Finite Group Databasebeta

← #13156016

Subgroups — #13156016

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
1Dic6×C3⋊Dic3C211yesmaximal, characteristic
2C3×#63873C211yesmaximal, characteristic
3C2×C3×#3718C212yesmaximal
4#11722078C211yesmaximal, characteristic
5#11722038C212yesmaximal
6#10591633—31nomaximal
7#10591574—32nomaximal
8C2×C32×Dic6C2212yes—
9C2×C32×Dic6C2211yescharacteristic
10C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C2×C3×C3⋊(C3⋊C8)C2212yes—
13Dic3⋊(C3⋊Dic3)—21no—
14Dic3×C3⋊Dic3—21no—
15C3×#3724—21no—
16C3×#3724—21no—
17C3×#3718—24no—
18Q8×C3⋊Dic3—31no—
19C3×#944—32no—
20C2×C3×#216—32no—
21C2×C3×#216—32no—
22#63873D311yescharacteristic
23Dic3×Dic6—32no—
24Dic3×Dic6—61no—
25C2×#3718D312yes—
26#63662—32no—
27#63572—31no—
28Dic3⋊(C3⋊C8)—32no—
29#63566—32no—
30#63566—32no—
31#63566—62no—
32C2×C33×C4C2311yescharacteristic
33C2×C3⋊(C3⋊Dic3)—21no—
34C2×C3×C3⋊Dic3D411yescharacteristic
35C2×C3×C3⋊Dic3D411yescharacteristic
36C2×C32×Dic3—21no—
37C2×C32×Dic3—22no—
38C32×Dic6—21no—
39C32×Dic6—22no—
40C32×Dic6—22no—
41C32×Dic6D412yes—
42C3×C3⋊(C3⋊C8)—22no—
43#5182—92no—
44#5090—91no—
45C2×C32×Q8—31no—
46C2×C32×Q8—32no—
47C2×C3×Dic6—21no—
48C2×C3×Dic6—22no—
49C2×C3×Dic6C2×D312yes—
50C2×C3×Dic6C2×D312yes—
51C2×C3×Dic6C2×D312yes—
52C4⋊(C3⋊Dic3)—31no—
53C4⋊(C3⋊Dic3)—32no—
54C4⋊(C3⋊Dic3)—61no—
55C4⋊(C3⋊Dic3)—61no—
56C4⋊(C3⋊Dic3)—62no—
57C4⋊(C3⋊Dic3)—121no—
58C4×C3⋊Dic3—61no—
59C4×C3⋊Dic3C2×D311yescharacteristic
60C2×C3⋊(C3⋊C8)C2×D312yes—
61C3×#216—64no—
62C3×#216—64no—
63C3×Dic3⋊C4—62no—
64C3×Dic3⋊C4—62no—
65C3×C4×Dic3—32no—
66C3×C4×Dic3—61no—
67C2×C3×C3⋊C8—32no—
68C2×C3×C3⋊C8—32no—
69C2×C3×C3⋊C8—62no—
70#3724—21no—
71#3724—21no—
72Dic3⋊Dic3—62no—
73Dic3⋊Dic3—62no—
74Dic32—62no—
75Dic32—62no—
76#3718—24no—
77C22×C33C2×D411yescharacteristic
78C33×C4C2×D411yescharacteristic
79C33×C4#4111yescharacteristic
80C3⋊(C3⋊Dic3)—41no—
81C3×C3⋊Dic3—21no—
82C3×C3⋊Dic3Q1612yescomplemented
83C32×Dic3—22no—
84C32×Dic3—42no—
85C2×C3×Q16—92no—
86C3×C4⋊Q8—91no—
87Q8×Dic3—92no—
88Q8×Dic3—181no—
89#944—32no—
90C2×#216—32no—
91C2×#216—32no—
92C4×Dic6—91no—
93#835—92no—
94#835—92no—
95#835—182no—
96#816—92no—
97Dic3⋊C8—92no—
98C4⋊(C3⋊C8)—91no—
99C32×Q8—32no—
100C32×Q8—61no—
101C32×Q8—62no—
102C32×Q8—62no—
103C2×C32×C4—22no—
104C2×C32×C4—21no—
105C2×C32×C4—41no—
106C2×C32×C4—61no—
107C2×C32×C4—62no—
108C2×C32×C4C22×D311yescharacteristic
109C2×C32×C4C22×D312yes—
110C2×C3⋊Dic3—61no—
111C2×C3⋊Dic3—62no—
112C2×C3⋊Dic3—62no—
113C2×C3⋊Dic3—122no—
114C2×C3⋊Dic3—122no—
115C2×C3⋊Dic3#6711yescharacteristic
