Finite Group Databasebeta

← #13155549

Subgroups — #13155549

Order 1728.

754 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 152 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×D3×C3⋊(C3⋊C8)C212yesmaximal, complemented
2C3×C4×#625C211yesmaximal, characteristic
3#11722078C211yesmaximal, characteristic, complemented
4C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
5C3×(C2×D3)⋊(C3⋊C8)C212yesmaximal, complemented
6#67⋊(C3⋊C8)—32nomaximal
7#10590718—31nomaximal
8C2×C32×C4×D3C2212yes—
9C2×C3×#625C411yescharacteristic
10C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C2×C3×C3⋊(C3⋊C8)C2212yes—
13C2×C32×C3⋊C8C2211yescharacteristic
14D3×C3⋊(C3⋊C8)—24no—
15C3×#3724C411yescharacteristic
16C3×(C2×D3)⋊Dic3C412yes—
17C22×C3⋊(C3⋊C8)—32no—
18C3×C4×#67—32no—
19C4×#625D311yescharacteristic
20C2×D3×C3⋊C8—32no—
21C2×D3×C3⋊C8—32no—
22C2×D3×C3⋊C8—62no—
23C8×C3⋊Dic3—31no—
24C3×#830—32no—
25C3×(C2×D3)⋊C8—32no—
26#63572—31no—
27Dic3⋊(C3⋊C8)—32no—
28(C2×D3)⋊(C3⋊C8)D312yescomplemented
29Dic3×C3⋊C8—32no—
30Dic3×C3⋊C8—61no—
31C22×C32×D3C2×C412yes—
32C2×C33×C4C2311yescharacteristic
33C2×C3×C3⋊Dic3C2×C411yescharacteristic
34C2×C3×C3⋊Dic3C2×C411yescharacteristic
35C2×C32×Dic3C2×C412yes—
36C32×C4×D3—22no—
37C32×C4×D3—22no—
38C3×#625C814yescomplemented
39C3⋊(C3⋊(C3⋊C8))D412yescomplemented
40C3×C3⋊(C3⋊C8)—22no—
41C32×C3⋊C8—21no—
42D4×C3⋊C8—91no—
43#67⋊C8—92no—
44C22×C32×C4—32no—
45C2×C3×#67—32no—
46C2×C3×C4×D3—22no—
47C2×C3×C4×D3C2×D312yes—
48C2×C3×C4×D3C2×D312yes—
49C2×#625Dic311yescharacteristic
50C2×C32×C8—31no—
51C4×C3⋊Dic3C2×D311yescharacteristic
52C2×C3⋊(C3⋊C8)—32no—
53C2×C3⋊(C3⋊C8)—31no—
54C2×C3⋊(C3⋊C8)—64no—
55C2×C3⋊(C3⋊C8)—61no—
56C2×C3⋊(C3⋊C8)—62no—
57C2×C3⋊(C3⋊C8)—121no—
58C2×C3⋊(C3⋊C8)C2×D312yes—
59C3×#217—32no—
60C3×(C2×D3)⋊C4—32no—
61C3×Dic3⋊C4—32no—
62C3×C4×Dic3—32no—
63C3×C4×Dic3—61no—
64C2×C3×C3⋊C8—21no—
65C2×C3×C3⋊C8—32no—
66C2×C3×C3⋊C8—32no—
67C2×C3×C3⋊C8—62no—
68C2×C3×C3⋊C8C2×D312yes—
69#3724Dic311yescharacteristic
70(C2×D3)⋊Dic3Dic312yes—
71D3×C3⋊C8—64no—
72D3×C3⋊C8—64no—
73D3×C3⋊C8—68no—
74C22×C33C22×C411yescharacteristic
75C2×C32×D3—22no—
76C2×C32×D3—22no—
77C2×C32×D3C2×C814yescomplemented
78C33×C4C2×D411yescharacteristic
79C33×C4#4111yescharacteristic
80C3×C3⋊Dic3—21no—
81C3×C3⋊Dic3C2×C812yescomplemented
82C32×Dic3—22no—
83C3×C4×D4—91no—
84C4×#67—32no—
85C22×C3⋊C8—92no—
86C22×C3⋊C8—92no—
87C22×C3⋊C8—182no—
88C2×D3×C8—92no—
89C3×C22⋊C8—92no—
90#830—32no—
91(C2×D3)⋊C8—32no—
92Dic3⋊C8—92no—
93C8×Dic3—92no—
94C8×Dic3—181no—
95C4⋊(C3⋊C8)—91no—
96C4×C3⋊C8—91no—
97C23×C32—32no—
98C22×C3×D3—22no—
99C22×C3×D3C4×D312yes—
100C22×C3×D3C2×Dic312yes—
101C2×C32×C4—21no—
102C2×C32×C4—22no—
103C2×C32×C4—32no—
104C2×C32×C4—41no—
105C2×C32×C4—62no—
106C2×C32×C4—62no—
107C2×C32×C4C22×D312yes—
108C2×C32×C4C22×D311yescharacteristic
109C2×C3⋊Dic3C2×Dic311yescharacteristic
110C2×C3⋊Dic3C2×Dic311yescharacteristic
111C3×#67—38no—
112C2×C3×Dic3—22no—
113C2×C3×Dic3—32no—
