Finite Group Databasebeta

← #13162017

Subgroups — #13162017

Order 1728.

582 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C3×#64153C211yesmaximal, characteristic
2C4⋊#6296C211yesmaximal, characteristic
3C3×#63662C212yesmaximal
4#11722076C212yesmaximal
5#11722070C211yesmaximal, characteristic
6#10591623—32nomaximal
7#10591538—31nomaximal
8C2×C33×Q8C2211yescharacteristic
9C2×C3×#634C2211yescharacteristic
10C2×#6296—21no—
11C3×C4⋊(C3⋊Dic3)C2212yes—
12C3×#3750—21no—
13C3×#3750—21no—
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2212yes—
16#88815—21no—
17#64153D311yescharacteristic
18C3×#944—32no—
19C3×#944—61no—
20#63919—31no—
21#63918—32no—
22#63662D312yes—
23C3×#835—32no—
24C3×#835—32no—
25C3×#835—62no—
26#63567—32no—
27#63566—32no—
28#63565—32no—
29#63553—31no—
30Dic3×C3⋊C8—32no—
31C33×Q8—22no—
32C2×C33×C4—21no—
33C2×C33×C4C2311yescharacteristic
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3—22no—
36C2×C3×C3⋊Dic3D411yescharacteristic
37C2×C3×C3⋊Dic3D411yescharacteristic
38C3×#634—22no—
39#6296—42no—
40C3×C3⋊(C3⋊C8)—22no—
41#5079—91no—
42#4792—92no—
43C2×C32×Q8—21no—
44C2×C32×Q8—22no—
45C2×C32×Q8—41no—
46C2×C32×Q8C2×D312yes—
47C2×C32×Q8C2×D311yescharacteristic
48C2×#634C2×D311yescharacteristic
49C2×C3×Dic6—32no—
50C2×C3×Dic6—61no—
51C2×#627—62no—
52C2×#627—61no—
53C4⋊(C3⋊Dic3)C2×D312yes—
54#3750—21no—
55#3750—21no—
56C4×C3⋊Dic3C2×D311yescharacteristic
57C2×C3⋊(C3⋊C8)C2×D312yes—
58C3×C4⋊Dic3—32no—
59C3×C4⋊Dic3—32no—
60C3×C4⋊Dic3—62no—
61C3×Dic3⋊C4—62no—
62C3×Dic3⋊C4—62no—
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—61no—
71Dic3⋊Dic3—62no—
72C22×C33C2×D411yescharacteristic
73C33×C4—41no—
74C33×C4#4112yes—
75C3×C3⋊Dic3—41no—
76C3×C3⋊Dic3—42no—
77C3×C3⋊Dic3SD1612yes—
78C3×C3⋊Dic3Q1612yes—
79C3×C4⋊Q8—91no—
80#944—32no—
81#944—61no—
82#888—92no—
83#869—91no—
84C3×#103—92no—
85#835—32no—
86#835—32no—
87#835—62no—
88#835—92no—
89#816—92no—
90C8×Dic3—92no—
91#810—92no—
92C4×C3⋊C8—91no—
93C32×Q8—22no—
94C32×Q8—24no—
95C32×Q8—22no—
96C32×Q8—22no—
97C32×Q8—42no—
98C32×Q8—42no—
99C32×Q8—44no—
100C2×C32×C4—21no—
101C2×C32×C4—22no—
102C2×C32×C4—22no—
103C2×C32×C4—22no—
104C2×C32×C4—21no—
105C2×C32×C4—41no—
106C2×C32×C4—42no—
107C2×C32×C4—42no—
108C2×C32×C4C22×D312yes—
109C2×C32×C4C22×D311yescharacteristic
110C2×C3⋊Dic3—21no—
111C2×C3⋊Dic3—22no—
112C2×C3⋊Dic3#6711yescharacteristic
113C2×C3⋊Dic3#6711yescharacteristic
114#634—22no—
115C2×C3×Dic3—32no—
116C2×C3×Dic3—32no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—62no—
120C2×C3×Dic3—62no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—62no—
123C2×C3×Dic3—122no—
124C3×Dic6—62no—
125C3×Dic6—62no—
126C3×Dic6—122no—
127#627—124no—
128#627—122no—
129C3⋊(C3⋊C8)—22no—
130C3×C3⋊C8—62no—
131C3×C3⋊C8—62no—
132C3×C3⋊C8—122no—
133#487—271noSylow
134C2×C33#12411yescharacteristic
135C2×C33C2×SD1611yescharacteristic
136C2×C33C2×Q1611yescharacteristic
137C2×C3×Q8—21no—
138C2×C3×Q8—22no—
139C2×C3×Q8—41no—
140C2×C3×Q8—91no—
141C2×C3×Q8D3211yescharacteristic
142C2×C3×Q8D3212yes—
143C2×Dic6—32no—
144C2×Dic6—61no—
145C2×Dic6—182no—
146C2×Dic6—181no—
147C2×C3×C8—92no—
148C3×C4⋊C4—92no—
149C3×C4⋊C4—181no—
150C3×C4⋊C4—181no—
151C3×C42—91no—
152C4⋊Dic3—32no—
153C4⋊Dic3—32no—
154C4⋊Dic3—62no—
155C4⋊Dic3—181no—
156Dic3⋊C4—62no—
157Dic3⋊C4—62no—
158Dic3⋊C4—62no—
159Dic3⋊C4—62no—
160Dic3⋊C4—182no—
161C4×Dic3—32no—
162C4×Dic3—61no—
163C2×C3⋊C8—32no—
164C2×C3⋊C8—32no—
165C2×C3⋊C8—62no—
166C22×C32—22no—
167C22×C32—21no—
168C22×C32—41no—
169C22×C32D3×D412yes—
170C22×C32C2×#6711yescharacteristic
171C32×C4—22no—
