Finite Group Databasebeta

← #13161204

Subgroups — #13161204

Order 1728.

1080 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 1080 conjugacy classes of subgroups — too many to draw. Only its 172 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×#3832C211yesmaximal, characteristic
2C2×C32×#216C211yesmaximal, characteristic, complemented
3#11723435C211yesmaximal, characteristic, complemented
4C2×#88793C211yesmaximal, characteristic, complemented
5#11722205C211yesmaximal, characteristic, complemented
6#11722034C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic, complemented
8#10592886—31nomaximal
9#10591597—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C3×#3832—22no—
12C3×#3832—22no—
13C2×C3×C4⋊(C3⋊D3)—21no—
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C3×C4×C3⋊D3—21no—
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×C32×Dic6C2211yescharacteristic
18C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C32×#216—22no—
21C2×C32×C3⋊C8C2211yescharacteristic
22#88806—21no—
23#88803—21no—
24#88793—22no—
25C2×#3832D311yescharacteristic
26C2×C3×#239—34no—
27C2×C32×Q16—31no—
28#64155—31no—
29C2×#3755—31no—
30C2×C3×#216D314yescomplemented
31#63856—34no—
32C2×#3715—34no—
33#63662—31no—
34#63662—34no—
35#63662—64no—
36#63648—31no—
37#63647—31no—
38#63563—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C22×C3×C3⋊D3—21no—
41C22×C3×C3⋊D3D411yescharacteristic
42C33×Q8—21no—
43C33×Q8D412yes—
44C2×C33×C4—21no—
45C2×C33×C4C2311yescharacteristic
46C2×C3⋊(C3⋊Dic3)—21no—
47C2×C3×C3⋊Dic3D411yescharacteristic
48C3×C4⋊(C3⋊D3)—24no—
49C3×C4⋊(C3⋊D3)—21no—
50C3×C4⋊(C3⋊D3)D412yes—
51C3×C4×C3⋊D3—24no—
52C3×C4×C3⋊D3—21no—
53C3×C4×C3⋊D3—21no—
54C2×C32×Dic3—21no—
55C32×Dic6—21no—
56C32×Dic6—21no—
57C3⋊(C3⋊(C3⋊C8))—21no—
58C32×C3⋊C8—21no—
59#5239—91no—
60#5187—94no—
61C2×C32×Q8—24no—
62C2×C32×Q8—31no—
63C2×C32×Q8C2×D314yes—
64C2×C32×Q8C2×D311yescharacteristic
65#3832—22no—
66#3832—22no—
67C2×C4⋊(C3⋊D3)—21no—
68C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
69C2×C4×C3⋊D3—21no—
70C2×C4×C3⋊D3C2×D311yescharacteristic
71C3×#239—68no—
72C3×#239—68no—
73C2×C3×D12—34no—
74C2×C3×D12—64no—
75C2×C3×C4×D3—34no—
76C2×C3×C4×D3—64no—
77C2×C3×Dic6C2×D314yes—
78C32×Q16—62no—
79C2×C32×C8—31no—
80#3755—62no—
81#3752—61no—
82C4⋊(C3⋊Dic3)—31no—
83C4⋊(C3⋊Dic3)—34no—
84C4⋊(C3⋊Dic3)—64no—
85#3750—61no—
86C2×C3⋊(C3⋊C8)—31no—
87C2×C3⋊(C3⋊C8)—34no—
88C2×C3⋊(C3⋊C8)—64no—
89C3×#216—28no—
90C2×C3×C3⋊C8C2×D314yes—
91#3724—64no—
92(C2×D3)⋊Dic3—64no—
93#3715—68no—
94C22×C33C2×D411yescharacteristic
95C2×C3×C3⋊D3—21no—
96C2×C3×C3⋊D3—22no—
97C2×C3×C3⋊D3—21no—
98C2×C3×C3⋊D3—21no—
99C2×C3×C3⋊D3—41no—
100C2×C3×C3⋊D3—41no—
101C33×C4—22no—
102C33×C4C2×D411yescharacteristic
103C33×C4C2×D411yescharacteristic
104C3⋊(C3⋊Dic3)—41no—
105C3×C3⋊Dic3—21no—
106C32×Dic3—41no—
107C2×C3×#41—91no—
108C2×#239—34no—
109C2×C3×Q16—34no—
110#946—94no—
111C2×#216C3⋊D311yescharacteristic, complemented
112C2×#215—94no—
113C2×#214—91no—
114#924—91no—
115#835—94no—
116#835—91no—
117#835—184no—
118#832—91no—
