×C4×D3—62no—
128C3×C4×D3—122no—
129C3×C4×D3—124no—
130#624—42no—
131C3⋊(C3⋊C8)—22no—
132C3×C3⋊C8—62no—
133C3×C3⋊C8—62no—
134C3×C3⋊C8—122no—
135#483—271noSylow
136C2×C33#12311yescharacteristic
137C2×C33C2×SD1611yescharacteristic
138C2×C33#13711yescharacteristic
139C3×C3⋊D3—22no—
140C22×C3×C4—91no—
141C22×C3×C4—181no—
142C2×C4×D3—32no—
143C2×C4×D3—61no—
144C2×C4×D3—62no—
145C2×Dic6—91no—
146C2×Dic6—92no—
147C2×Dic6—182no—
148C2×Dic6—181no—
149C2×Dic6—361no—
150C2×C3×C8—92no—
151C3×C4⋊C4—32no—
152C3×C4⋊C4—32no—
153C3×C4⋊C4—32no—
154C3×C4⋊C4—61no—
155C3×C4⋊C4—62no—
156C3×C4⋊C4—181no—
157#217—181no—
158(C2×D3)⋊C4—182no—
159(C2×D3)⋊C4—182no—
160C4⋊Dic3—182no—
161C4⋊Dic3—182no—
162C4⋊Dic3D3211yescharacteristic
163C4⋊Dic3D3212yes—
164Dic3⋊C4—62no—
165Dic3⋊C4—181no—
166C2×C3⋊C8—32no—
167C2×C3⋊C8—32no—
168C2×C3⋊C8—62no—
169C22×C32—22no—
170C22×C32—21no—
171C22×C32—41no—
172C22×C32D3×D412yes—
173C22×C32C2×#6711yescharacteristic
174C2×C3⋊D3—21no—
175C2×C3⋊D3—21no—
176C2×C3⋊D3#21412yes—
177C2×C3×D3—34no—
178C2×C3×D3—62no—
179C2×C3×D3—62no—
180C2×C3×D3—62no—
181C2×C3×D3—121no—
182C2×C3×D3—121no—
183C32×C4—22no—
184C32×C4—22no—
185C32×C4—22no—
186C32×C4—42no—
187C32×C4—121no—
188C32×C4—122no—
189C32×C4#23512yes—
190C32×C4#23512yes—
191C32×C4#23912yes—
192C3⋊Dic3—21no—
193C3⋊Dic3—124no—
194C3⋊Dic3—122no—
195C3⋊Dic3—124no—
196C3⋊Dic3—122no—
197C3⋊Dic3—121no—
198C3×Dic3—42no—
199C3×Dic3—42no—
200C3×Dic3—42no—
201C3×Dic3—42no—
202C3×Dic3—62no—
203C3×Dic3—82no—
204C3×Dic3—121no—
205C22⋊Q8—271no—
206C2×C4⋊C4—91no—
207#106—92no—
208#103—272no—
209C22⋊C8—271no—
210C33#48311yescharacteristic, complemented, Sylow
211C23×C3—91no—
212C22×D3—32no—
213C22×D3—61no—
214C2×C3×C4—22no—
215C2×C3×C4—21no—
216C2×C3×C4—41no—
217C2×C3×C4—62no—
218C2×C3×C4—62no—
219C2×C3×C4—62no—
220C2×C3×C4—62no—
221C2×C3×C4—91no—
222C2×C3×C4—122no—
223C2×C3×C4—182no—
224C2×C3×C4—362no—
225C2×C3×C4C2×D3212yes—
226C2×C3×C4C2×D3211yescharacteristic
227C2×Dic3—22no—
228C2×Dic3—21no—
229C2×Dic3—32no—
230C2×Dic3—61no—
231C2×Dic3—182no—
232C2×Dic3—182no—
233C2×Dic3—181no—
234C2×Dic3—362no—
235C2×Dic3—362no—
236C4×D3—62no—
237C4×D3—62no—
238C4×D3—122no—
239C4×D3—124no—
240Dic6—184no—
241Dic6—182no—
242Dic6—182no—
243Dic6—182no—
244Dic6—362no—
245Dic6—362no—
246Dic6—364no—
247C3×C8—182no—
248C3⋊C8—62no—
249C3⋊C8—62no—
250C3⋊C8—122no—
251C2×C32—22no—
252C2×C32—21no—
253C2×C32—22no—
254C2×C32—21no—
255C2×C32—21no—
256C2×C32—22no—
257C2×C32—41no—
258C2×C32—41no—
259C2×C32—41no—