172C32×C4—22no—
173C32×C4—22no—
174C32×C4—42no—
175C32×C4—42no—
176C32×C4—44no—
177C32×C4—44no—
178C32×C4—42no—
179C32×C4—41no—
180C32×C4#23512yes—
181C32×C4#23712yes—
182C32×C4#23712yes—
183C3⋊Dic3—41no—
184C3⋊Dic3—42no—
185C3⋊Dic3#21512yes—
186C3⋊Dic3#21612yes—
187C3×Dic3—34no—
188C3×Dic3—34no—
189C3×Dic3—62no—
190C3×Dic3—62no—
191C3×Dic3—122no—
192C3×Dic3—122no—
193C3×Dic3—122no—
194C3×Dic3—122no—
195C3×Dic3—242no—
196C4⋊Q8—91no—
197C4⋊Q8—271no—
198#103—92no—
199#103—272no—
200C4×C8—271no—
201C33#48711yescharacteristic, complemented, Sylow
202C3×Q8—24no—
203C3×Q8—22no—
204C3×Q8—22no—
205C3×Q8—22no—
206C3×Q8—42no—
207C3×Q8—42no—
208C3×Q8—44no—
209C3×Q8—182no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—21no—
213C2×C3×C4—22no—
214C2×C3×C4—21no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—41no—
218C2×C3×C4—91no—
219C2×C3×C4—91no—
220C2×C3×C4—182no—
221C2×C3×C4—181no—
222C2×C3×C4C2×D3211yescharacteristic
223C2×C3×C4C2×D3212yes—
224C2×Dic3—32no—
225C2×Dic3—32no—
226C2×Dic3—61no—
227C2×Dic3—61no—
228C2×Dic3—62no—
229C2×Dic3—62no—
230C2×Dic3—62no—
231C2×Dic3—62no—
232C2×Dic3—122no—
233Dic6—62no—
234Dic6—62no—
235Dic6—122no—
236Dic6—364no—
237Dic6—362no—
238C3×C8—182no—
239C3⋊C8—62no—
240C3⋊C8—62no—
241C3⋊C8—122no—
242C2×C32—22no—
243C2×C32—21no—
244C2×C32—22no—
245C2×C32—21no—
246C2×C32—21no—
247C2×C32—22no—
248C2×C32—41no—
249C2×C32—41no—
250C2×C32—41no—
251C2×C32#88512yes—
252C2×C32D3×SD1612yes—
253C2×C32D3×Q1612yes—
254C2×C32#93611yescharacteristic
255C2×C32C2×#21511yescharacteristic
256C2×C32C2×#21611yescharacteristic
257C2×Q8—91no—
258C2×Q8—541no—
259C2×Q8#111311yescharacteristic
260C2×C8—92no—
261C4⋊C4—92no—
262C4⋊C4—181no—
263C4⋊C4—181no—
264C4⋊C4—541no—
265C42—91no—
266C22×C3—21no—
267C22×C3—22no—
268C22×C3—41no—
269C22×C3D3×#6712yes—
270C22×C3#381111yescharacteristic
271C3×C4—22no—
272C3×C4—22no—
273C3×C4—22no—
274C3×C4—42no—
275C3×C4—44no—
276C3×C4—44no—
277C3×C4—41no—
278C3×C4—42no—
279C3×C4—42no—
280C3×C4—92no—
281C3×C4—92no—
282C3×C4—362no—
283C3×C4—361no—
284C3×C4#380412yes—
285C3×C4#380412yes—
286C3×C4#380512yes—
287Dic3—34no—
288Dic3—34no—
289Dic3—62no—
290Dic3—62no—
291Dic3—122no—
292Dic3—122no—
293Dic3—122no—
294Dic3—122no—
295Dic3—242no—
296C32—21no—
297C32—22no—
298C32—41no—
299C32#479212yescomplemented
300C32#507911yescharacteristic, complemented
301Q8—22no—
302Q8—182no—
303Q8—1082no—
304C2×C4—21no—
305C2×C4—91no—
306C2×C4—91no—
307C2×C4—182no—
308C2×C4—181no—
309C2×C4C2×#111311yescharacteristic
310C8—182no—
311C2×C3—21no—
312C2×C3—22no—
313C2×C3—21no—
314C2×C3—22no—
315C2×C3—21no—
316C2×C3—22no—
317C2×C3—41no—
318C2×C3—41no—
319C2×C3—41no—
320C2×C3D3×#21512yes—
321C2×C3#6394011yescharacteristic
322C2×C3D3×#21612yes—
323C2×C3#6394411yescharacteristic
324C2×C3#6395311yescharacteristic
325C2×C3#6395912yes—
326C22#8904911yescharacteristic
327C4—41no—
328C4—92no—
329C4—92no—
330C4—362no—
331C4—361no—
332C4#8904612yes—
333C3—22no—
334C3—21no—
335C3—41no—
336C3#1059153811yescharacteristic, complemented
337C3#1059162312yescomplemented
338C2#1172370811yescharacteristic
339C2#1172371011yescharacteristic
340C2#1172371811yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×Q8#1113
hypocenterC33#487
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
201. C33#487semidirect
299. C32#4792semidirect
300. C32#5079semidirect
336. C3#10591538semidirect
337. C3#10591623semidirect

Maximal subgroups

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

subgroupindex
C3×#641532
C4⋊#62962
C3×#636622
#117220762
#117220702
#105916233
#105915383