Finite Group Databasebeta

← #13161182

Subgroups — #13161182

Order 1728.

1116 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 176 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×Q8×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×C32×#215C211yesmaximal, characteristic, complemented
3#11723522C211yesmaximal, characteristic
4C2×#88792C211yesmaximal, characteristic, complemented
5#11722205C211yesmaximal, characteristic, complemented
6#11722040C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic, complemented
8#10592869—31nomaximal
9#10591583—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C3×Q8×C3⋊D3—22no—
12C3×Q8×C3⋊D3—22no—
13C2×C3×C4×C3⋊D3—21no—
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C3×#634—21no—
16C2×C3×#634C2211yescharacteristic
17C2×C32×D12C2211yescharacteristic
18C2×#6290—21no—
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C32×#215—22no—
22C2×C32×C3⋊C8C2211yescharacteristic
23(C2×D3)⋊(C3⋊Dic3)—21no—
24#88792—22no—
25C2×Q8×C3⋊D3D311yescharacteristic, complemented
26C2×C3×D3×Q8—34no—
27C2×C32×SD16—31no—
28#64147—31no—
29C2×#3754—31no—
30C2×C3×#215D314yescomplemented
31#63908—34no—
32C2×#3715—34no—
33#63662—31no—
34#63662—34no—
35#63662—64no—
36#63647—31no—
37#63643—31no—
38#63566—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C22×C3×C3⋊D3D411yescharacteristic
41C22×C32×D3—21no—
42C33×Q8—21no—
43C33×Q8D412yescomplemented
44C2×C33×C4—21no—
45C2×C33×C4C2311yescharacteristic
46C2×C3⋊(C3⋊Dic3)—21no—
47C2×C3×C3⋊Dic3—21no—
48C2×C3×C3⋊Dic3D411yescharacteristic
49C3×C4×C3⋊D3—24no—
50C3×C4×C3⋊D3—21no—
51C3×C4×C3⋊D3—21no—
52C3×#634—24no—
53C3×#634—21no—
54C3×#634D412yescomplemented
55C32×D12—21no—
56C32×D12—21no—
57#6290—42no—
58C3⋊(C3⋊(C3⋊C8))—21no—
59C32×C3⋊C8—21no—
60#5227—91no—
61#5170—94no—
62C2×C32×Q8—24no—
63C2×C32×Q8C2×D314yes—
64C2×C32×Q8C2×D311yescharacteristic
65C2×C32×D4—31no—
66C2×#638—61no—
67Q8×C3⋊D3—22no—
68Q8×C3⋊D3—22no—
69C2×C4×C3⋊D3—21no—
70C2×C4×C3⋊D3C2×D311yescharacteristic
71C2×#634—21no—
72C2×#634C2×D311yescharacteristic
73C3×D3×Q8—68no—
74C3×D3×Q8—68no—
75C2×C3×D12C2×D314yes—
76C2×C3×C4×D3—34no—
77C2×C3×C4×D3—64no—
78C2×C3×Dic6—34no—
79C2×C3×Dic6—64no—
80C2×#625—64no—
81C32×SD16—62no—
82C2×C32×C8—31no—
83#3757—61no—
84#3754—62no—
85C4⋊(C3⋊Dic3)—31no—
86C4⋊(C3⋊Dic3)—34no—
87C4⋊(C3⋊Dic3)—64no—
88C2×C3⋊(C3⋊C8)—31no—
89C2×C3⋊(C3⋊C8)—34no—
90C2×C3⋊(C3⋊C8)—64no—
91C3×#215—28no—
92C2×C3×C3⋊C8C2×D314yes—
93(C2×D3)⋊Dic3—64no—
94#3715—68no—
95C22×C33C2×D411yescharacteristic
96C2×C3×C3⋊D3—21no—
97C2×C3×C3⋊D3—21no—
98C2×C3×C3⋊D3SD1612yescomplemented
99C2×C32×D3—41no—
100C2×C32×D3—41no—
101C2×C32×D3—41no—
102C33×C4—22no—
103C33×C4C2×D411yescharacteristic
104C33×C4C2×D411yescharacteristic
105C3⋊(C3⋊Dic3)—41no—
106C3×C3⋊Dic3—21no—
107C3×C3⋊Dic3—22no—
108C22×C3×Q8—91no—
109C2×D3×Q8—34no—
110C2×C3×SD16—34no—
111C2×#215—91no—
112C2×#215C3⋊D311yescharacteristic, complemented
113#938—94no—
114C2×#214—94no—
115#930—91no—
116#835—94no—
117#835—91no—
118#835—91no—
119#835—184no—
120#830—91no—
121(C2×D3)⋊C8—94no—
122#816—94no—
123C23×C32—61no—
124C22×C3⋊D3#6711yescharacteristic
125C22×C3×D3—24no—
126C22×C3×D3—34no—
127C32×Q8—21no—
128C32×Q8—28no—
