Finite Group Databasebeta

← #13167384

Subgroups — #13167384

Order 1728.

504 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 69 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#11723689C211yesmaximal, characteristic
2#11723682C211yesmaximal, characteristic
3C3×#63781C211yesmaximal, characteristic
4#11722791C212yesmaximal
5#11722791C212yesmaximal
6#10593343—31nomaximal
7C22×C3×#157C2212yes—
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×#6294C2211yescharacteristic
10C2×#6294C2212yes—
11C2×#6294C2211yescharacteristic
12C3×C4⋊(C3⋊Dic3)—21no—
13C3×#3750—21no—
14#88815—21no—
15#88815—21no—
16#63900—31no—
17D3×C4⋊Dic3—61no—
18#63883—31no—
19D3×Dic3⋊C4—61no—
20#63781D311yescharacteristic
21#63737—32no—
22#63737—32no—
23C22×C3×C3⋊D3C2311yescharacteristic
24C2×C3×#157—22no—
25C2×C3×#157—22no—
26C2×C3×#157—22no—
27C2×C33×C4D411yescharacteristic
28C2×C3×C3⋊Dic3—21no—
29C2×C3×C3⋊Dic3—21no—
30C2×C3×C3⋊Dic3—22no—
31C2×C3×C3⋊Dic3D411yescharacteristic
32C3×C4×C3⋊D3—22no—
33#6294—21no—
34#6294—22no—
35#6294—24no—
36#6294Q812yes—
37#4659—91nomaximal
38C22×#157C2×D312yes—
39C2×C4×C3⋊D3C2×D311yescharacteristic
40C2×C3×C4×D3—61no—
41C2×C3×C4×D3—61no—
42C2×#624—31no—
43C2×#624—31no—
44C2×#624—32no—
45C2×D3×Dic3—61no—
46C2×D3×Dic3—61no—
47C2×D3×Dic3—62no—
48C4⋊(C3⋊Dic3)—21no—
49#3750—21no—
50C3×C4⋊Dic3—61no—
51C3×C4⋊Dic3—61no—
52C3×C4⋊Dic3—121no—
53C3×Dic3⋊C4—61no—
54C3×Dic3⋊C4—61no—
55C3×Dic3⋊C4—121no—
56#3724—61no—
57#3724—61no—
58Dic3⋊Dic3—61no—
59Dic3⋊Dic3—62no—
60Dic3⋊Dic3—61no—
61C22×C33C2×D411yescharacteristic
62C2×C3×C3⋊D3C2×Q811yescharacteristic
63C2×C3×C3⋊D3#4112yes—
64C2×C3×C3⋊D3#4112yes—
65C2×C3×C3⋊D3#4111yescharacteristic
66C3×#157—42no—
67C33×C4—21no—
68C3×C3⋊Dic3—22no—
69C3×C3⋊Dic3—21no—
70C3×C3⋊Dic3—41no—
71C3×C3⋊Dic3—42no—
72C2×C4⋊Dic3—91no—
73C2×Dic3⋊C4—91no—
74D3×C4⋊C4—181no—
75D3×C4⋊C4—181no—
76C3×#95—91no—
77C3⋊#95—92no—
78C3⋊#95—92no—
79C22×C3⋊D3C22×D311yescharacteristic
80C22×C3×D3—61no—
81C22×C3×D3—61no—
82C2×#157—22no—
83C2×#157—22no—
84C2×#157—22no—
85C2×C32×C4—21no—
86C2×C32×C4—21no—
87C2×C32×C4—41no—
88C2×C32×C4—41no—
89C2×C32×C4D1211yescharacteristic
90C2×C3⋊Dic3—21no—
91C2×C3⋊Dic3—21no—
92C2×C3⋊Dic3—22no—
93C2×C3⋊Dic3D1211yescharacteristic
94C4×C3⋊D3—22no—
95C2×C3×Dic3—61no—
96C2×C3×Dic3—61no—
97C2×C3×Dic3—61no—
98C2×C3×Dic3—61no—
99C2×C3×Dic3—62no—
100C2×C3×Dic3—61no—
101C2×C3×Dic3—61no—
102C2×C3×Dic3—62no—
103C2×C3×Dic3—121no—
104C2×C3×Dic3—121no—
105C2×C3×Dic3—122no—
