Finite Group Databasebeta

← #13167281

Subgroups — #13167281

Order 1728.

860 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 860 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C22×#6295C211yesmaximal, characteristic
2C2×#88993C211yesmaximal, characteristic, complemented
3C3×#63957C211yesmaximal, characteristic, complemented
4#11722772C214yesmaximal, complemented
5#10593493—31nomaximal
6C23×C3×C3⋊D3—21no—
7C22×C3×C3⋊Dic3C2211yescharacteristic
8C2×#6295—21no—
9C2×#6295—22no—
10C2×#6295C414yescomplemented
11C2×#6295C2212yes—
12#88993—22no—
13C2×C3⋊#622C2212yescomplemented
14C32×#217—21no—
15C3×#3724C2212yescomplemented
16C3×Dic32—21no—
17C22×#626—61no—
18C22×#625—31no—
19C2×#3792—31no—
20C3×#877—61no—
21#63957D311yescharacteristic, complemented
22#63739—34no—
23C23×C33D411yescharacteristic
24C22×C3×C3⋊D3—24no—
25C22×C3×C3⋊D3—22no—
26C22×C3×C3⋊D3—44no—
27C2×C3×C3⋊Dic3C2×C412yes—
28C2×C3×C3⋊Dic3D411yescharacteristic
29C2×C3×C3⋊Dic3D412yes—
30C2×C3×C3⋊Dic3C2311yescharacteristic
31C2×C32×Dic3—22no—
32#6295—24no—
33#6295—22no—
34#6295C2×C414yes—
35C3⋊#622—22no—
36#5001—91nomaximal
37C23×C3⋊D3—21no—
38C23×C3×D3—61no—
39C23×C3×D3—61no—
40C23×C3×D3—121no—
41C22×C3⋊Dic3C2×D311yescharacteristic
42C22×C3×Dic3—61no—
43C22×C3×Dic3—61no—
44C2×#626—64no—
45C2×#626—62no—
46C2×#626—62no—
47C2×#626—64no—
48C2×#625—34no—
49C2×#625—32no—
50C2×#625—62no—
51C2×#625—61no—
52#3792—62no—
53C2×#622—32no—
54C32×C22⋊C4—61no—
55C3×#217—21no—
56C3×#217—21no—
57C3×#217—41no—
58C3×Dic3⋊C4—62no—
59C3×C4×Dic3—62no—
60#3724C2×D312yescomplemented
61Dic32—21no—
62C22×C33—22no—
63C22×C33C22⋊C412yes—
64C22×C33C2×D411yescharacteristic
65C2×C3×C3⋊D3—24no—
66C2×C3×C3⋊D3—48no—
67C2×C3×C3⋊D3—44no—
68C3×C3⋊Dic3C22⋊C412yes—
69C3×C3⋊Dic3C22×C411yescharacteristic
70C3×C3⋊Dic3C2×D411yescharacteristic
71C32×Dic3—42no—
72C22×#67—181no—
73C22×D12—91no—
74C3×#117—91no—
75C2×#208—91no—
76#877—61no—
77#809—94no—
78C23×C32—21no—
79C23×C32—21no—
80C23×C32—41no—
81C23×C32—41no—
82C23×C32#6711yescharacteristic
83C22×C3⋊D3—22no—
84C22×C3⋊D3—24no—
85C22×C3⋊D3—44no—
86C22×C3×D3—62no—
87C22×C3×D3—64no—
88C22×C3×D3—64no—
89C22×C3×D3—62no—
90C22×C3×D3—124no—
91C22×C3×D3—122no—
92C22×C3×D3—124no—
93C22×C3×D3—124no—
94C22×C3×D3—128no—
95C2×C32×C4—62no—
96C2×C3⋊Dic3C4×D312yes—
97C2×C3⋊Dic3D1212yes—
98C2×C3⋊Dic3#6711yescharacteristic
99C2×C3⋊Dic3C22×D311yescharacteristic
100C2×C3×Dic3—22no—
101C2×C3×Dic3—22no—
102C2×C3×Dic3—42no—
103C2×C3×Dic3—61no—
104C2×C3×Dic3—61no—
105C2×C3×Dic3—62no—
106C2×C3×Dic3—62no—
107C2×C3×Dic3—62no—
