Finite Group Databasebeta

← #13161305

Subgroups — #13161305

Order 1728.

1080 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 1080 conjugacy classes of subgroups — too many to draw. Only its 188 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×C32×#239C211yesmaximal, characteristic
2C2×C3×#3756C211yesmaximal, characteristic, complemented
3#11723425C211yesmaximal, characteristic, complemented
4C2×#88792C211yesmaximal, characteristic, complemented
5#11722205C211yesmaximal, characteristic, complemented
6#11722032C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592938—31nomaximal
9#10591597—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C32×#239—22no—
13C32×#239—22no—
14C2×C32×D12—21no—
15C2×C32×D12C2211yescharacteristic
16C2×C32×C4×D3—21no—
17C2×C32×C4×D3C2211yescharacteristic
18C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C3×#3756—22no—
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22#88806—21no—
23(C2×D3)⋊(C3⋊Dic3)—21no—
24#88792—22no—
25C2×C32×#41—31no—
26C2×C3×#239D314yes—
27C2×#3756D311yescharacteristic, complemented
28C2×#3754—31no—
29#64133—31no—
30C2×C3×#216—34no—
31#63856—34no—
32C2×#3715—34no—
33#63662—31no—
34#63662—34no—
35#63662—64no—
36#63659—31no—
37#63657—31no—
38#63563—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C22×C32×D3—21no—
41C22×C32×D3D411yescharacteristic
42C33×Q8—21no—
43C33×Q8D412yes—
44C2×C33×C4—21no—
45C2×C33×C4C2311yescharacteristic
46C2×C3⋊(C3⋊Dic3)—21no—
47C2×C3×C3⋊Dic3—21no—
48C3×#634—21no—
49C3×#634—21no—
50C2×C32×Dic3D411yescharacteristic
51C32×D12—24no—
52C32×D12—21no—
53C32×D12D412yes—
54C32×C4×D3—24no—
55C32×C4×D3—21no—
56C32×C4×D3—21no—
57C3⋊(C3⋊(C3⋊C8))—21no—
58C3×C3⋊(C3⋊C8)—21no—
59#5239—94no—
60#5187—91no—
61C32×#41—62no—
62C32×#41—62no—
63C2×C32×Q8—24no—
64C2×C32×Q8C2×D314yes—
65C2×C32×Q8C2×D311yescharacteristic
66C2×C32×D4—31no—
67C2×C32×D4—61no—
68C22×C32×C4—31no—
69C22×C32×C4—61no—
70C2×#634C2×D311yescharacteristic
71C3×#239—28no—
72C3×#239—28no—
73C2×C3×D12—24no—
74C2×C3×D12C2×D314yes—
75C2×C3×C4×D3—24no—
76C2×C3×C4×D3C2×D314yes—
77C2×C3×Dic6—34no—
78#3757—61no—
79#3756—22no—
80#3754—62no—
81C4⋊(C3⋊Dic3)—31no—
82C4⋊(C3⋊Dic3)—34no—
83C4⋊(C3⋊Dic3)—64no—
84#3750—61no—
85C2×C3⋊(C3⋊C8)—31no—
86C2×C3⋊(C3⋊C8)—34no—
87C2×C3⋊(C3⋊C8)—64no—
88C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
89C3×#216—68no—
90C2×C3×C3⋊C8—34no—
91#3724—64no—
92(C2×D3)⋊Dic3—64no—
93#3715—68no—
94C22×C33C2×D411yescharacteristic
95C2×C32×D3—21no—
96C2×C32×D3—22no—
97C2×C32×D3—21no—
98C2×C32×D3—21no—
99C2×C32×D3—41no—
100C2×C32×D3—41no—
101C33×C4—22no—
102C33×C4C2×D411yescharacteristic
103C33×C4C2×D411yescharacteristic
104C3⋊(C3⋊Dic3)—41no—
105C3×C3⋊Dic3—41no—
106C32×Dic3—21no—
107C2×C3×#41—34no—
108C2×#239C3⋊D311yescharacteristic
109C2×C3×Q16—91no—
110#946—91no—
111C2×#216—34no—
112C2×#215—91no—
113C2×#214—94no—
114#924—94no—
115#835—94no—
116#835—91no—
117#835—184no—
118#832—94no—
119#830—94no—
120#821—91no—
121(C2×D3)⋊C8—91no—
