Finite Group Databasebeta

← #13156018

Subgroups — #13156018

Order 1728.

612 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 110 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
2Dic6×C3⋊Dic3C211yesmaximal, characteristic
3C2×C3×#3719C212yesmaximal
4#11722040C212yesmaximal
5C3×#63572C211yesmaximal, characteristic
6#10591633—31nomaximal
7#10590800—32nomaximal
8C2×C3×#634C2212yes—
9C2×C32×Dic6C2211yescharacteristic
10C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C2×C32×C3⋊C8C2212yes—
13#88815—21no—
14#88815—21no—
15Dic3⋊(C3⋊Dic3)—21no—
16Dic3×C3⋊Dic3—21no—
17C3×#3719—24no—
18Q8×C3⋊Dic3—31no—
19C2×C3×#216—32no—
20C2×C3×Dic12—32no—
21#63873—31no—
22#63866—32no—
23Dic3×Dic6—32no—
24Dic3×Dic6—61no—
25C2×#3719D312yes—
26#63643—32no—
27C3×Dic3⋊C8—32no—
28#63572D311yescharacteristic
29#63566—32no—
30#63566—32no—
31#63566—62no—
32C2×C33×C4C2311yescharacteristic
33C2×C3⋊(C3⋊Dic3)—21no—
34C2×C3×C3⋊Dic3—22no—
35C2×C3×C3⋊Dic3D411yescharacteristic
36C2×C3×C3⋊Dic3D411yescharacteristic
37C3×#634—22no—
38C3×#634—22no—
39C2×C32×Dic3—21no—
40C32×Dic6—21no—
41C32×Dic6D412yes—
42C32×C3⋊C8—22no—
43#5182—92no—
44#4849—91no—
45C2×C32×Q8—31no—
46C2×#634C2×D312yes—
47C2×C3×Dic6—21no—
48C2×C3×Dic6—32no—
49C2×C3×Dic6—32no—
50C2×C3×Dic6—62no—
51C2×C3×Dic6C2×D312yes—
52C2×C32×C8—32no—
53C4⋊(C3⋊Dic3)—31no—
54C4⋊(C3⋊Dic3)—32no—
55C4⋊(C3⋊Dic3)—61no—
56C4⋊(C3⋊Dic3)—61no—
57C4⋊(C3⋊Dic3)—62no—
58C4⋊(C3⋊Dic3)—121no—
59C4×C3⋊Dic3—61no—
60C4×C3⋊Dic3C2×D311yescharacteristic
61C3×#216—64no—
62C3×C4×Dic3—32no—
63C3×C4×Dic3—61no—
64C2×C3×C3⋊C8—22no—
65C2×C3×C3⋊C8C2×D312yes—
66C2×C3×C3⋊C8C2×D312yes—
67C3×Dic12—64no—
68#3724—61no—
69#3724—61no—
70Dic3⋊Dic3—62no—
71Dic3⋊Dic3—62no—
72Dic3⋊Dic3—62no—
73Dic3⋊Dic3—62no—
74Dic32—62no—
75Dic32—62no—
76#3719—24no—
77C22×C33C2×D411yescharacteristic
78C33×C4C2×D411yescharacteristic
79C33×C4#4111yescharacteristic
80C3⋊(C3⋊Dic3)—41no—
81C3×C3⋊Dic3—21no—
82C3×C3⋊Dic3—42no—
83C3×C3⋊Dic3Q1612yescomplemented
84C32×Dic3—22no—
85C2×C3×Q16—92no—
86Q8×Dic3—92no—
87Q8×Dic3—181no—
88#944—92no—
89C2×#216—32no—
90C2×Dic12—32no—
91#869—91no—
92C4×Dic6—91no—
93C3×C4⋊C8—91no—
94#835—92no—
95#816—92no—
96#816—92no—
97#816—182no—
98Dic3⋊C8—32no—
99C32×Q8—32no—
100C32×Q8—61no—
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⋊Dic3—61no—
109C2×C3⋊Dic3—62no—
110C2×C3⋊Dic3—62no—
111C2×C3⋊Dic3—122no—
112C2×C3⋊Dic3—122no—
113C2×C3⋊Dic3D1211yescharacteristic
114C2×C3⋊Dic3D1211yescharacteristic
115#634—22no—
116#634—22no—
117C2×C3×Dic3—22no—
118C2×C3×Dic3—22no—