129C32×Q8—24no—
130C32×Q8—44no—
131C32×Q8#6718yescomplemented
132C32×Q8#6712yescomplemented
133C32×D4—61no—
134C32×D4—61no—
135C2×C32×C4—21no—
136C2×C32×C4—24no—
137C2×C32×C4—24no—
138C2×C32×C4—44no—
139C2×C32×C4C22×D314yes—
140C2×C32×C4C22×D311yescharacteristic
141#638—122no—
142C2×C3⋊Dic3—21no—
143C2×C3⋊Dic3—61no—
144C2×C3⋊Dic3—64no—
145C2×C3⋊Dic3—124no—
146C2×C3⋊Dic3#6711yescharacteristic
147C4×C3⋊D3—24no—
148C4×C3⋊D3—21no—
149C4×C3⋊D3—21no—
150#634—24no—
151#634—21no—
152#634#6712yescomplemented
153C2×C3×Dic3—34no—
154C2×C3×Dic3—64no—
155C3×D12—24no—
156C3×D12—24no—
157C3×C4×D3—616no—
158C3×C4×D3—64no—
159C3×C4×D3—64no—
160C3×Dic6—38no—
161C3×Dic6—616no—
162C3×Dic6—64no—
163#625—128no—
164C32×C8—61no—
165C3⋊(C3⋊C8)—61no—
166C3⋊(C3⋊C8)—64no—
167C3⋊(C3⋊C8)—124no—
168C3×C3⋊C8—24no—
169#448—271noSylow
170C2×C33#12011yescharacteristic
171C2×C33C2×SD1611yescharacteristic
172C2×C33#13711yescharacteristic
173C3×C3⋊D3—22no—
174C32×D3—81no—
175C2×C3×Q8—24no—
176C2×C3×Q8—91no—
177C2×C3×Q8—181no—
178C2×C3×Q8—182no—
179C2×C3×Q8—182no—
180C2×C3×Q8D3214yes—
181C2×C3×Q8C2×C3⋊D311yescharacteristic
182C2×C3×D4—34no—
183C22×C3×C4—91no—
184C22×C3×C4—181no—
185C2×#67—181no—
186C2×#67—184no—
187D3×Q8—68no—
188D3×Q8—68no—
189C2×D12C2×C3⋊D311yescharacteristic
190C2×C4×D3—34no—
191C2×C4×D3—64no—
192C2×Dic6—34no—
193C2×Dic6—64no—
194C3×SD16—68no—
195C2×C3×C8—34no—
196#217—184no—
197#215—22no—
198#215—182no—
199#214—188no—
200(C2×D3)⋊C4—181no—
201C4⋊Dic3—94no—
202C4⋊Dic3—91no—
203C4⋊Dic3—184no—
204C2×C3⋊C8—94no—
205C2×C3⋊C8—91no—
206C2×C3⋊C8—184no—
207C2×C3⋊C8C2×C3⋊D311yescharacteristic
208C22×C32—24no—
209C22×C32—121no—
210C22×C32—121no—
211C22×C32—121no—
212C22×C32D3×D414yes—
213C22×C32C2×#6711yescharacteristic
214C2×C3⋊D3—21no—
215C2×C3⋊D3—21no—
216C2×C3⋊D3#21512yescomplemented
217C2×C3×D3—38no—
218C2×C3×D3—44no—
219C2×C3×D3—44no—
220C2×C3×D3—44no—
221C2×C3×D3—64no—
222C2×C3×D3—64no—
223C32×C4—22no—
224C32×C4—24no—
225C32×C4—24no—
226C32×C4—28no—
227C32×C4—48no—
228C32×C4D3×D414yes—
229C32×C4C2×#6714yes—
230C32×C4C2×#6711yescharacteristic
231C32×C4C2×#6711yescharacteristic
232C3⋊Dic3—21no—
233C3⋊Dic3—22no—
234C3⋊Dic3—121no—
235C3⋊Dic3—124no—
236C3⋊Dic3—244no—
237C3×Dic3—68no—
238C3×Dic3—64no—
239C22×Q8—91no—
240C2×SD16—31no—
241C2×SD16—271no—
242#121—271no—
243#103—271no—
244#103—271no—
245C22⋊C8—271no—
246C33#44811yescharacteristic, complemented, Sylow
247C23×C3—64no—
248C23×C3—91no—
249C22×D3—21no—
250C22×D3—34no—
251C3×Q8—24no—
252C3×Q8—21no—
253C3×Q8—28no—
254C3×Q8—44no—
255C3×Q8—92no—
256C3×Q8—184no—
257C3×Q8—181no—
258C3×Q8#62518yescomplemented
259C3×Q8#63812yescomplemented
260C3×D4—64no—
261C3×D4—64no—
262C2×C3×C4—24no—
263C2×C3×C4—21no—
264C2×C3×C4—24no—
265C2×C3×C4—44no—
266C2×C3×C4—91no—
267C2×C3×C4—184no—
268C2×C3×C4—181no—
269C2×C3×C4—181no—
270C2×C3×C4—181no—
271C2×C3×C4C2×D3214yes—
272C2×C3×C4C22×C3⋊D311yescharacteristic
273#67—368no—
274#67—362no—
275C2×Dic3—34no—
276C2×Dic3—64no—
277C2×Dic3—184no—
278C2×Dic3—181no—
279C2×Dic3—364no—
280D12—21no—
281D12—21no—
282C4×D3—616no—
283C4×D3—64no—
284C4×D3—64no—
285Dic6—38no—
286Dic6—616no—
287Dic6—64no—
288C3×C8—64no—
289C3⋊C8—21no—
290C3⋊C8—184no—
291C3⋊C8—181no—