119#830—91no—
120#821—94no—
121(C2×D3)⋊C8—94no—
122C22×C3⋊D3—21no—
123C22×C3⋊D3#6711yescharacteristic
124C22×C3×D3—34no—
125C22×C3×D3—64no—
126C32×Q8—21no—
127C32×Q8—28no—
128C32×Q8—24no—
129C32×Q8—44no—
130C32×Q8—61no—
131C32×Q8—61no—
132C32×Q8#6718yes—
133C32×Q8#6712yes—
134C2×C32×C4—21no—
135C2×C32×C4—24no—
136C2×C32×C4—24no—
137C2×C32×C4—44no—
138C2×C32×C4—61no—
139C2×C32×C4C22×D314yes—
140C2×C32×C4C22×D311yescharacteristic
141C2×C3⋊Dic3—61no—
142C2×C3⋊Dic3—64no—
143C2×C3⋊Dic3—124no—
144C2×C3⋊Dic3#6711yescharacteristic
145C4⋊(C3⋊D3)—24no—
146C4⋊(C3⋊D3)—21no—
147C4⋊(C3⋊D3)#6712yes—
148C4×C3⋊D3—24no—
149C4×C3⋊D3—21no—
150C4×C3⋊D3—21no—
151C2×C3×Dic3—24no—
152C2×C3×Dic3—34no—
153C3×D12—38no—
154C3×D12—616no—
155C3×D12—64no—
156C3×C4×D3—616no—
157C3×C4×D3—64no—
158C3×C4×D3—64no—
159C3×Dic6—24no—
160C3×Dic6—24no—
161C32×C8—61no—
162C3⋊(C3⋊C8)—61no—
163C3⋊(C3⋊C8)—64no—
164C3⋊(C3⋊C8)—124no—
165C3×C3⋊C8—24no—
166#452—271noSylow
167C2×C33#12011yescharacteristic
168C2×C33#13511yescharacteristic
169C2×C33#13711yescharacteristic
170C3×C3⋊D3—41no—
171C3×C3⋊D3—42no—
172C3×#41—182no—
173C3×#41—182no—
174C2×C3×Q8—24no—
175C2×C3×Q8—34no—
176C2×C3×Q8D3214yes—
177C2×C3×Q8C2×C3⋊D311yescharacteristic
178C2×C3×D4—91no—
179C2×C3×D4—181no—
180C22×C3×C4—91no—
181C22×C3×C4—181no—
182#239—68no—
183#239—68no—
184C2×D12—34no—
185C2×D12—64no—
186C2×C4×D3—34no—
187C2×C4×D3—64no—
188C2×Dic6C2×C3⋊D311yescharacteristic
189C3×Q16—68no—
190C2×C3×C8—34no—
191#217—181no—
192#216—22no—
193#215—188no—
194#214—182no—
195(C2×D3)⋊C4—184no—
196C4⋊Dic3—94no—
197C4⋊Dic3—91no—
198C4⋊Dic3—184no—
199Dic3⋊C4—181no—
200Dic3⋊C4—184no—
201C2×C3⋊C8—94no—
202C2×C3⋊C8—91no—
203C2×C3⋊C8—184no—
204C2×C3⋊C8C2×C3⋊D311yescharacteristic
205C22×C32—24no—
206C22×C32D3×D414yes—
207C22×C32C2×#6711yescharacteristic
208C2×C3⋊D3—21no—
209C2×C3⋊D3—21no—
210C2×C3⋊D3—21no—
211C2×C3⋊D3—22no—
212C2×C3⋊D3—41no—
213C2×C3⋊D3—41no—
214C2×C3×D3—64no—
215C2×C3×D3—64no—
216C2×C3×D3—68no—
217C2×C3×D3—64no—
218C2×C3×D3—124no—
219C2×C3×D3—124no—
220C32×C4—22no—
221C32×C4—24no—
222C32×C4—24no—
223C32×C4—28no—
224C32×C4—48no—
225C32×C4—121no—
226C32×C4D3×D414yes—
227C32×C4C2×#6714yes—
228C32×C4C2×#6711yescharacteristic
229C32×C4C2×#6711yescharacteristic
230C3⋊Dic3—21no—
231C3⋊Dic3—121no—
232C3⋊Dic3—124no—
233C3⋊Dic3—244no—
234C3×Dic3—44no—
235C3×Dic3—64no—
236C2×#41—91no—
237C2×Q16—31no—
238C2×SD16—271no—
239C22⋊Q8—271no—
240#103—271no—
241#102—271no—
242C22⋊C8—271no—
243C33#45211yescharacteristic, complemented, Sylow
244C23×C3—91no—
245C23×C3—181no—
246C22×D3—34no—
247C22×D3—64no—
248C3×Q8—24no—
249C3×Q8—21no—
250C3×Q8—28no—
251C3×Q8—44no—
252C3×Q8—64no—
253C3×Q8—64no—
254C3×Q8#62518yes—
255C3×Q8#63812yes—
256C3×D4—92no—
257C3×D4—184no—
258C3×D4—181no—
259C2×C3×C4—24no—
260C2×C3×C4—21no—
261C2×C3×C4—24no—
262C2×C3×C4—44no—
263C2×C3×C4—64no—
264C2×C3×C4—91no—
265C2×C3×C4—184no—
266C2×C3×C4—181no—
267C2×C3×C4—181no—
268C2×C3×C4C2×D3214yes—
269C2×C3×C4C22×C3⋊D311yescharacteristic
270C2×Dic3—21no—
271C2×Dic3—34no—
272C2×Dic3—184no—
273C2×Dic3—181no—
