Finite Group Databasebeta

← #13161208

Subgroups — #13161208

Order 1728.

1152 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 1152 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×#88863C211yesmaximal, characteristic, complemented
3#11723426C211yesmaximal, characteristic, complemented
4C2×#88794C211yesmaximal, characteristic, complemented
5C3×#63662C211yesmaximal, characteristic, complemented
6#11722031C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592938—31nomaximal
9#10591599—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C32×#239—22no—
13C32×#239—22no—
14C2×C32×D12—21no—
15C2×C32×D12C2211yescharacteristic
16C2×C32×C4×D3—21no—
17C2×C32×C4×D3C2211yescharacteristic
18#88863—22no—
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22#88807—21no—
23(C2×D3)⋊(C3⋊Dic3)—21no—
24#88794—22no—
25C2×C32×#41—31no—
26C2×C3×#239D314yes—
27C2×#3756—31no—
28C2×#3756—34no—
29C2×#3756—64no—
30C2×#3754—31no—
31#64133—31no—
32#63857—34no—
33C2×#3716—34no—
34#63662D311yescharacteristic, complemented
35#63659—31no—
36#63657—31no—
37C3×#835—34no—
38#63562—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—
47#6308—21no—
48#6308—21no—
49C2×C3×C3⋊Dic3—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#4808—91no—
61C32×#41—62no—
62C32×#41—62no—
63C2×C32×Q8—24no—
64C2×C32×Q8C2×D311yescharacteristic
65C2×C32×Q8C2×D314yes—
66C2×C32×D4—31no—
67C2×C32×D4—61no—
68C22×C32×C4—31no—
69C22×C32×C4—61no—
70C2×#634—31no—
71C2×#634—34no—
72C2×#634—64no—
73C3×#239—28no—
74C3×#239—28no—
75C2×C3×D12—24no—
76C2×C3×D12C2×D314yes—
77C2×C3×C4×D3—24no—
78C2×C3×C4×D3C2×D314yes—
79#3757—61no—
80#3756—616no—
81#3756—62no—
82#3756—68no—
83#3754—62no—
84C4⋊(C3⋊Dic3)C2×D311yescharacteristic
85#3750—61no—
86C2×C3⋊(C3⋊C8)—31no—
87C2×C3⋊(C3⋊C8)—34no—
88C2×C3⋊(C3⋊C8)—64no—
89C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
90C3×C4⋊Dic3—34no—
91C2×C3×C3⋊C8—34no—
92Dic3⋊Dic3—64no—
93(C2×D3)⋊Dic3—64no—
94#3716—68no—
95C22×C33C2×D411yescharacteristic
96C2×C32×D3—21no—
97C2×C32×D3—22no—
98C2×C32×D3—21no—
99C2×C32×D3—21no—
100C2×C32×D3—41no—
101C2×C32×D3—41no—
102C33×C4—22no—
103C33×C4C2×D411yescharacteristic
104C33×C4C2×D411yescharacteristic
105C3⋊(C3⋊Dic3)—41no—
106C3×C3⋊Dic3—41no—
107C32×Dic3—21no—
108C2×C3×#41—34no—
109C2×#239C3⋊D311yescharacteristic
110C2×#216—94no—
111C2×#216—91no—
112C2×#216—184no—
113C2×#214—94no—
114#924—94no—
115C2×#204—91no—
116#897—91no—
117C3×#103—91no—
118#835—34no—
119#832—94no—
120#830—94no—
121(C2×D3)⋊C8—91no—
122#809—91no—
123C23×C32—31no—
124C23×C32—61no—
125C22×C3×D3—24no—
126C22×C3×D3#6714yes—
127C32×Q8—21no—
128C32×Q8—28no—
129C32×Q8—24no—
130C32×Q8—44no—
131C32×Q8D1212yes—
132C32×Q8#6718yes—
133C32×D4—32no—
134C32×D4—64no—
135C32×D4—61no—
136C2×C32×C4—21no—
137C2×C32×C4—24no—
138C2×C32×C4—24no—
139C2×C32×C4—31no—
140C2×C32×C4—44no—