292C3⋊C8—364no—
293C2×C32—24no—
294C2×C32—24no—
295C2×C32—24no—
296C2×C32—241no—
297C2×C32D3×SD1614yes—
298C2×C32#91514yes—
299C2×C32#93714yes—
300C2×C32C2×#21511yescharacteristic
301C2×C32#94211yescharacteristic
302C2×C32#95311yescharacteristic
303C3⋊D3—22no—
304C3×D3—68no—
305C3×D3—84no—
306C2×Q8—91no—
307C2×Q8—181no—
308C2×Q8—182no—
309C2×Q8—182no—
310C2×Q8D3×C3⋊D311yescharacteristic
311C2×D4—31no—
312C2×D4—541no—
313C22×C4—91no—
314C22×C4—181no—
315SD16—62no—
316SD16—542no—
317C2×C8—31no—
318C2×C8—271no—
319C4⋊C4—271no—
320C22⋊C4—541no—
321C22×C3—24no—
322C22×C3—92no—
323C22×C3—124no—
324C22×C3—124no—
325C22×C3—124no—
326C22×C3—181no—
327C22×C3—181no—
328C22×C3D3×#6714yes—
329C22×C3D4×C3⋊D311yescharacteristic
330C2×D3—38no—
331C2×D3—41no—
332C2×D3—41no—
333C2×D3—41no—
334C2×D3—64no—
335C2×D3—64no—
336C3×C4—22no—
337C3×C4—28no—
338C3×C4—24no—
339C3×C4—24no—
340C3×C4—48no—
341C3×C4—181no—
342C3×C4—182no—
343C3×C4C2×#62514yes—
344C3×C4D3×#6714yes—
345C3×C4D4×C3⋊D311yescharacteristic
346C3×C4C2×#63811yescharacteristic
347Dic3—68no—
348Dic3—64no—
349Dic3—361no—
350Dic3—364no—
351Dic3—724no—
352C32—24no—
353C32#517014yescomplemented
354C32#522711yescharacteristic, complemented
355C23—61no—
356C23—91no—
357Q8—21no—
358Q8—92no—
359Q8—184no—
360Q8—181no—
361Q8#629012yescomplemented
362D4—61no—
363D4—61no—
364D4—1082no—
365C2×C4—21no—
366C2×C4—91no—
367C2×C4—184no—
368C2×C4—181no—
369C2×C4—181no—
370C2×C4—181no—
371C2×C4—541no—
372C2×C4C2×D3×C3⋊D311yescharacteristic
373C8—61no—
374C8—541no—
375C2×C3—24no—
376C2×C3—24no—
377C2×C3—24no—
378C2×C3—182no—
379C2×C3—244no—
380C2×C3D3×#21514yes—
381C2×C3#6394614yes—
382C2×C3#6397614yes—
383C2×C3SD16×C3⋊D311yescharacteristic
384C2×C3#6412411yescharacteristic
385C2×C3#6414611yescharacteristic
386D3—68no—
387D3—81no—
388C22—92no—
389C22—121no—
390C22—121no—
391C22—121no—
392C22—181no—
393C22—181no—
394C22C3⋊D3×#6711yescharacteristic
395C4—22no—
396C4—181no—
397C4—182no—
398C4—1081no—
399C4C2×#629011yescharacteristic
400C4C3⋊D3×#6711yescharacteristic
401C3—24no—
402C3#1059158314yescomplemented
403C3#1059286911yescharacteristic, complemented
404C2—182no—
405C2—241no—
406C2C3⋊D3×#21511yescharacteristic
407C2#1172358411yescharacteristic
408C2#1172363411yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radical#11723522C2
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#448
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×Q8×C3⋊D3C2semidirect
2. C2×C32×#215C2semidirect
4. C2×#88792C2semidirect
5. #11722205C2semidirect
6. #11722040C2semidirect
7. #11722023C2semidirect
25. C2×Q8×C3⋊D3D3semidirect
30. C2×C3×#215D3semidirect
43. C33×Q8D4semidirect
54. C3×#634D4semidirect
98. C2×C3×C3⋊D3SD16semidirect
112. C2×#215C3⋊D3semidirect
131. C32×Q8#67semidirect
132. C32×Q8#67semidirect
152. #634#67semidirect
216. C2×C3⋊D3#215semidirect
246. C33#448semidirect
258. C3×Q8#625semidirect
259. C3×Q8#638semidirect
353. C32#5170semidirect
354. C32#5227semidirect
361. Q8#6290semidirect
402. C3#10591583semidirect
403. C3#10592869semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×C3⋊D32
C2×C32×#2152
#117235222
C2×#887922
#117222052
#117220402
#117220232
#105928693
#105915833