274C2×Dic3—364no—
275D12—38no—
276D12—616no—
277D12—64no—
278C4×D3—616no—
279C4×D3—64no—
280C4×D3—64no—
281Dic6—21no—
282Dic6—21no—
283C3×C8—64no—
284C3⋊C8—21no—
285C3⋊C8—184no—
286C3⋊C8—181no—
287C3⋊C8—364no—
288C2×C32—24no—
289C2×C32—24no—
290C2×C32—24no—
291C2×C32#91814yes—
292C2×C32#91914yes—
293C2×C32#93714yes—
294C2×C32#95011yescharacteristic
295C2×C32#95111yescharacteristic
296C2×C32#95311yescharacteristic
297C3⋊D3—41no—
298C3⋊D3—42no—
299C3×D3—124no—
300C3×D3—128no—
301#41—182no—
302#41—182no—
303C2×Q8—31no—
304C2×Q8D3×C3⋊D311yescharacteristic
305C2×D4—91no—
306C2×D4—181no—
307C22×C4—91no—
308C22×C4—181no—
309Q16—62no—
310SD16—542no—
311C2×C8—31no—
312C2×C8—271no—
313C4⋊C4—271no—
314C4⋊C4—541no—
315C22⋊C4—541no—
316C22×C3—24no—
317C22×C3—181no—
318C22×C3—181no—
319C22×C3—182no—
320C22×C3—181no—
321C22×C3—361no—
322C22×C3—361no—
323C22×C3D3×#6714yes—
324C22×C3D4×C3⋊D311yescharacteristic
325C2×D3—64no—
326C2×D3—64no—
327C2×D3—68no—
328C2×D3—64no—
329C2×D3—124no—
330C2×D3—124no—
331C3×C4—22no—
332C3×C4—28no—
333C3×C4—24no—
334C3×C4—24no—
335C3×C4—48no—
336C3×C4—124no—
337C3×C4—181no—
338C3×C4C2×#62514yes—
339C3×C4D3×#6714yes—
340C3×C4D4×C3⋊D311yescharacteristic
341C3×C4C2×#63811yescharacteristic
342Dic3—41no—
343Dic3—64no—
344Dic3—361no—
345Dic3—364no—
346Dic3—724no—
347C32—24no—
348C32#518714yescomplemented
349C32#523911yescharacteristic, complemented
350C23—91no—
351C23—181no—
352Q8—21no—
353Q8—61no—
354Q8—61no—
355Q8#629012yes—
356D4—92no—
357D4—184no—
358D4—181no—
359C2×C4—21no—
360C2×C4—61no—
361C2×C4—91no—
362C2×C4—184no—
363C2×C4—181no—
364C2×C4—181no—
365C2×C4—541no—
366C2×C4C2×D3×C3⋊D311yescharacteristic
367C8—61no—
368C8—541no—
369C2×C3—24no—
370C2×C3—24no—
371C2×C3—24no—
372C2×C3—361no—
373C2×C3—362no—
374C2×C3#6394314yes—
375C2×C3#6394914yes—
376C2×C3#6397614yes—
377C2×C3#6412711yescharacteristic
378C2×C3#6412811yescharacteristic
379C2×C3#6414611yescharacteristic
380D3—124no—
381D3—128no—
382C22—181no—
383C22—181no—
384C22—182no—
385C22—181no—
386C22—361no—
387C22—361no—
388C22C3⋊D3×#6711yescharacteristic
389C4—22no—
390C4—121no—
391C4—181no—
392C4—1081no—
393C4C2×#629011yescharacteristic
394C4C3⋊D3×#6711yescharacteristic
395C3—24no—
396C3#1059159714yescomplemented
397C3#1059288611yescharacteristic, complemented
398C2—361no—
399C2—362no—
400C2#1172357611yescharacteristic
401C2#1172358511yescharacteristic
402C2#1172363411yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C3×#3832C2
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#452
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
2. C2×C32×#216C2semidirect
3. #11723435C2semidirect
4. C2×#88793C2semidirect
5. #11722205C2semidirect
6. #11722034C2semidirect
7. #11722023C2semidirect
30. C2×C3×#216D3semidirect
111. C2×#216C3⋊D3semidirect
243. C33#452semidirect
348. C32#5187semidirect
349. C32#5239semidirect
396. C3#10591597semidirect
397. C3#10592886semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38322
C2×C32×#2162
#117234352
C2×#887932
#117222052
#117220342
#117220232
#105928863
#105915973