141C2×C32×C4—64no—
142C2×C32×C4—61no—
143C2×C32×C4—61no—
144C2×C32×C4C22×D314yes—
145C2×C32×C4C22×D311yescharacteristic
146C2×C3⋊Dic3—21no—
147C2×C3⋊Dic3—61no—
148C2×C3⋊Dic3—64no—
149C2×C3⋊Dic3—124no—
150#634—68no—
151#634—61no—
152#634—61no—
153#634—64no—
154#634—64no—
155#634—124no—
156C2×C3×Dic3—64no—
157C2×C3×Dic3#6714yes—
158C3×D12—216no—
159C3×D12—24no—
160C3×D12#6718yes—
161C3×C4×D3—216no—
162C3×C4×D3—24no—
163C3×C4×D3—24no—
164C3⋊(C3⋊C8)—21no—
165C3⋊(C3⋊C8)—68no—
166C3⋊(C3⋊C8)—61no—
167C3⋊(C3⋊C8)—64no—
168C3×C3⋊C8—64no—
169#452—271noSylow
170C2×C33#12011yescharacteristic
171C2×C33#13511yescharacteristic
172C2×C33#13711yescharacteristic
173C32×D3—41no—
174C32×D3—42no—
175C3×#41—68no—
176C3×#41—68no—
177C2×C3×Q8—24no—
178C2×C3×Q8D3214yes—
179C2×C3×Q8C2×C3⋊D311yescharacteristic
180C2×C3×D4—34no—
181C2×C3×D4—64no—
182C22×C3×C4—34no—
183C22×C3×C4—64no—
184#239—22no—
185#239—22no—
186C2×D12—21no—
187C2×D12C2×C3⋊D311yescharacteristic
188C2×C4×D3—21no—
189C2×C4×D3C2×C3⋊D311yescharacteristic
190C2×Dic6—94no—
191C2×Dic6—91no—
192C2×Dic6—184no—
193C2×C3×C8—91no—
194C3×C4⋊C4—91no—
195#217—184no—
196#216—1816no—
197#216—188no—
198#216—182no—
199#214—188no—
200(C2×D3)⋊C4—181no—
201C4⋊Dic3—34no—
202C4⋊Dic3—181no—
203Dic3⋊C4—184no—
204C2×C3⋊C8—34no—
205C2×C3⋊C8—94no—
206C2×C3⋊C8—91no—
207C2×C3⋊C8—184no—
208#204—182no—
209C22×C32—24no—
210C22×C32—62no—
211C22×C32—61no—
212C22×C32—61no—
213C22×C32—61no—
214C22×C32—121no—
215C22×C32—121no—
216C22×C32D3×D411yescharacteristic
217C22×C32C2×#6714yes—
218C2×C3×D3—24no—
219C2×C3×D3—24no—
220C2×C3×D3—24no—
221C2×C3×D3—28no—
222C2×C3×D3—44no—
223C2×C3×D3—44no—
224C32×C4—22no—
225C32×C4—24no—
226C32×C4—24no—
227C32×C4—28no—
228C32×C4—48no—
229C32×C4—61no—
230C32×C4C2×D1211yescharacteristic
231C32×C4D3×D411yescharacteristic
232C32×C4C2×#6714yes—
233C32×C4C2×#6714yes—
234C3⋊Dic3—41no—
235C3⋊Dic3—128no—
236C3⋊Dic3—121no—
237C3⋊Dic3—124no—
238C3×Dic3—24no—
239C3×Dic3—124no—
240C2×#41—31no—
241C2×Q16—271no—
242C2×SD16—271no—
243C22⋊Q8—271no—
244#103—91no—
245#102—271no—
246C22⋊C8—271no—
247C33#45211yescharacteristic, complemented, Sylow
248C23×C3—34no—
249C23×C3—64no—
250C22×D3—21no—
251C22×D3#63811yescharacteristic
252C3×Q8—24no—
253C3×Q8—21no—
254C3×Q8—28no—
255C3×Q8—44no—
256C3×Q8#62618yes—
257C3×Q8#63812yes—
258C3×D4—38no—
259C3×D4—616no—
260C3×D4—64no—
261C2×C3×C4—24no—
262C2×C3×C4—21no—
263C2×C3×C4—24no—
264C2×C3×C4—34no—
265C2×C3×C4—44no—
266C2×C3×C4—616no—
267C2×C3×C4—64no—
268C2×C3×C4—64no—
269C2×C3×C4—181no—
270C2×C3×C4C2×D3214yes—
271C2×C3×C4C22×C3⋊D311yescharacteristic
272C2×Dic3—64no—
273C2×Dic3—184no—
274C2×Dic3—181no—
275C2×Dic3—364no—
276C2×Dic3#63811yescharacteristic
277D12—24no—
278D12—21no—
279D12#63812yes—
280C4×D3—24no—
281C4×D3—21no—
282C4×D3—21no—
283Dic6—188no—
284Dic6—184no—
285Dic6—184no—