260C2×C32#89412yes—
261C2×C32D3×SD1612yes—
262C2×C32#91512yes—
263C2×C32#92911yescharacteristic
264C2×C32C2×#21411yescharacteristic
265C2×C32#94211yescharacteristic
266C3⋊D3—22no—
267C3×D3—64no—
268C3×D3—122no—
269C2×Q8—271no—
270C22×C4—91no—
271C22×C4—181no—
272C2×C8—92no—
273C4⋊C4—32no—
274C4⋊C4—31no—
275C4⋊C4—181no—
276C4⋊C4—541no—
277C22⋊C4—541no—
278C22×C3—21no—
279C22×C3—22no—
280C22×C3—41no—
281C22×C3—92no—
282C22×C3—181no—
283C22×C3—181no—
284C22×C3D3×#6712yes—
285C22×C3#381111yescharacteristic
286C2×D3—34no—
287C2×D3—62no—
288C2×D3—62no—
289C2×D3—62no—
290C2×D3—121no—
291C2×D3—121no—
292C3×C4—22no—
293C3×C4—22no—
294C3×C4—22no—
295C3×C4—42no—
296C3×C4—122no—
297C3×C4—122no—
298C3×C4—122no—
299C3×C4—122no—
300C3×C4—181no—
301C3×C4—242no—
302C3×C4#379812yes—
303C3×C4#379912yes—
304C3×C4#379912yes—
305Dic3—42no—
306Dic3—41no—
307Dic3—62no—
308Dic3—121no—
309Dic3—364no—
310Dic3—362no—
311Dic3—364no—
312Dic3—362no—
313Dic3—361no—
314C32—22no—
315C32—21no—
316C32—41no—
317C32#486212yescomplemented
318C32#497111yescharacteristic, complemented
319C23—91no—
320Q8—542no—
321C2×C4—61no—
322C2×C4—62no—
323C2×C4—91no—
324C2×C4—182no—
325C2×C4—362no—
326C2×C4—541no—
327C2×C4D3311yescharacteristic
328C8—182no—
329C2×C3—22no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3—22no—
333C2×C3—21no—
334C2×C3—22no—
335C2×C3—41no—
336C2×C3—41no—
337C2×C3—41no—
338C2×C3—182no—
339C2×C3#6385012yes—
340C2×C3#6389711yescharacteristic
341C2×C3D3×#21412yes—
342C2×C3#6393111yescharacteristic
343C2×C3#6394011yescharacteristic
344C2×C3#6394612yes—
345D3—64no—
346D3—122no—
347C22—92no—
348C22—181no—
349C22—181no—
350C22#8895211yescharacteristic
351C4—122no—
352C4—121no—
353C4—181no—
354C4—1081no—
355C4#8896512yes—
356C3—21no—
357C3—22no—
358C3—41no—
359C3#1059042612yescomplemented
360C3#1059044311yescharacteristic, complemented
361C2—182no—
362C2#1172283711yescharacteristic
363C2#1172288911yescharacteristic
364C2#1172299311yescharacteristic
subgroupG/H
centerC22#88952
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypocenterC33#483
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
4. #11722041C2semidirect
5. C3×#63574C2semidirect
25. #63574D3semidirect
210. C33#483semidirect
317. C32#4862semidirect
318. C32#4971semidirect
359. C3#10590426semidirect
360. C3#10590443semidirect

Maximal subgroups

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

subgroupindex
#117234312
C3×#639002
#117220722
#117220412
C3×#635742
#105904433
#105904263