106C3×C4×D3—64no—
107C3×C4×D3—122no—
108#624—32no—
109#624—61no—
110#624—64no—
111#624—62no—
112D3×Dic3—62no—
113D3×Dic3—62no—
114D3×Dic3—124no—
115D3×Dic3—122no—
116#400—271noSylow
117C2×C33C22⋊Q811yescharacteristic
118C2×C33#12311yescharacteristic
119C2×C33#12411yescharacteristic
120C3×C3⋊D3#12512yes—
121C3×C3⋊D3#12612yes—
122C22×C3×C4—91no—
123C22×C3×C4—92no—
124C22×Dic3—91no—
125C22×Dic3—91no—
126C22×Dic3—92no—
127C2×C4×D3—61no—
128C2×C4×D3—61no—
129C2×C4×D3—181no—
130C2×C4×D3—181no—
131C2×C4×D3—182no—
132C3×C4⋊C4—181no—
133C3×C4⋊C4—181no—
134C4⋊Dic3—61no—
135C4⋊Dic3—61no—
136C4⋊Dic3—121no—
137C4⋊Dic3—181no—
138C4⋊Dic3—181no—
139Dic3⋊C4—61no—
140Dic3⋊C4—61no—
141Dic3⋊C4—121no—
142Dic3⋊C4—181no—
143Dic3⋊C4—182no—
144Dic3⋊C4—181no—
145C22×C32—21no—
146C22×C32—21no—
147C22×C32—41no—
148C22×C32—41no—
149C22×C32C2×D1211yescharacteristic
150C2×C3⋊D3#23512yes—
151C2×C3⋊D3#23711yescharacteristic
152C2×C3⋊D3#23712yes—
153C2×C3⋊D3D3×Q811yescharacteristic
154C2×C3×D3—61no—
155C2×C3×D3—61no—
156C2×C3×D3—61no—
157C2×C3×D3—62no—
158C2×C3×D3—62no—
159C2×C3×D3—62no—
160C2×C3×D3—62no—
161C2×C3×D3—61no—
162#157—42no—
163C32×C4—22no—
164C32×C4—21no—
165C32×C4—41no—
166C32×C4—42no—
167C32×C4—81no—
168C3⋊Dic3—22no—
169C3⋊Dic3—21no—
170C3⋊Dic3—42no—
171C3⋊Dic3—41no—
172C3×Dic3—62no—
173C3×Dic3—62no—
174C3×Dic3—62no—
175C3×Dic3—121no—
176C3×Dic3—122no—
177C3×Dic3—122no—
178C3×Dic3—121no—
179C3×Dic3—122no—
180C3×Dic3—121no—
181C3×Dic3—241no—
182C3×Dic3—242no—
183C2×C4⋊C4—271no—
184C2×C4⋊C4—271no—
185#95—91no—
186#95—272no—
187#95—272no—
188C33#40011yescharacteristic, complemented, Sylow
189C23×C3—91no—
190C22×D3—61no—
191C22×D3—61no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—91no—
197C2×C3×C4—182no—
198C2×C3×C4—181no—
199C2×C3×C4—182no—
200C2×C3×C4—181no—
201C2×C3×C4—182no—
202C2×C3×C4—182no—
203C2×C3×C4—182no—
204C2×C3×C4GO+(4,𝔽2)11yescharacteristic
205C2×Dic3—61no—
206C2×Dic3—61no—
207C2×Dic3—61no—
208C2×Dic3—61no—
209C2×Dic3—61no—
210C2×Dic3—62no—
211C2×Dic3—61no—
212C2×Dic3—62no—
213C2×Dic3—92no—
214C2×Dic3—121no—
215C2×Dic3—121no—
216C2×Dic3—122no—
217C2×Dic3—181no—
218C2×Dic3—184no—
219C2×Dic3—182no—
220C4×D3—64no—
221C4×D3—122no—
222C4×D3—182no—
223C4×D3—182no—
224C4×D3—362no—
225C4×D3—364no—
226C2×C32—21no—
227C2×C32—21no—
228C2×C32—21no—
229C2×C32—21no—
230C2×C32—21no—
231C2×C32—21no—
232C2×C32—41no—
233C2×C32—41no—
234C2×C32—41no—
235C2×C32—41no—
236C2×C32—41no—
237C2×C32—41no—