286Dic6—181no—
287Dic6—181no—
288Dic6—364no—
289C3×C8—181no—
290C3⋊C8—64no—
291C3⋊C8—188no—
292C3⋊C8—184no—
293C3⋊C8—181no—
294C2×C32—24no—
295C2×C32—24no—
296C2×C32—24no—
297C2×C32—121no—
298C2×C32—122no—
299C2×C32#88211yescharacteristic
300C2×C32#91611yescharacteristic
301C2×C32#91811yescharacteristic
302C2×C32#95014yes—
303C2×C32#95114yes—
304C2×C32#95314yes—
305C3×D3—44no—
306C3×D3—48no—
307#41—62no—
308#41—62no—
309C2×Q8—271no—
310C2×Q8D3×C3⋊D311yescharacteristic
311C2×D4—31no—
312C2×D4—61no—
313C22×C4—31no—
314C22×C4—61no—
315Q16—542no—
316SD16—542no—
317C2×C8—91no—
318C2×C8—271no—
319C4⋊C4—91no—
320C4⋊C4—541no—
321C22⋊C4—541no—
322C22×C3—24no—
323C22×C3—64no—
324C22×C3—64no—
325C22×C3—68no—
326C22×C3—64no—
327C22×C3—124no—
328C22×C3—124no—
329C22×C3D3×#6714yes—
330C22×C3C2×#63811yescharacteristic
331C2×D3—21no—
332C2×D3—21no—
333C2×D3—21no—
334C2×D3—22no—
335C2×D3—41no—
336C2×D3—41no—
337C3×C4—22no—
338C3×C4—28no—
339C3×C4—24no—
340C3×C4—24no—
341C3×C4—48no—
342C3×C4—64no—
343C3×C4—361no—
344C3×C4C2×#62614yes—
345C3×C4D3×#6714yes—
346C3×C4C2×#63811yescharacteristic
347C3×C4C2×#63811yescharacteristic
348Dic3—21no—
349Dic3—124no—
350Dic3—368no—
351Dic3—361no—
352Dic3—364no—
353C32—24no—
354C32#480811yescharacteristic, complemented
355C32#523914yescomplemented
356C23—31no—
357C23—61no—
358Q8—21no—
359Q8—541no—
360Q8—541no—
361Q8#629112yes—
362D4—32no—
363D4—64no—
364D4—61no—
365C2×C4—21no—
366C2×C4—31no—
367C2×C4—64no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—181no—
371C2×C4—541no—
372C2×C4C2×D3×C3⋊D311yescharacteristic
373C8—181no—
374C8—541no—
375C2×C3—24no—
376C2×C3—24no—
377C2×C3—24no—
378C2×C3—124no—
379C2×C3—128no—
380C2×C3#6394314yes—
381C2×C3#6394714yes—
382C2×C3#6397714yes—
383C2×C3#6415911yescharacteristic
384C2×C3#6416011yescharacteristic
385C2×C3#6416211yescharacteristic
386D3—41no—
387D3—42no—
388C22—61no—
389C22—61no—
390C22—62no—
391C22—61no—
392C22—121no—
393C22—121no—
394C22D3×#63811yescharacteristic
395C4—22no—
396C4—61no—
397C4—361no—
398C4—1081no—
399C4C2×#629111yescharacteristic
400C4D3×#63811yescharacteristic
401C3—24no—
402C3#1059159914yescomplemented
403C3#1059293811yescharacteristic, complemented
404C2—121no—
405C2—122no—
406C2#1172357411yescharacteristic
407C2#1172358011yescharacteristic
408C2#1172363011yescharacteristic
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×#88863C2semidirect
3. #11723426C2semidirect
4. C2×#88794C2semidirect
5. C3×#63662C2semidirect
6. #11722031C2semidirect
7. (C2×D3)⋊(C3⋊(C3⋊C8))C2semidirect
34. #63662D3semidirect
247. C33#452semidirect
354. C32#4808semidirect
355. C32#5239semidirect
402. C3#10591599semidirect
403. 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×#888632
#117234262
C2×#887942
C3×#636622
#117220312
(C2×D3)⋊(C3⋊(C3⋊C8))2
#105929383
#105915993