108C2×C3×Dic3—61no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—61no—
111#626—64no—
112#626—68no—
113#626—64no—
114#625—34no—
115#625—64no—
116#625—62no—
117#622—62no—
118#418—271noSylow
119C2×C33—22no—
120C2×C33C2×C22⋊C411yescharacteristic
121C2×C33AGL(1,ℤ/8)12yes—
122C3×C3⋊D3—44no—
123C24×C3—181no—
124C23×D3—61no—
125C23×D3—61no—
126C23×D3—121no—
127C22×C3×C4—91no—
128C2×#67—182no—
129C2×#67—184no—
130C2×#67—184no—
131C2×#67—182no—
132C22×Dic3—61no—
133C22×Dic3—61no—
134C2×D12—92no—
135C2×D12—94no—
136C2×D12—181no—
137C2×D12—182no—
138C3×C4⋊C4—92no—
139C3×C22⋊C4—61no—
140C3×C22⋊C4—61no—
141C3×C22⋊C4—121no—
142C3×C42—181no—
143#217—21no—
144Dic3⋊C4—62no—
145C4×Dic3—62no—
146#208—182no—
147C2×C3⋊C8—92no—
148C22×C32—24no—
149C22×C32—22no—
150C22×C32—21no—
151C22×C32—21no—
152C22×C32—22no—
153C22×C32—22no—
154C22×C32—42no—
155C22×C32—41no—
156C22×C32—41no—
157C22×C32—42no—
158C22×C32—42no—
159C22×C32—44no—
160C22×C32—82no—
161C22×C32(C2×D3)⋊C412yes—
162C22×C32C2×#6711yescharacteristic
163C2×C3⋊D3—24no—
164C2×C3⋊D3—48no—
165C2×C3⋊D3—44no—
166C2×C3×D3—64no—
167C2×C3×D3—64no—
168C2×C3×D3—124no—
169C2×C3×D3—128no—
170C2×C3×D3—128no—
171C2×C3×D3—124no—
172C2×C3×D3—128no—
173C2×C3×D3—1216no—
174C2×C3×D3—124no—
175C32×C4—122no—
176C3⋊Dic3(C2×D3)⋊C412yes—
177C3⋊Dic3C2×C4×D311yescharacteristic
178C3⋊Dic3C2×D1211yescharacteristic
179C3×Dic3—42no—
180C3×Dic3—42no—
181C3×Dic3—62no—
182C3×Dic3—62no—
183C3×Dic3—61no—
184C3×Dic3—61no—
185C3×Dic3—61no—
186C3×Dic3—61no—
187C3×Dic3—82no—
188C22×D4—271no—
189C2×M16—271no—
190#117—91no—
191#102—274no—
192C33#41811yescharacteristic, complemented, Sylow
193C23×C3—21no—
194C23×C3—21no—
195C23×C3—41no—
196C23×C3—41no—
197C23×C3—182no—
198C23×C3—184no—
199C23×C3—364no—
200C23×C3GO+(4,𝔽2)11yescharacteristic
201C22×D3—62no—
202C22×D3—64no—
203C22×D3—62no—
204C22×D3—64no—
205C22×D3—122no—
206C22×D3—124no—
207C22×D3—124no—
208C22×D3—124no—
209C22×D3—128no—
210C2×C3×C4—62no—
211C2×C3×C4—62no—
212C2×C3×C4—91no—
213C2×C3×C4—91no—
214C2×C3×C4—92no—
215C2×C3×C4—92no—
216C2×C3×C4—122no—
217#67—184no—
218#67—188no—
219#67—184no—
220C2×Dic3—22no—
221C2×Dic3—62no—
222C2×Dic3—62no—
223C2×Dic3—62no—
224C2×Dic3—62no—
225C2×Dic3—61no—
226C2×Dic3—61no—
227C2×Dic3—61no—
228C2×Dic3—61no—
229D12—94no—
230D12—182no—
231D12—184no—
232C3⋊C8—182no—
233C2×C32—24no—
234C2×C32—22no—
235C2×C32—21no—
236C2×C32—21no—
237C2×C32—22no—
238C2×C32—22no—
239C2×C32—41no—
240C2×C32—41no—
241C2×C32—42no—
242C2×C32—42no—
243C2×C32—42no—
244C2×C32—44no—
245C2×C32—82no—
246C2×C32C2×(C2×D3)⋊C411yescharacteristic
