Finite Group Databasebeta

← #13159643

Subgroups — #13159643

Order 1728.

944 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 168 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
1C2×C3×#3743C211yesmaximal, characteristic, complemented
2C4×#6292C211yesmaximal, characteristic
3C2×#88785C211yesmaximal, characteristic, complemented
4#11722190C211yesmaximal, characteristic, complemented
5C32×Dic3⋊C8C211yesmaximal, characteristic, complemented
6#11722023C211yesmaximal, characteristic, complemented
7#11722019C211yesmaximal, characteristic, complemented
8#10592535—31nomaximal
9#10590738—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×#6292C411yescharacteristic
13C2×C33×C8C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C3×#3743—22no—
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19Dic3⋊(C3⋊Dic3)C411yescharacteristic
20#88804C411yescharacteristic
21#88803C411yescharacteristic
22#88785—22no—
23C2×#3743D311yescharacteristic, complemented
24C2×C8×C3⋊D3—31no—
25C4×C4⋊(C3⋊D3)—31no—
26C2×C3×ΓL(1,𝔽25)—34no—
27C4×#626—34no—
28C2×#3710—34no—
29C32×C4⋊C8—31no—
30#63647—31no—
31#63647—31no—
32#63647—34no—
33#63647—64no—
34#63630—31no—
35C3×Dic3⋊C8D314yescomplemented
36(C2×D3)⋊(C3⋊C8)—34no—
37#63554—34no—
38C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
39C22×C3×C3⋊D3C2×C411yescharacteristic
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
42C4×C3⋊(C3⋊D3)—21no—
43C4×C3⋊(C3⋊D3)—21no—
44C2×C3×C3⋊Dic3C2×C411yescharacteristic
45C3×C4×C3⋊D3—21no—
46C3×C4×C3⋊D3—21no—
47C2×C32×Dic3C2×C411yescharacteristic
48C2×C32×Dic3C2×C411yescharacteristic
49#6292—22no—
50C33×C8—21no—
51C3⋊(C3⋊(C3⋊C8))—21no—
52C3×C3⋊(C3⋊C8)D412yescomplemented
53C32×C3⋊C8—21no—
54#5127—91no—
55#4836—94no—
56C2×C4⋊(C3⋊D3)—31no—
57C2×C4×C3⋊D3—31no—
58C2×C4×C3⋊D3—34no—
59C2×C4×C3⋊D3—64no—
60C2×C4×C3⋊D3C2×D311yescharacteristic
61C2×C3×C4×D3—34no—
62C2×#626—34no—
63C2×C32×C8—24no—
64C2×C32×C8—31no—
65C2×C32×C8C2×D314yes—
66C2×C32×C8C2×D311yescharacteristic
67C32×C42—31no—
68#3752—31no—
69#3752—31no—
70C4⋊(C3⋊Dic3)—31no—
71C2×C3⋊(C3⋊C8)—31no—
72C2×C3⋊(C3⋊C8)—34no—
73C2×C3⋊(C3⋊C8)—64no—
74C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
75#3743—22no—
76C8×C3⋊D3—62no—
77C3×C4×Dic3C2×D314yes—
78C2×C3×C3⋊C8—34no—
79C2×C3×C3⋊C8C2×D314yes—
80C3×ΓL(1,𝔽25)—68no—
81Dic3⋊Dic3—34no—
82#3722—34no—
83(C2×D3)⋊Dic3—34no—
84#3710—68no—
85C22×C33C22×C411yescharacteristic
86C2×C3⋊(C3⋊D3)—21no—
87C2×C3⋊(C3⋊D3)—21no—
88C2×C3⋊(C3⋊D3)M1612yescomplemented
89C2×C3×C3⋊D3—21no—
90C2×C3×C3⋊D3—21no—
91C2×C3×C3⋊D3—21no—
92C33×C4C2×D411yescharacteristic
93C33×C4#4111yescharacteristic
94C3⋊(C3⋊Dic3)—21no—
95C3×C3⋊Dic3—21no—
96C32×Dic3—21no—
97C32×Dic3—21no—
98C2×C3×M16—91no—
99C4×#67—91no—
100C2×ΓL(1,𝔽25)—34no—
101C2×D3×C8—94no—
102C2×D3×C8—91no—
103C4×D12—94no—
104C3×C4⋊C8—34no—
105#830—91no—
106(C2×D3)⋊C8—94no—
107(C2×D3)⋊C8—94no—
108(C2×D3)⋊C8—91no—
109(C2×D3)⋊C8—184no—
110#815—91no—
111Dic3⋊C8C3⋊D311yescharacteristic, complemented
112#803—94no—
113C22×C3⋊D3—31no—
114C22×C3⋊D3—34no—