119C2×C3×Dic3—32no—
120C2×C3×Dic3—32no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—62no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—122no—
126C3×Dic6—22no—
127C3×Dic6—22no—
128C3×Dic6—22no—
129C3×Dic6—64no—
130C3×Dic6—62no—
131C3×Dic6—62no—
132C3×Dic6—62no—
133C3×Dic6—62no—
134C3×Dic6—122no—
135C3×Dic6#6714yes—
136C32×C8—62no—
137C3×C3⋊C8—22no—
138C3×C3⋊C8—22no—
139C3×C3⋊C8—42no—
140#462—271noSylow
141C2×C33#12111yescharacteristic
142C2×C33C2×Q1611yescharacteristic
143C2×C33#13711yescharacteristic
144C2×C3×Q8—32no—
145C2×C3×Q8—61no—
146C2×C3×Q8—92no—
147C2×Dic6—32no—
148C2×Dic6—32no—
149C2×Dic6—62no—
150C2×Dic6D3211yescharacteristic
151C3×Q16—184no—
152C2×C3×C8—32no—
153C2×C3×C8—32no—
154C2×C3×C8—62no—
155C3×C42—91no—
156#216—64no—
157C4⋊Dic3—91no—
158C4⋊Dic3—92no—
159C4⋊Dic3—181no—
160C4⋊Dic3—182no—
161C4⋊Dic3—182no—
162C4⋊Dic3—181no—
163C4⋊Dic3—181no—
164C4⋊Dic3—182no—
165C4⋊Dic3—361no—
166Dic3⋊C4—181no—
167Dic3⋊C4—182no—
168Dic3⋊C4—182no—
169C4×Dic3—32no—
170C4×Dic3—61no—
171C4×Dic3—181no—
172C4×Dic3—182no—
173C4×Dic3—182no—
174C2×C3⋊C8D3212yes—
175Dic12—64no—
176C22×C32—22no—
177C22×C32—21no—
178C22×C32—41no—
179C22×C32C2×D1211yescharacteristic
180C22×C32D3×D412yes—
181C32×C4—22no—
182C32×C4—21no—
183C32×C4—21no—
184C32×C4—22no—
185C32×C4—41no—
186C32×C4—41no—
187C32×C4—62no—
188C32×C4D3×D411yescharacteristic
189C32×C4#23712yes—
190C32×C4#23911yescharacteristic
191C32×C4C2×#6712yes—
192C3⋊Dic3—21no—
193C3⋊Dic3—42no—
194C3⋊Dic3—121no—
195C3⋊Dic3—122no—
196C3⋊Dic3—122no—
197C3⋊Dic3—242no—
198C3⋊Dic3—242no—
199C3⋊Dic3Dic1212yescomplemented
200C3×Dic3—24no—
201C3×Dic3—24no—
202C3×Dic3—34no—
203C3×Dic3—62no—
204C3×Dic3—62no—
205C3×Dic3—121no—
206C3×Dic3—124no—
207C3×Dic3—122no—
208C3×Dic3—122no—
209C2×Q16—92no—
210C4⋊Q8—271no—
211C4×Q8—271no—
212C4⋊C8—91no—
213#103—272no—
214C33#46211yescharacteristic, complemented, Sylow
215C3×Q8—34no—
216C3×Q8—62no—
217C3×Q8—62no—
218C3×Q8—62no—
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—91no—
227C2×C3×C4—91no—
228C2×C3×C4—182no—
229C2×C3×C4C2×D3212yes—
230C2×C3×C4C2×D3211yescharacteristic
231C2×Dic3—21no—
232C2×Dic3—32no—
233C2×Dic3—32no—
234C2×Dic3—61no—
235C2×Dic3—61no—
236C2×Dic3—62no—
237C2×Dic3—62no—
238C2×Dic3—122no—
239C2×Dic3—182no—
240C2×Dic3—182no—
241C2×Dic3—181no—
242C2×Dic3—362no—
243C2×Dic3—362no—
244Dic6—21no—
245Dic6—64no—
246Dic6—62no—
247Dic6—62no—
248Dic6—62no—
249Dic6—62no—
250Dic6—122no—
251Dic6#62512yes—
252C3×C8—62no—
253C3×C8—62no—
254C3×C8—122no—
255C3⋊C8—22no—