238C2×C32#87511yescharacteristic
239C2×C32#88611yescharacteristic
240C2×C32#89711yescharacteristic
241C3⋊D3#87912yes—
242C3⋊D3#88912yes—
243C3×D3—62no—
244C3×D3—62no—
245C3×D3—62no—
246C3×D3—62no—
247C22×C4—91no—
248C22×C4—92no—
249C22×C4—271no—
250C22×C4—271no—
251C22×C4—272no—
252C4⋊C4—181no—
253C4⋊C4—181no—
254C4⋊C4—541no—
255C4⋊C4—541no—
256C22×C3—21no—
257C22×C3—21no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—91no—
261C22×C3—91no—
262C22×C3—92no—
263C22×C3—92no—
264C22×C3C2×GO+(4,𝔽2)11yescharacteristic
265C2×D3—62no—
266C2×D3—62no—
267C2×D3—61no—
268C2×D3—61no—
269C2×D3—62no—
270C2×D3—62no—
271C2×D3—61no—
272C2×D3—61no—
273C3×C4—22no—
274C3×C4—21no—
275C3×C4—41no—
276C3×C4—42no—
277C3×C4—81no—
278C3×C4—182no—
279C3×C4—181no—
280C3×C4—362no—
281C3×C4—361no—
282C3×C4—362no—
283Dic3—62no—
284Dic3—62no—
285Dic3—62no—
286Dic3—121no—
287Dic3—122no—
288Dic3—122no—
289Dic3—121no—
290Dic3—122no—
291Dic3—121no—
292Dic3—242no—
293Dic3—241no—
294C32—21no—
295C32—21no—
296C32—41no—
297C32—41no—
298C32#465911yescharacteristic, complemented
299C23—91no—
300C2×C4—91no—
301C2×C4—182no—
302C2×C4—181no—
303C2×C4—182no—
304C2×C4—182no—
305C2×C4—182no—
306C2×C4—182no—
307C2×C4—181no—
308C2×C4—272no—
309C2×C4—541no—
310C2×C4—544no—
311C2×C4—542no—
312C2×C4#157⋊D311yescharacteristic
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—41no—
320C2×C3—41no—
321C2×C3—41no—
322C2×C3—41no—
323C2×C3—41no—
324C2×C3—41no—
325C2×C3—92no—
326C2×C3—92no—
327C2×C3D32⋊Q811yescharacteristic
328C2×C3#6422111yescharacteristic
329C2×C3#6423311yescharacteristic
330D3—62no—
331D3—62no—
332D3—62no—
333D3—62no—
334C22—91no—
335C22—91no—
336C22—92no—
337C22—92no—
338C22C2×#157⋊D311yescharacteristic
339C4—21no—
340C4—182no—
341C4—181no—
342C4—362no—
343C4—362no—
344C4—361no—
345C3—21no—
346C3—21no—
347C3—41no—
348C3—41no—
349C3#1059334311yescharacteristic, complemented
350C2—92no—
351C2—92no—
352C2#1172433311yescharacteristic
353C2#1172433411yescharacteristic
354C2#1172434611yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, prime radical, cosocleC22×C3×C3⋊D3C23
FittingC2×C33×C4D4
socleC22×C33C2×D4
hypercenterC2×C4#157⋊D3
hypocenterC33#400
LCSG ▷ C22×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
188. C33#400semidirect
298. C32#4659semidirect
349. C3#10593343semidirect

Maximal subgroups

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

subgroupindex
#117236892
#117236822
C3×#637812
#117227912
#105933433
#46599