122C23×C32—31no—
123C23×C32—61no—
124C22×C3×D3—24no—
125C22×C3×D3#6714yes—
126C32×Q8—21no—
127C32×Q8—28no—
128C32×Q8—24no—
129C32×Q8—44no—
130C32×Q8#6718yes—
131C32×Q8#6712yes—
132C32×D4—32no—
133C32×D4—64no—
134C32×D4—61no—
135C2×C32×C4—21no—
136C2×C32×C4—24no—
137C2×C32×C4—24no—
138C2×C32×C4—31no—
139C2×C32×C4—44no—
140C2×C32×C4—64no—
141C2×C32×C4—61no—
142C2×C32×C4—61no—
143C2×C32×C4C22×D314yes—
144C2×C32×C4C22×D311yescharacteristic
145C2×C3⋊Dic3—21no—
146C2×C3⋊Dic3—61no—
147C2×C3⋊Dic3—64no—
148C2×C3⋊Dic3—124no—
149#634—21no—
150#634—21no—
151C2×C3×Dic3—64no—
152C2×C3×Dic3#6714yes—
153C3×D12—216no—
154C3×D12—24no—
155C3×D12#6718yes—
156C3×C4×D3—216no—
157C3×C4×D3—24no—
158C3×C4×D3—24no—
159C3×Dic6—64no—
160C3×Dic6—64no—
161C3⋊(C3⋊C8)—21no—
162C3⋊(C3⋊C8)—61no—
163C3⋊(C3⋊C8)—64no—
164C3⋊(C3⋊C8)—124no—
165C3×C3⋊C8—64no—
166#452—271noSylow
167C2×C33#12011yescharacteristic
168C2×C33#13511yescharacteristic
169C2×C33#13711yescharacteristic
170C32×D3—41no—
171C32×D3—42no—
172C3×#41—68no—
173C3×#41—68no—
174C2×C3×Q8—24no—
175C2×C3×Q8—91no—
176C2×C3×Q8D3214yes—
177C2×C3×Q8C2×C3⋊D311yescharacteristic
178C2×C3×D4—34no—
179C2×C3×D4—64no—
180C22×C3×C4—34no—
181C22×C3×C4—64no—
182#239—22no—
183#239—22no—
184C2×D12—21no—
185C2×D12C2×C3⋊D311yescharacteristic
186C2×C4×D3—21no—
187C2×C4×D3C2×C3⋊D311yescharacteristic
188C2×Dic6—34no—
189C3×Q16—182no—
190C2×C3×C8—91no—
191#217—184no—
192#216—68no—
193#215—182no—
194#214—188no—
195(C2×D3)⋊C4—181no—
196C4⋊Dic3—94no—
197C4⋊Dic3—91no—
198C4⋊Dic3—184no—
199Dic3⋊C4—184no—
200Dic3⋊C4—181no—
201C2×C3⋊C8—34no—
202C2×C3⋊C8—94no—
203C2×C3⋊C8—91no—
204C2×C3⋊C8—184no—
205C22×C32—24no—
206C22×C32—61no—
207C22×C32—61no—
208C22×C32—62no—
209C22×C32—61no—
210C22×C32—121no—
211C22×C32—121no—
212C22×C32D3×D411yescharacteristic
213C22×C32C2×#6714yes—
214C2×C3×D3—24no—
215C2×C3×D3—24no—
216C2×C3×D3—24no—
217C2×C3×D3—28no—
218C2×C3×D3—44no—
219C2×C3×D3—44no—
220C32×C4—22no—
221C32×C4—24no—
222C32×C4—24no—
223C32×C4—28no—
224C32×C4—48no—
225C32×C4—61no—
226C32×C4D3×D411yescharacteristic
227C32×C4C2×#6714yes—
228C32×C4C2×#6714yes—
229C32×C4C2×#6711yescharacteristic
230C3⋊Dic3—41no—
231C3⋊Dic3—121no—
232C3⋊Dic3—124no—
233C3⋊Dic3—244no—
234C3×Dic3—24no—
235C3×Dic3—124no—
236C2×#41—31no—
237C2×Q16—91no—
238C2×SD16—271no—
239C22⋊Q8—271no—
240#103—271no—
241#102—271no—
242C22⋊C8—271no—
243C33#45211yescharacteristic, complemented, Sylow
244C23×C3—34no—
245C23×C3—64no—
246C22×D3—21no—
247C22×D3#63811yescharacteristic
248C3×Q8—24no—
249C3×Q8—21no—
250C3×Q8—28no—
251C3×Q8—44no—
252C3×Q8—181no—
253C3×Q8—181no—
254C3×Q8#62518yes—
255C3×Q8#63812yes—
256C3×D4—38no—
257C3×D4—616no—
258C3×D4—64no—
259C2×C3×C4—24no—
260C2×C3×C4—21no—
261C2×C3×C4—24no—
262C2×C3×C4—34no—
263C2×C3×C4—44no—
264C2×C3×C4—616no—
265C2×C3×C4—64no—
266C2×C3×C4—64no—
267C2×C3×C4—181no—
268C2×C3×C4C2×D3214yes—
269C2×C3×C4C22×C3⋊D311yescharacteristic
270C2×Dic3—64no—
271C2×Dic3—184no—
272C2×Dic3—181no—
273C2×Dic3—364no—