247C2×C32#94912yes—
248C3⋊D3—44no—
249C3×D3—124no—
250C3×D3—128no—
251C3×D3—124no—
252C24—181no—
253C2×D4—272no—
254C2×D4—274no—
255C2×D4—541no—
256C2×D4—542no—
257C22×C4—91no—
258M16—542no—
259C2×C8—272no—
260C4⋊C4—92no—
261C22⋊C4—61no—
262C42—181no—
263C22×C3—24no—
264C22×C3—22no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—22no—
268C22×C3—22no—
269C22×C3—42no—
270C22×C3—44no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—42no—
274C22×C3—42no—
275C22×C3—82no—
276C22×C3—184no—
277C22×C3—368no—
278C22×C3—364no—
279C22×C3D32⋊C412yes—
280C22×C3C2×GO+(4,𝔽2)11yescharacteristic
281C2×D3—64no—
282C2×D3—64no—
283C2×D3—124no—
284C2×D3—128no—
285C2×D3—128no—
286C2×D3—124no—
287C2×D3—1216no—
288C2×D3—128no—
289C2×D3—124no—
290C3×C4—91no—
291C3×C4—91no—
292C3×C4—92no—
293C3×C4—122no—
294C3×C4—122no—
295C3×C4—242no—
296Dic3—42no—
297Dic3—62no—
298Dic3—61no—
299Dic3—61no—
300Dic3—62no—
301Dic3—61no—
302Dic3—61no—
303C32—21no—
304C32—21no—
305C32—41no—
306C32—41no—
307C32#500111yescharacteristic, complemented
308C23—182no—
309C23—184no—
310C23—364no—
311C23D32⋊D311yescharacteristic
312D4—274no—
313D4—542no—
314D4—544no—
315C2×C4—62no—
316C2×C4—91no—
317C2×C4—91no—
318C2×C4—92no—
319C2×C4—92no—
320C8—542no—
321C2×C3—24no—
322C2×C3—22no—
323C2×C3—21no—
324C2×C3—21no—
325C2×C3—22no—
326C2×C3—22no—
327C2×C3—42no—
328C2×C3—44no—
329C2×C3—41no—
330C2×C3—41no—
331C2×C3—42no—
332C2×C3—42no—
333C2×C3—82no—
334C2×C3—364no—
335C2×C3C2×D32⋊C411yescharacteristic
336C2×C3#6423612yes—
337D3—124no—
338D3—128no—
339D3—124no—
340C22—22no—
341C22—184no—
342C22—368no—
343C22—364no—
344C22#8893212yes—
345C22C2×D32⋊D311yescharacteristic
346C4—91no—
347C4—91no—
348C4—92no—
349C4—122no—
350C3—21no—
351C3—21no—
352C3—41no—
353C3—41no—
354C3#1059349311yescharacteristic, complemented
355C2—22no—
356C2—364no—
357C2C2×#8893211yescharacteristic
358C2#1172432212yes—
subgroupG/H
center, FrattiniC22C2×D32⋊D3
commutatorC3×C3⋊Dic3C22×C4
FittingC23×C33D4
socleC22×C33C2×D4
prime radicalC22×#6295C2
cosocleC2×C3×C3⋊Dic3C23
hypercenterC23D32⋊D3
hypocenterC33#418
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#88993C2semidirect
3. C3×#63957C2semidirect
4. #11722772C2semidirect
10. C2×#6295C4semidirect
13. C2×C3⋊#622C22semidirect
15. C3×#3724C22semidirect
21. #63957D3semidirect
60. #3724C2×D3semidirect
192. C33#418semidirect
307. C32#5001semidirect
354. C3#10593493semidirect

Maximal subgroups

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

subgroupindex
C22×#62952
C2×#889932
C3×#639572
#117227722
#105934933
#50019