Finite Group Databasebeta

← #13166992

Subgroups — #13166992

Order 1728.

434 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 55 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#11723679C211yesmaximal, characteristic, complemented
2C3×C4×#625C211yesmaximal, characteristic
3#11723104C211yesmaximal, characteristic, complemented
4#11722779C211yesmaximal, characteristic, complemented
5#11722778C211yesmaximal, characteristic, complemented
6#11722773C211yesmaximal, characteristic
7#11722773C211yesmaximal, characteristic
8#10593350—31nomaximal
9C2×C32×C4×D3—21no—
10C2×C3×#625C2211yescharacteristic
11C2×C3⋊#622C2211yescharacteristic
12C2×C3⋊#622C2211yescharacteristic
13C3×#3750—21no—
14C3×C4×C3⋊Dic3C2211yescharacteristic
15#88815—21no—
16#88815C2211yescharacteristic
17#88815C2211yescharacteristic
18C3×#3724C2211yescharacteristic
19C3×(C2×D3)⋊Dic3—21no—
20C3×C4×#67—61no—
21C4×#625D311yescharacteristic
22#63870—31no—
23#63867—61no—
24#63779—31no—
25#63743—31no—
26#63742—31no—
27#63739—31no—
28#63739—31no—
29C22×C32×D3—21no—
30C2×C33×C4D411yescharacteristic
31C2×C3×C3⋊Dic3—21no—
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3D411yescharacteristic
34C2×C3×C3⋊Dic3C2311yescharacteristic
35C2×C32×Dic3—21no—
36C32×C4×D3—42no—
37C3×#625—21no—
38C3×#625Q812yes—
39C3⋊#622—21no—
40C3⋊#622—21no—
41#5009—91nomaximal
42C22×C32×C4—61no—
43C2×C3×#67—61no—
44C2×C3×C4×D3—21no—
45C2×C3×C4×D3—21no—
46C2×C3×C4×D3—41no—
47C2×#625C2×D311yescharacteristic
48C2×#622—31no—
49C2×#622—31no—
50#3750—21no—
51C4×C3⋊Dic3C2×D311yescharacteristic
52C3×#217—61no—
53C3×(C2×D3)⋊C4—61no—
54C3×Dic3⋊C4—61no—
55C3×Dic3⋊C4—61no—
56C3×Dic3⋊C4—61no—
57C3×Dic3⋊C4—121no—
58C3×C4×Dic3—61no—
59C3×C4×Dic3—61no—
60#3724—31no—
61#3724—31no—
62#3724—61no—
63#3724C2×D311yescharacteristic
64Dic3⋊Dic3—61no—
65Dic3⋊Dic3—61no—
66Dic3⋊Dic3—61no—
67Dic3⋊Dic3—61no—
68(C2×D3)⋊Dic3—21no—
69C22×C33C2×D411yescharacteristic
70C2×C32×D3—22no—
71C2×C32×D3—41no—
72C2×C32×D3—41no—
73C33×C4—21no—
74C3×C3⋊Dic3—21no—
75C3×C3⋊Dic3—41no—
76C3×C3⋊Dic3—41no—
77C3×C3⋊Dic3C2×Q811yescharacteristic
78C3×C3⋊Dic3#4111yescharacteristic
79C32×Dic3—41no—
80C3×C4×D4—91no—
81C4×#67—61no—
82#889—181no—
83#870—91no—
84#832—91no—
85#832—91no—
86#808—91no—
87#807—91no—
88C4⋊(C3⋊C8)—91no—
89C23×C32—61no—
90C22×C3×D3—21no—
91C22×C3×D3—21no—
92C22×C3×D3—41no—
93C2×C32×C4—21no—
94C2×C32×C4—21no—
95C2×C32×C4—41no—
96C2×C32×C4—41no—
97C2×C32×C4—61no—
98C2×C32×C4—122no—
99C2×C32×C4#6711yescharacteristic
100C2×C3⋊Dic3—21no—
101C2×C3⋊Dic3—21no—
102C2×C3⋊Dic3#6711yescharacteristic
103C2×C3⋊Dic3C22×D311yescharacteristic
104C3×#67—62no—
105C3×#67—62no—
106C2×C3×Dic3—21no—