114C2×C3×Dic3—32no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3C4×D312yes—
118C2×C3×Dic3C2×Dic312yes—
119C3×C4×D3—24no—
120C3×C4×D3—24no—
121C3×C4×D3—22no—
122C3×C4×D3—22no—
123C3×C4×D3—22no—
124C3×C4×D3—22no—
125#625C3⋊C814yescomplemented
126C32×C8—61no—
127C3⋊(C3⋊C8)—22no—
128C3⋊(C3⋊C8)—32no—
129C3⋊(C3⋊C8)—34no—
130C3⋊(C3⋊C8)—64no—
131C3⋊(C3⋊C8)—62no—
132C3⋊(C3⋊C8)—122no—
133C3×C3⋊C8—22no—
134C3×C3⋊C8—22no—
135C3×C3⋊C8—64no—
136C3×C3⋊C8—62no—
137C3×C3⋊C8—62no—
138C8×D4—271noSylow
139C2×C33C4×D411yescharacteristic
140C2×C33C22×C811yescharacteristic
141C2×C33#13111yescharacteristic
142C32×D3—24no—
143C2×C3×D4—91no—
144C22×C3×C4—32no—
145C22×C3×C4—32no—
146C22×C3×C4—62no—
147C2×#67—32no—
148C2×C4×D3D3212yes—
149C2×C3×C8—32no—
150C2×C3×C8—61no—
151C2×C3×C8—92no—
152C3×C4⋊C4—91no—
153C3×C22⋊C4—92no—
154C3×C42—91no—
155#217—32no—
156(C2×D3)⋊C4—32no—
157Dic3⋊C4—32no—
158C4×Dic3—32no—
159C4×Dic3—61no—
160C2×C3⋊C8—32no—
161C2×C3⋊C8—32no—
162C2×C3⋊C8—62no—
163C2×C3⋊C8—91no—
164C2×C3⋊C8—92no—
165C2×C3⋊C8—181no—
166C2×C3⋊C8—182no—
167C2×C3⋊C8—184no—
168C2×C3⋊C8—184no—
169C2×C3⋊C8—188no—
170C2×C3⋊C8—361no—
171C2×C3⋊C8D3211yescharacteristic
172D3×C8—184no—
173C22×C32—21no—
174C22×C32—22no—
175C22×C32—34no—
176C22×C32—41no—
177C22×C32—62no—
178C22×C32—62no—
179C22×C32C2×C4×D312yes—
180C22×C32C22×Dic311yescharacteristic
181C2×C3×D3—24no—
182C2×C3×D3—24no—
183C2×C3×D3—22no—
184C2×C3×D3—22no—
185C2×C3×D3—22no—
186C2×C3×D3—22no—
187C2×C3×D3—24no—
188C2×C3×D3D3×C814yescomplemented
189C2×C3×D3C2×C3⋊C814yescomplemented
190C32×C4—21no—
191C32×C4—21no—
192C32×C4—22no—
193C32×C4—22no—
194C32×C4—41no—
195C32×C4—41no—
196C32×C4—62no—
197C32×C4D3×D412yes—
198C32×C4D3×D411yescharacteristic
199C32×C4#23712yes—
200C32×C4#23711yescharacteristic
201C3⋊Dic3—21no—
202C3⋊Dic3C2×C3⋊C812yescomplemented
203C3×Dic3—24no—
204C3×Dic3—22no—
205C3×Dic3—22no—
206C3×Dic3—34no—
207C3×Dic3—62no—
208C3×Dic3—62no—
209C3×Dic3—62no—
210C22×C8—272no—
211C4×D4—91no—
212C4⋊C8—271no—
213C22⋊C8—92no—
214C4×C8—271no—
215C33C8×D411yescharacteristic, complemented, Sylow
216C23×C3—32no—
217C23×C3—32no—
218C23×C3—62no—
219C22×D3D3×Dic312yes—
220C3×D4—94no—
221C2×C3×C4—22no—
222C2×C3×C4—21no—
223C2×C3×C4—32no—
224C2×C3×C4—32no—
225C2×C3×C4—41no—
226C2×C3×C4—62no—
227C2×C3×C4—64no—
228C2×C3×C4—64no—
229C2×C3×C4—62no—
230C2×C3×C4—62no—
231C2×C3×C4—62no—
232C2×C3×C4—62no—
233C2×C3×C4—91no—
234C2×C3×C4—91no—
235C2×C3×C4C2×D3211yescharacteristic
236C2×C3×C4C2×D3212yes—
237#67—38no—
238C2×Dic3—32no—
239C2×Dic3—32no—
240C2×Dic3—61no—
241C2×Dic3—61no—
242C2×Dic3D3×Dic312yes—
243C4×D3—22no—
244C4×D3—22no—
245C3×C8—62no—
246C3×C8—62no—
247C3×C8—182no—
248C3⋊C8—21no—
249C3⋊C8—64no—
250C3⋊C8—62no—
251C3⋊C8—62no—
252C3⋊C8—92no—
253C3⋊C8—94no—
254C3⋊C8—182no—
255C3⋊C8—184no—
256C3⋊C8—362no—
257C2×C32—21no—
258C2×C32—21no—
259C2×C32—22no—
260C2×C32—22no—
261C2×C32—21no—
262C2×C32—22no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32—41no—
266C2×C32—64no—
267C2×C32#67⋊C412yes—