116C2×C3⋊Dic3#6711yescharacteristic
117C2×C3×Dic3—22no—
118C2×C3×Dic3—22no—
119C2×C3×Dic3—22no—
120C2×C3×Dic3—22no—
121C2×C3×Dic3—32no—
122C2×C3×Dic3—32no—
123C2×C3×Dic3—42no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C3×Dic6—22no—
127C3×Dic6—22no—
128C3×Dic6—24no—
129C3×Dic6—22no—
130C3×Dic6—22no—
131C3×Dic6—22no—
132C3×Dic6—22no—
133C3×Dic6—22no—
134C3×Dic6—42no—
135C3×Dic6#6714yes—
136C3⋊(C3⋊C8)—22no—
137C3×C3⋊C8—62no—
138C3×C3⋊C8—62no—
139C3×C3⋊C8—122no—
140#462—271noSylow
141C2×C33#12111yescharacteristic
142C2×C33C2×Q1611yescharacteristic
143C2×C33#13711yescharacteristic
144C2×C3×Q8—32no—
145C2×C3×Q8—32no—
146C2×C3×Q8—32no—
147C2×C3×Q8—61no—
148C2×C3×Q8—62no—
149C2×Dic6D3212yes—
150C2×Dic6D3211yescharacteristic
151C3×Q16—184no—
152C2×C3×C8—92no—
153C3×C4⋊C4—181no—
154C3×C4⋊C4—181no—
155C3×C42—91no—
156#216—64no—
157#216—64no—
158C4⋊Dic3—92no—
159C4⋊Dic3—91no—
160C4⋊Dic3—181no—
161C4⋊Dic3—182no—
162C4⋊Dic3—182no—
163C4⋊Dic3—182no—
164C4⋊Dic3—361no—
165Dic3⋊C4—62no—
166Dic3⋊C4—62no—
167Dic3⋊C4—181no—
168C4×Dic3—32no—
169C4×Dic3—61no—
170C4×Dic3—181no—
171C4×Dic3—182no—
172C4×Dic3—182no—
173C2×C3⋊C8—32no—
174C2×C3⋊C8—32no—
175C2×C3⋊C8—62no—
176C22×C32—22no—
177C22×C32—21no—
178C22×C32—41no—
179C22×C32D3×D412yes—
180C22×C32C2×#6711yescharacteristic
181C32×C4—22no—
182C32×C4—21no—
183C32×C4—21no—
184C32×C4—22no—
185C32×C4—41no—
186C32×C4—41no—
187C32×C4—62no—
188C32×C4—122no—
189C32×C4D3×D411yescharacteristic
190C32×C4#23711yescharacteristic
191C32×C4#23712yes—
192C32×C4C2×#6712yes—
193C3⋊Dic3—21no—
194C3⋊Dic3—121no—
195C3⋊Dic3—122no—
196C3⋊Dic3—122no—
197C3⋊Dic3—242no—
198C3⋊Dic3—242no—
199C3⋊Dic3#21612yescomplemented
200C3×Dic3—24no—
201C3×Dic3—24no—
202C3×Dic3—34no—
203C3×Dic3—44no—
204C3×Dic3—42no—
205C3×Dic3—42no—
206C3×Dic3—62no—
207C3×Dic3—62no—
208C3×Dic3—121no—
209C2×Q16—92no—
210C4⋊Q8—91no—
211C4×Q8—271no—
212C4⋊C8—271no—
213#103—272no—
214C33#46211yescharacteristic, complemented, Sylow
215C3×Q8—34no—
216C3×Q8—62no—
217C3×Q8—62no—
218C3×Q8—64no—
219C3×Q8—62no—
220C3×Q8—62no—
221C3×Q8—62no—
222C3×Q8—62no—
223C3×Q8—62no—
224C3×Q8—122no—
225C2×C3×C4—22no—
226C2×C3×C4—21no—
227C2×C3×C4—41no—
228C2×C3×C4—62no—
229C2×C3×C4—62no—
230C2×C3×C4—62no—
231C2×C3×C4—62no—
232C2×C3×C4—91no—
233C2×C3×C4—91no—
234C2×C3×C4—122no—
235C2×C3×C4C2×D3211yescharacteristic
236C2×C3×C4C2×D3212yes—
237C2×Dic3—21no—
238C2×Dic3—22no—
239C2×Dic3—32no—
240C2×Dic3—32no—
241C2×Dic3—61no—
242C2×Dic3—61no—
243C2×Dic3—182no—
244C2×Dic3—182no—
245C2×Dic3—181no—
246C2×Dic3—362no—
247C2×Dic3—362no—
248Dic6—21no—
249Dic6—22no—