274C2×Dic3#63811yescharacteristic
275D12—24no—
276D12—21no—
277D12#63812yes—
278C4×D3—24no—
279C4×D3—21no—
280C4×D3—21no—
281Dic6—64no—
282Dic6—64no—
283C3×C8—181no—
284C3⋊C8—64no—
285C3⋊C8—184no—
286C3⋊C8—181no—
287C3⋊C8—364no—
288C2×C32—24no—
289C2×C32—24no—
290C2×C32—24no—
291C2×C32—121no—
292C2×C32—122no—
293C2×C32#91811yescharacteristic
294C2×C32#91911yescharacteristic
295C2×C32#93711yescharacteristic
296C2×C32#95014yes—
297C2×C32#95114yes—
298C2×C32#95314yes—
299C3×D3—44no—
300C3×D3—48no—
301#41—62no—
302#41—62no—
303C2×Q8—91no—
304C2×Q8D3×C3⋊D311yescharacteristic
305C2×D4—31no—
306C2×D4—61no—
307C22×C4—31no—
308C22×C4—61no—
309Q16—182no—
310SD16—542no—
311C2×C8—91no—
312C2×C8—271no—
313C4⋊C4—271no—
314C4⋊C4—541no—
315C22⋊C4—541no—
316C22×C3—24no—
317C22×C3—64no—
318C22×C3—64no—
319C22×C3—68no—
320C22×C3—64no—
321C22×C3—124no—
322C22×C3—124no—
323C22×C3D3×#6714yes—
324C22×C3C2×#63811yescharacteristic
325C2×D3—21no—
326C2×D3—21no—
327C2×D3—21no—
328C2×D3—22no—
329C2×D3—41no—
330C2×D3—41no—
331C3×C4—22no—
332C3×C4—28no—
333C3×C4—24no—
334C3×C4—24no—
335C3×C4—48no—
336C3×C4—64no—
337C3×C4—361no—
338C3×C4C2×#62514yes—
339C3×C4D3×#6714yes—
340C3×C4C2×#63811yescharacteristic
341C3×C4C2×#63811yescharacteristic
342Dic3—21no—
343Dic3—124no—
344Dic3—361no—
345Dic3—364no—
346Dic3—724no—
347C32—24no—
348C32#518711yescharacteristic, complemented
349C32#523914yescomplemented
350C23—31no—
351C23—61no—
352Q8—21no—
353Q8—181no—
354Q8—181no—
355Q8#629012yes—
356D4—32no—
357D4—64no—
358D4—61no—
359C2×C4—21no—
360C2×C4—31no—
361C2×C4—64no—
362C2×C4—61no—
363C2×C4—61no—
364C2×C4—181no—
365C2×C4—541no—
366C2×C4C2×D3×C3⋊D311yescharacteristic
367C8—181no—
368C8—541no—
369C2×C3—24no—
370C2×C3—24no—
371C2×C3—24no—
372C2×C3—124no—
373C2×C3—128no—
374C2×C3#6394314yes—
375C2×C3#6394914yes—
376C2×C3#6397614yes—
377C2×C3#6415911yescharacteristic
378C2×C3#6416011yescharacteristic
379C2×C3#6416211yescharacteristic
380D3—41no—
381D3—42no—
382C22—61no—
383C22—61no—
384C22—62no—
385C22—61no—
386C22—121no—
387C22—121no—
388C22D3×#63811yescharacteristic
389C4—22no—
390C4—61no—
391C4—361no—
392C4—1081no—
393C4C2×#629011yescharacteristic
394C4D3×#63811yescharacteristic
395C3—24no—
396C3#1059159714yescomplemented
397C3#1059293811yescharacteristic, complemented
398C2—121no—
399C2—122no—
400C2#1172357411yescharacteristic
401C2#1172358211yescharacteristic
402C2#1172362911yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C32×#239C2
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×C3×#3756C2semidirect
3. #11723425C2semidirect
4. C2×#88792C2semidirect
5. #11722205C2semidirect
6. #11722032C2semidirect
7. (C2×D3)⋊(C3⋊(C3⋊C8))C2semidirect
27. C2×#3756D3semidirect
243. C33#452semidirect
348. C32#5187semidirect
349. C32#5239semidirect
396. C3#10591597semidirect
397. C3#10592938semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2392
C2×C3×#37562
#117234252
C2×#887922
#117222052
#117220322
(C2×D3)⋊(C3⋊(C3⋊C8))2
#105929383
#105915973