256C2×C32—22no—
257C2×C32—22no—
258C2×C32—22no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32—41no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32#89511yescharacteristic
266C2×C32C2×Dic1211yescharacteristic
267C2×C32#90911yescharacteristic
268C2×C32D3×Q1612yes—
269C2×C32#91812yes—
270C2×C32#93912yes—
271C2×Q8—31no—
272C2×Q8—92no—
273Q16—184no—
274C2×C8—32no—
275C4⋊C4—271no—
276C4⋊C4—541no—
277C4⋊C4—541no—
278C4⋊C4—541no—
279C42—91no—
280C42—541no—
281C22×C3—21no—
282C22×C3—22no—
283C22×C3—41no—
284C22×C3D3×D1212yes—
285C22×C3#381111yescharacteristic
286C3×C4—22no—
287C3×C4—21no—
288C3×C4—22no—
289C3×C4—21no—
290C3×C4—41no—
291C3×C4—41no—
292C3×C4—64no—
293C3×C4—64no—
294C3×C4—92no—
295C3×C4—181no—
296C3×C4—362no—
297C3×C4#379612yes—
298C3×C4#380511yescharacteristic
299C3×C4C2×#62511yescharacteristic
300C3×C4D3×#6712yes—
301Dic3—22no—
302Dic3—34no—
303Dic3—62no—
304Dic3—62no—
305Dic3—121no—
306Dic3—124no—
307Dic3—122no—
308Dic3—122no—
309Dic3—362no—
310Dic3—362no—
311Dic3—361no—
312Dic3—722no—
313Dic3—722no—
314C32—22no—
315C32—21no—
316C32—41no—
317C32#484911yescharacteristic, complemented
318C32#518212yescomplemented
319Q8—32no—
320Q8—61no—
321Q8—182no—
322Q8—182no—
323C2×C4—61no—
324C2×C4—91no—
325C2×C4—91no—
326C2×C4—182no—
327C2×C4—541no—
328C2×C4D3311yescharacteristic
329C8—62no—
330C2×C3—21no—
331C2×C3—22no—
332C2×C3—21no—
333C2×C3—22no—
334C2×C3—22no—
335C2×C3—21no—
336C2×C3—41no—
337C2×C3—41no—
338C2×C3—41no—
339C2×C3D3×Dic1212yes—
340C2×C3#6380112yes—
341C2×C3#6388612yes—
342C2×C3#6394411yescharacteristic
343C2×C3#6394511yescharacteristic
344C2×C3#6398011yescharacteristic
345C22#8895511yescharacteristic
346C4—62no—
347C4—92no—
348C4—181no—
349C4—362no—
350C4—1081no—
351C4D3×#62511yescharacteristic
352C4#8896411yescharacteristic
353C3—22no—
354C3—21no—
355C3—41no—
356C3#1059080012yescomplemented
357C3#1059163311yescharacteristic, complemented
358C2#1172284911yescharacteristic
359C2#1172289711yescharacteristic
360C2#1172295511yescharacteristic
subgroupG/H
centerC22#88955
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#462
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
83. C3×C3⋊Dic3Q16semidirect
199. C3⋊Dic3Dic12semidirect
214. C33#462semidirect
317. C32#4849semidirect
318. C32#5182semidirect
356. C3#10590800semidirect
357. C3#10591633semidirect

Maximal subgroups

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

subgroupindex
#117236782
Dic6×C3⋊Dic32
C2×C3×#37192
#117220402
C3×#635722
#105916333
#105908003