268C2×C32C2×D3×C812yes—
269C2×C32#90712yes—
270C2×C32C22×C3⋊C811yescharacteristic
271C2×C32D4×Dic311yescharacteristic
272C2×C32#94811yescharacteristic
273C3×D3—28no—
274C3×D3—24no—
275C3×D3—24no—
276C2×D4—91no—
277C22×C4—32no—
278C2×C8—31no—
279C2×C8—92no—
280C2×C8—271no—
281C2×C8—544no—
282C4⋊C4—91no—
283C22⋊C4—92no—
284C42—91no—
285C22×C3—21no—
286C22×C3—22no—
287C22×C3—34no—
288C22×C3—34no—
289C22×C3—41no—
290C22×C3—64no—
291C22×C3—64no—
292C22×C3—64no—
293C22×C3—62no—
294C22×C3—62no—
295C22×C3—62no—
296C22×C3—62no—
297C22×C3C4×D3211yescharacteristic
298C22×C3C2×D3×Dic312yes—
299C2×D3—22no—
300C2×D3—22no—
301C2×D3D3×C3⋊C814yescomplemented
302C3×C4—22no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—22no—
306C3×C4—41no—
307C3×C4—41no—
308C3×C4—64no—
309C3×C4—62no—
310C3×C4—62no—
311C3×C4—92no—
312C3×C4—181no—
313C3×C4#380211yescharacteristic
314C3×C4#380511yescharacteristic
315C3×C4#380512yes—
316C3×C4#381112yes—
317Dic3—22no—
318Dic3—34no—
319Dic3—62no—
320Dic3—62no—
321Dic3—62no—
322C32—22no—
323C32—21no—
324C32—41no—
325C32#67⋊C812yescomplemented
326C32D4×C3⋊C811yescharacteristic, complemented
327C23—32no—
328D4—94no—
329C2×C4—32no—
330C2×C4—62no—
331C2×C4—62no—
332C2×C4—91no—
333C2×C4—91no—
334C2×C4D3311yescharacteristic
335C8—61no—
336C8—182no—
337C8—272no—
338C2×C3—21no—
339C2×C3—22no—
340C2×C3—21no—
341C2×C3—22no—
342C2×C3—22no—
343C2×C3—21no—
344C2×C3—41no—
345C2×C3—41no—
346C2×C3—41no—
347C2×C3—68no—
348C2×C3—64no—
349C2×C3—64no—
350C2×C3D32×C811yescharacteristic
351C2×C3#6380511yescharacteristic
352C2×C3C2×D3×C3⋊C812yes—
353C2×C3#6381512yes—
354C2×C3#6390211yescharacteristic
355C2×C3#67⋊Dic312yes—
356D3—24no—
357C22—34no—
358C22—62no—
359C22—62no—
360C22D32×Dic311yescharacteristic
361C4—62no—
362C4—92no—
363C4—181no—
364C4#8895411yescharacteristic
365C4#8896311yescharacteristic
366C3—22no—
367C3—21no—
368C3—41no—
369C3#1059071811yescharacteristic, complemented
370C3#67⋊(C3⋊C8)12yescomplemented
371C2—64no—
372C2D32×C3⋊C811yescharacteristic
373C2#1172282311yescharacteristic
374C2#1172292511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D33
commutatorC2×C33C22×C8
Fitting, cosocleC2×C33×C4C23
socleC22×C33C22×C4
prime radicalC2×C3×#625C4
hypocenterC33C8×D4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×D3×C3⋊(C3⋊C8)C2semidirect
3. #11722078C2semidirect
4. C3⋊C8×C3⋊Dic3C2semidirect
5. C3×(C2×D3)⋊(C3⋊C8)C2semidirect
28. (C2×D3)⋊(C3⋊C8)D3semidirect
38. C3×#625C8semidirect
39. C3⋊(C3⋊(C3⋊C8))D4semidirect
77. C2×C32×D3C2×C8semidirect
81. C3×C3⋊Dic3C2×C8semidirect
125. #625C3⋊C8semidirect
188. C2×C3×D3D3×C8semidirect
189. C2×C3×D3C2×C3⋊C8semidirect
202. C3⋊Dic3C2×C3⋊C8semidirect
215. C33C8×D4semidirect
301. C2×D3D3×C3⋊C8semidirect
325. C32#67⋊C8semidirect
326. C32D4×C3⋊C8semidirect
369. C3#10590718semidirect
370. C3#67⋊(C3⋊C8)semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C3⋊(C3⋊C8)2
C3×C4×#6252
#117220782
C3⋊C8×C3⋊Dic32
C3×(C2×D3)⋊(C3⋊C8)2
#67⋊(C3⋊C8)3
#105907183