107C2×C3×Dic3—21no—
108C2×C3×Dic3—41no—
109C2×C3×Dic3—61no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—61no—
112C2×C3×Dic3—61no—
113C2×C3×Dic3—61no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—121no—
118C2×C3×Dic3—121no—
119C3×C4×D3—42no—
120C3×C4×D3—42no—
121C3×C4×D3—44no—
122#625—21no—
123#625Dic612yes—
124#622—61no—
125#622—61no—
126#478—271noSylow
127C2×C33C22⋊Q811yescharacteristic
128C2×C33#13511yescharacteristic
129C2×C33AGL(1,ℤ/8)11yescharacteristic
130C32×D3—42no—
131C2×C3×D4—91no—
132C22×C3×C4—61no—
133C22×C3×C4—61no—
134C22×C3×C4—121no—
135C2×#67—61no—
136C2×C4×D3—21no—
137C3×C4⋊C4—91no—
138C3×C4⋊C4—181no—
139C3×C22⋊C4—181no—
140C3×C42—91no—
141#217—61no—
142(C2×D3)⋊C4—61no—
143C4⋊Dic3—91no—
144C4⋊Dic3—91no—
145C4⋊Dic3—181no—
146Dic3⋊C4—61no—
147Dic3⋊C4—61no—
148Dic3⋊C4—61no—
149Dic3⋊C4—121no—
150Dic3⋊C4—181no—
151Dic3⋊C4—181no—
152Dic3⋊C4—181no—
153Dic3⋊C4—181no—
154C4×Dic3—61no—
155C4×Dic3—61no—
156C2×C3⋊C8—91no—
157C2×C3⋊C8—91no—
158C22×C32—21no—
159C22×C32—21no—
160C22×C32—41no—
161C22×C32—41no—
162C22×C32—62no—
163C22×C32—121no—
164C22×C32—121no—
165C22×C32C2×#6711yescharacteristic
166C2×C3×D3—22no—
167C2×C3×D3—22no—
168C2×C3×D3—42no—
169C2×C3×D3—41no—
170C2×C3×D3—41no—
171C2×C3×D3—42no—
172C2×C3×D3—42no—
173C2×C3×D3—41no—
174C2×C3×D3—41no—
175C32×C4—22no—
176C32×C4—21no—
177C32×C4—41no—
178C32×C4—42no—
179C32×C4—81no—
180C32×C4—121no—
181C3⋊Dic3—21no—
182C3⋊Dic3—41no—
183C3⋊Dic3—41no—
184C3⋊Dic3C2×Dic611yescharacteristic
185C3⋊Dic3#23511yescharacteristic
186C3×Dic3—41no—
187C3×Dic3—42no—
188C3×Dic3—41no—
189C3×Dic3—61no—
190C3×Dic3—61no—
191C3×Dic3—61no—
192C3×Dic3—61no—
193C3×Dic3—121no—
194C3×Dic3—121no—
195C3×Dic3—121no—
196C3×Dic3—121no—
197C3×Dic3—122no—
198C3×Dic3—121no—
199C3×Dic3—121no—
200C3×Dic3—241no—
201#125—271no—
202C4×D4—91no—
203#107—271no—
204#106—271no—
205C4⋊C8—271no—
206#102—271no—
207#102—271no—
208C33#47811yescharacteristic, complemented, Sylow
209C23×C3—61no—
210C23×C3—61no—
211C23×C3—121no—
212C22×D3—21no—
213C3×D4—92no—
214C3×D4—181no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—61no—
220C2×C3×C4—61no—
221C2×C3×C4—91no—
222C2×C3×C4—91no—
223C2×C3×C4—121no—
224C2×C3×C4—122no—
225C2×C3×C4—122no—
226C2×C3×C4—124no—
227C2×C3×C4—181no—
228C2×C3×C4—181no—
229C2×C3×C4GO+(4,𝔽2)11yescharacteristic
230#67—62no—
231#67—62no—
232C2×Dic3—21no—
233C2×Dic3—61no—
234C2×Dic3—61no—
235C2×Dic3—61no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—61no—
239C2×Dic3—61no—
240C2×Dic3—61no—
241C2×Dic3—121no—
242C2×Dic3—121no—
243C4×D3—42no—
244C3⋊C8—181no—
245C3⋊C8—181no—
246C2×C32—21no—
247C2×C32—21no—
248C2×C32—21no—
249C2×C32—21no—
250C2×C32—21no—