115C22×C3⋊D3—64no—
116C22×C3⋊D3C4×D311yescharacteristic
117C22×C3×D3—34no—
118C2×C32×C4—24no—
119C2×C32×C4—31no—
120C2×C32×C4—31no—
121C2×C32×C4C22×D314yes—
122C2×C32×C4C22×D311yescharacteristic
123C2×C3⋊Dic3—31no—
124C2×C3⋊Dic3—34no—
125C2×C3⋊Dic3—64no—
126C2×C3⋊Dic3C4×D311yescharacteristic
127C4⋊(C3⋊D3)—62no—
128C4×C3⋊D3—21no—
129C4×C3⋊D3—21no—
130C4×C3⋊D3—61no—
131C4×C3⋊D3—61no—
132C4×C3⋊D3—64no—
133C4×C3⋊D3—64no—
134C4×C3⋊D3—124no—
135C4×C3⋊D3—124no—
136C2×C3×Dic3—34no—
137C2×C3×Dic3C4×D314yes—
138C2×C3×Dic3C4×D314yes—
139C3×C4×D3—64no—
140C3×C4×D3—64no—
141#626—68no—
142C32×C8—21no—
143C32×C8—24no—
144C32×C8—44no—
145C32×C8—61no—
146C3⋊(C3⋊C8)—61no—
147C3⋊(C3⋊C8)—64no—
148C3⋊(C3⋊C8)—124no—
149C3⋊(C3⋊C8)#6712yescomplemented
150C3×C3⋊C8—24no—
151C3×C3⋊C8—38no—
152#435—271noSylow
153C2×C33C4×D411yescharacteristic
154C2×C33C2×M1611yescharacteristic
155C2×C33#13111yescharacteristic
156C3⋊(C3⋊D3)—22no—
157C3×C3⋊D3—41no—
158C22×C3×C4—91no—
159C2×#67—91no—
160C2×D12—94no—
161C2×C4×D3—34no—
162C2×C4×D3—94no—
163C2×C4×D3—91no—
164C2×C4×D3—184no—
165C3×M16—182no—
166C2×C3×C8—24no—
167C2×C3×C8—34no—
168C2×C3×C8—91no—
169C2×C3×C8D3214yes—
170C2×C3×C8C2×C3⋊D311yescharacteristic
171C3×C42—34no—
172#217—91no—
173(C2×D3)⋊C4—94no—
174(C2×D3)⋊C4—94no—
175(C2×D3)⋊C4—91no—
176C4⋊Dic3—94no—
177Dic3⋊C4—91no—
178C4×Dic3C2×C3⋊D311yescharacteristic
179C2×C3⋊C8—34no—
180C2×C3⋊C8—94no—
181C2×C3⋊C8—91no—
182C2×C3⋊C8—184no—
183C2×C3⋊C8C2×C3⋊D311yescharacteristic
184ΓL(1,𝔽25)—68no—
185D3×C8—182no—
186D3×C8—188no—
187C22×C32—24no—
188C22×C32C2×C4×D314yes—
189C22×C32C2×C4×D311yescharacteristic
190C2×C3⋊D3—21no—
191C2×C3⋊D3—21no—
192C2×C3⋊D3—21no—
193C2×C3⋊D3—32no—
194C2×C3⋊D3—38no—
195C2×C3⋊D3—68no—
196C2×C3⋊D3—61no—
197C2×C3⋊D3—61no—
198C2×C3⋊D3—64no—
199C2×C3⋊D3—64no—
200C2×C3⋊D3—124no—
201C2×C3⋊D3—124no—
202C2×C3×D3—64no—
203C2×C3×D3—64no—
204C2×C3×D3—64no—
205C32×C4—24no—
206C32×C4—24no—
207C32×C4—61no—
208C32×C4—61no—
209C32×C4#23511yescharacteristic
210C32×C4D3×D414yes—
211C32×C4#23914yes—
212C32×C4C2×#6711yescharacteristic
213C3⋊Dic3—21no—
214C3⋊Dic3—61no—
215C3⋊Dic3—64no—
216C3⋊Dic3—124no—
217C3×Dic3—24no—
218C3×Dic3—24no—
219C3×Dic3—64no—
220C2×M16—91no—
221C22×C8—271no—
222C4×D4—271no—
223C4⋊C8—31no—
224C22⋊C8—271no—
225C22⋊C8—271no—
226#97—271no—
227C33#43511yescharacteristic, complemented, Sylow
228C23×C3—91no—
229C22×D3—34no—
230C22×D3—94no—
231C22×D3—91no—
232C22×D3—184no—
233C2×C3×C4—24no—
234C2×C3×C4—34no—
235C2×C3×C4—34no—
236C2×C3×C4—91no—
237C2×C3×C4—181no—
238C2×C3×C4—181no—
239C2×C3×C4C2×D3214yes—
240C2×C3×C4C22×C3⋊D311yescharacteristic
241#67—182no—
242C2×Dic3—34no—
243C2×Dic3—94no—
244C2×Dic3—91no—
245C2×Dic3—184no—
246C2×Dic3C4×C3⋊D311yescharacteristic
247C2×Dic3C4×C3⋊D311yescharacteristic
248D12—188no—
249C4×D3—64no—
250C4×D3—64no—
251C4×D3—184no—
252C4×D3—184no—
253C4×D3—181no—
254C4×D3—181no—
255C4×D3—364no—
256C4×D3—364no—
257C3×C8—24no—
258C3×C8—21no—
259C3×C8—44no—
260C3×C8—64no—
261C3×C8—92no—
262C3⋊C8—21no—
263C3⋊C8—38no—
264C3⋊C8—184no—
265C3⋊C8—181no—