250Dic6—22no—
251Dic6#62512yes—
252C3×C8—182no—
253C3⋊C8—62no—
254C3⋊C8—62no—
255C3⋊C8—122no—
256C2×C32—22no—
257C2×C32—21no—
258C2×C32—21no—
259C2×C32—22no—
260C2×C32—22no—
261C2×C32—21no—
262C2×C32—41no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32D3×Q1612yes—
266C2×C32#91812yes—
267C2×C32#93811yescharacteristic
268C2×C32#93912yes—
269C2×C32#94211yescharacteristic
270C2×C32C2×#21611yescharacteristic
271C2×Q8—31no—
272C2×Q8—32no—
273Q16—184no—
274C2×C8—92no—
275C4⋊C4—181no—
276C4⋊C4—181no—
277C4⋊C4—271no—
278C4⋊C4—541no—
279C42—91no—
280C42—541no—
281C22×C3—21no—
282C22×C3—22no—
283C22×C3—41no—
284C22×C3D3×#6712yes—
285C22×C3#381111yescharacteristic
286C3×C4—22no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—22no—
290C3×C4—41no—
291C3×C4—41no—
292C3×C4—64no—
293C3×C4—64no—
294C3×C4—92no—
295C3×C4—124no—
296C3×C4—122no—
297C3×C4—122no—
298C3×C4—181no—
299C3×C4#380512yes—
300C3×C4#380511yescharacteristic
301C3×C4C2×#62511yescharacteristic
302C3×C4D3×#6712yes—
303Dic3—22no—
304Dic3—34no—
305Dic3—42no—
306Dic3—62no—
307Dic3—62no—
308Dic3—121no—
309Dic3—362no—
310Dic3—361no—
311Dic3—362no—
312Dic3—722no—
313Dic3—722no—
314C32—22no—
315C32—21no—
316C32—41no—
317C32#509011yescharacteristic, complemented
318C32#518212yescomplemented
319Q8—32no—
320Q8—61no—
321Q8—62no—
322Q8—62no—
323C2×C4—61no—
324C2×C4—62no—
325C2×C4—91no—
326C2×C4—91no—
327C2×C4—541no—
328C2×C4D3311yescharacteristic
329C8—182no—
330C2×C3—21no—
331C2×C3—22no—
332C2×C3—21no—
333C2×C3—22no—
334C2×C3—21no—
335C2×C3—22no—
336C2×C3—41no—
337C2×C3—41no—
338C2×C3—41no—
339C2×C3D3×#21612yes—
340C2×C3#6394411yescharacteristic
341C2×C3#6394511yescharacteristic
342C2×C3#6394812yes—
343C2×C3#6397012yes—
344C2×C3#6398011yescharacteristic
345C22#8895211yescharacteristic
346C4—62no—
347C4—92no—
348C4—122no—
349C4—181no—
350C4—1081no—
351C4D3×#62511yescharacteristic
352C4#8896311yescharacteristic
353C3—22no—
354C3—21no—
355C3—41no—
356C3#1059157412yescomplemented
357C3#1059163311yescharacteristic, complemented
358C2#1172284611yescharacteristic
359C2#1172289011yescharacteristic
360C2#1172295311yescharacteristic
subgroupG/H
centerC22#88952
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
82. C3×C3⋊Dic3Q16semidirect
199. C3⋊Dic3#216semidirect
214. C33#462semidirect
317. C32#5090semidirect
318. C32#5182semidirect
356. C3#10591574semidirect
357. C3#10591633semidirect

Maximal subgroups

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

subgroupindex
Dic6×C3⋊Dic32
C3×#638732
C2×C3×#37182
#117220782
#117220382
#105916333
#105915743