251C2×C32—21no—
252C2×C32—41no—
253C2×C32—41no—
254C2×C32—41no—
255C2×C32—41no—
256C2×C32—41no—
257C2×C32—41no—
258C2×C32—122no—
259C2×C32#92411yescharacteristic
260C2×C32#93211yescharacteristic
261C2×C32#95011yescharacteristic
262C3×D3—42no—
263C3×D3—44no—
264C3×D3—42no—
265C2×D4—91no—
266C22×C4—61no—
267C2×C8—271no—
268C2×C8—271no—
269C4⋊C4—91no—
270C4⋊C4—181no—
271C4⋊C4—271no—
272C4⋊C4—271no—
273C4⋊C4—541no—
274C22⋊C4—181no—
275C42—91no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—62no—
281C22×C3—62no—
282C22×C3—122no—
283C22×C3—121no—
284C22×C3—121no—
285C22×C3—122no—
286C22×C3—122no—
287C22×C3—121no—
288C22×C3—121no—
289C22×C3C2×GO+(4,𝔽2)11yescharacteristic
290C2×D3—22no—
291C2×D3—41no—
292C2×D3—41no—
293C3×C4—22no—
294C3×C4—21no—
295C3×C4—41no—
296C3×C4—42no—
297C3×C4—81no—
298C3×C4—91no—
299C3×C4—91no—
300C3×C4—121no—
301C3×C4—122no—
302C3×C4—121no—
303C3×C4—181no—
304C3×C4—361no—
305C3×C4—361no—
306Dic3—41no—
307Dic3—61no—
308Dic3—61no—
309Dic3—61no—
310Dic3—61no—
311Dic3—121no—
312Dic3—121no—
313Dic3—121no—
314Dic3—121no—
315Dic3—122no—
316Dic3—121no—
317Dic3—121no—
318Dic3—241no—
319C32—21no—
320C32—21no—
321C32—41no—
322C32—41no—
323C32#500911yescharacteristic, complemented
324C23—61no—
325D4—92no—
326D4—181no—
327C2×C4—61no—
328C2×C4—91no—
329C2×C4—91no—
330C2×C4—122no—
331C2×C4—181no—
332C2×C4—181no—
333C2×C4D32⋊D311yescharacteristic
334C8—541no—
335C8—541no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—21no—
342C2×C3—41no—
343C2×C3—41no—
344C2×C3—41no—
345C2×C3—41no—
346C2×C3—41no—
347C2×C3—41no—
348C2×C3—122no—
349C2×C3—124no—
350C2×C3—122no—
351C2×C3D32⋊Q811yescharacteristic
352C2×C3#6422311yescharacteristic
353C2×C3#6423611yescharacteristic
354D3—42no—
355C22—62no—
356C22—121no—
357C22—121no—
358C22C2×D32⋊D311yescharacteristic
359C4—21no—
360C4—91no—
361C4—91no—
362C4—121no—
363C4—181no—
364C4—361no—
365C4—361no—
366C3—21no—
367C3—21no—
368C3—41no—
369C3—41no—
370C3#1059335011yescharacteristic, complemented
371C2—122no—
372C2#1172430011yescharacteristic
373C2#1172430311yescharacteristic
374C2#1172432311yescharacteristic
subgroupG/H
center, FrattiniC22C2×D32⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×C3×#625C22
hypercenterC2×C4D32⋊D3
hypocenterC33#478
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723679C2semidirect
3. #11723104C2semidirect
4. #11722779C2semidirect
5. #11722778C2semidirect
208. C33#478semidirect
323. C32#5009semidirect
370. C3#10593350semidirect

Maximal subgroups

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

subgroupindex
#117236792
C3×C4×#6252
#117231042
#117227792
#117227782
#117227732
#105933503
#50099