266C3⋊C8—364no—
267C2×C32—24no—
268C2×C32—24no—
269C2×C32—24no—
270C2×C32#89314yes—
271C2×C32#90114yes—
272C2×C32D3×M1614yes—
273C2×C32D3×M1611yescharacteristic
274C2×C32#90711yescharacteristic
275C2×C32C4×#6711yescharacteristic
276C3⋊D3—41no—
277C3⋊D3—62no—
278C3⋊D3—68no—
279C3⋊D3—128no—
280C3×D3—124no—
281C2×D4—271no—
282C22×C4—91no—
283C22×C4—271no—
284M16—182no—
285C2×C8—31no—
286C2×C8—91no—
287C2×C8—271no—
288C2×C8—542no—
289C2×C8D3×C3⋊D311yescharacteristic
290C4⋊C4—271no—
291C22⋊C4—271no—
292C22⋊C4—271no—
293C42—31no—
294C22×C3—24no—
295C22×C3—181no—
296C22×C3—181no—
297C22×C3—181no—
298C22×C3C4×D3214yes—
299C22×C3C2×C4×C3⋊D311yescharacteristic
300C2×D3—64no—
301C2×D3—64no—
302C2×D3—64no—
303C2×D3—98no—
304C2×D3—92no—
305C2×D3—188no—
306C2×D3—184no—
307C2×D3—181no—
308C2×D3—181no—
309C2×D3—184no—
310C2×D3—364no—
311C2×D3—364no—
312C3×C4—24no—
313C3×C4—24no—
314C3×C4—64no—
315C3×C4—64no—
316C3×C4—181no—
317C3×C4#379914yes—
318C3×C4D3×#6714yes—
319C3×C4D4×C3⋊D311yescharacteristic
320C3×C4#383211yescharacteristic
321Dic3—21no—
322Dic3—21no—
323Dic3—64no—
324Dic3—181no—
325Dic3—184no—
326Dic3—364no—
327C32—24no—
328C32#483614yescomplemented
329C32#512711yescharacteristic, complemented
330C23—91no—
331C23—271no—
332D4—542no—
333C2×C4—31no—
334C2×C4—31no—
335C2×C4—91no—
336C2×C4—181no—
337C2×C4—181no—
338C2×C4—271no—
339C2×C4—541no—
340C2×C4—541no—
341C2×C4C2×D3×C3⋊D311yescharacteristic
342C8—21no—
343C8—61no—
344C8—92no—
345C8—541no—
346C2×C3—24no—
347C2×C3—24no—
348C2×C3—24no—
349C2×C3—361no—
350C2×C3#6379114yes—
351C2×C3#6380414yes—
352C2×C3#6390414yes—
353C2×C3#6410211yescharacteristic
354C2×C3#6411011yescharacteristic
355C2×C3M16×C3⋊D311yescharacteristic
356D3—124no—
357D3—188no—
358D3—182no—
359D3—368no—
360C22—181no—
361C22—181no—
362C22—181no—
363C22—272no—
364C22—541no—
365C22—541no—
366C22C4×D3×C3⋊D311yescharacteristic
367C4—61no—
368C4—61no—
369C4—181no—
370C4—541no—
371C4#8901811yescharacteristic
372C4C3⋊D3×#6711yescharacteristic
373C3—24no—
374C3#1059073814yescomplemented
375C3#1059253511yescharacteristic, complemented
376C2—361no—
377C2—542no—
378C2#1172331211yescharacteristic
379C2#1172333211yescharacteristic
380C2#1172351211yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×C3⋊D3
commutator, socleC22×C33C22×C4
FittingC2×C33×C8C22
prime radicalC2×#6292C4
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#435
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3743C2semidirect
3. C2×#88785C2semidirect
4. #11722190C2semidirect
5. C32×Dic3⋊C8C2semidirect
6. #11722023C2semidirect
7. #11722019C2semidirect
23. C2×#3743D3semidirect
35. C3×Dic3⋊C8D3semidirect
52. C3×C3⋊(C3⋊C8)D4semidirect
88. C2×C3⋊(C3⋊D3)M16semidirect
111. Dic3⋊C8C3⋊D3semidirect
149. C3⋊(C3⋊C8)#67semidirect
227. C33#435semidirect
328. C32#4836semidirect
329. C32#5127semidirect
374. C3#10590738semidirect
375. C3#10592535semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37432
C4×#62922
C2×#887852
#117221902
C32×Dic3⋊C82
#117220232
#117220192
#105925353
#105907383