Finite Group Databasebeta

← #13161211

Subgroups — #13161211

Order 1728.

900 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 154 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×#88863C211yesmaximal, characteristic, complemented
2C32×#946C211yesmaximal, characteristic
3#11723426C211yesmaximal, characteristic
4C2×#88786C211yesmaximal, characteristic
5C3×#63662C211yesmaximal, characteristic, complemented
6#11722053C211yesmaximal, characteristic, complemented
7#11722041C211yesmaximal, characteristic, complemented
8#10592751—31nomaximal
9#10591598—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13#88863—22no—
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×(C2×D3)⋊C4—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C32×Dic3⋊C4—21no—
20#88807—21no—
21(C2×D3)⋊(C3⋊Dic3)—21no—
22#88786—22no—
23C32×C22⋊Q8—31no—
24C2×#3756—31no—
25C2×#3756—34no—
26C2×#3756—64no—
27#64133—31no—
28C2×#3748—31no—
29C3×#946D314yes—
30#63857—34no—
31C2×#3711—34no—
32#63662D311yescharacteristic, complemented
33#63637—31no—
34#63635—31no—
35C3×#835—34no—
36#63574—34no—
37#63567—34no—
38C22×C32×D3D411yescharacteristic
39C33×Q8—21no—
40C33×Q8—21no—
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)—21no—
44#6308—21no—
45#6308—21no—
46C2×C3×C3⋊Dic3—21no—
47C2×C32×Dic3—21no—
48C2×C32×Dic3D411yescharacteristic
49C32×C4×D3—21no—
50C32×C4×D3—21no—
51C3⋊(C3⋊(C3⋊C8))D412yes—
52C3×C3⋊(C3⋊C8)—21no—
53#5052—94no—
54#4816—91no—
55C2×C32×Q8—24no—
56C2×C32×Q8C2×D314yes—
57C2×C32×Q8C2×D311yescharacteristic
58C22×C32×C4—31no—
59C2×#634—31no—
60C2×#634—34no—
61C2×#634—64no—
62C2×C3×C4×D3C2×D314yes—
63C32×C4⋊C4—31no—
64C32×C4⋊C4—61no—
65C32×C22⋊C4—61no—
66#3757—61no—
67#3756—62no—
68#3756—68no—
69#3756—616no—
70C4⋊(C3⋊Dic3)C2×D311yescharacteristic
71#3750—61no—
72#3748—62no—
73C2×C3⋊(C3⋊C8)—31no—
74C2×C3⋊(C3⋊C8)—34no—
75C2×C3⋊(C3⋊C8)—64no—
76C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
77C3×(C2×D3)⋊C4—24no—
78C3×C4⋊Dic3—34no—
79C3×C4⋊Dic3C2×D314yes—
80C3×Dic3⋊C4—24no—
81C2×C3×C3⋊C8—34no—
82Dic3⋊Dic3—64no—
83(C2×D3)⋊Dic3—64no—
84#3711—68no—
85C22×C33C2×D411yescharacteristic
86C2×C32×D3—21no—
87C2×C32×D3—21no—
88C2×C32×D3—21no—
89C33×C4—41no—
90C33×C4C2×D411yescharacteristic
91C33×C4#4111yescharacteristic
92C3⋊(C3⋊Dic3)—41no—
93C3×C3⋊Dic3—41no—
94C32×Dic3—21no—
95C32×Dic3—41no—
96C3×C22⋊Q8—34no—
97#946C3⋊D311yescharacteristic
98C2×#216—94no—
99C2×#216—91no—
100C2×#216—184no—
101#924—94no—
102C2×#208—94no—
103C2×ΓL(1,𝔽25)—91no—
104#897—91no—
105C3×#103—91no—
106#835—34no—
107#816—91no—
108#810—94no—
109#808—94no—
110#808—91no—
111C23×C32—31no—
112C22×C3×D3#6714yes—
113C32×Q8—28no—
114C32×Q8—21no—
115C32×Q8—24no—
116C32×Q8—21no—
117C32×Q8—24no—
118C32×Q8—44no—
119C2×C32×C4—21no—
120C2×C32×C4—24no—
121C2×C32×C4—24no—
122C2×C32×C4—31no—
123C2×C32×C4—44no—
124C2×C32×C4—61no—
125C2×C32×C4—61no—
126C2×C32×C4—61no—
127C2×C32×C4C22×D314yes—
128C2×C32×C4C22×D311yescharacteristic
129C2×C3⋊Dic3—21no—
130C2×C3⋊Dic3—61no—
131C2×C3⋊Dic3—64no—
132C2×C3⋊Dic3—124no—
133#634—68no—
134#634—61no—
135#634—64no—
136#634—61no—
137#634—64no—
138#634—124no—
139C2×C3×Dic3—24no—
140C2×C3×Dic3—64no—
141C2×C3×Dic3#6714yes—
142C3×C4×D3—24no—
143C3×C4×D3—24no—
144C3⋊(C3⋊C8)—21no—
145C3⋊(C3⋊C8)—32no—
146C3⋊(C3⋊C8)—38no—
147C3⋊(C3⋊C8)—68no—
148C3×C3⋊C8—64no—
149#470—271noSylow
150C2×C33#12111yescharacteristic
151C2×C33#13711yescharacteristic
152C2×C33#13711yescharacteristic
153C32×D3—41no—
154C2×C3×Q8—24no—
155C2×C3×Q8D3214yes—
156C2×C3×Q8C2×C3⋊D311yescharacteristic
157C22×C3×C4—34no—
158C2×C4×D3C2×C3⋊D311yescharacteristic
159C2×Dic6—94no—
160C2×Dic6—91no—
161C2×Dic6—184no—
162C2×C3×C8—91no—
163C3×C4⋊C4—34no—
164C3×C4⋊C4—64no—
165C3×C4⋊C4—91no—
166C3×C22⋊C4—64no—
167#217—184no—
168#216—188no—
169#216—182no—
170#216—1816no—
171(C2×D3)⋊C4—21no—
172(C2×D3)⋊C4—181no—
173C4⋊Dic3—34no—
174C4⋊Dic3—181no—
175C4⋊Dic3C2×C3⋊D311yescharacteristic
176Dic3⋊C4—21no—
177Dic3⋊C4—184no—
178#208—188no—
179C2×C3⋊C8—34no—
180C2×C3⋊C8—94no—
181C2×C3⋊C8—91no—
182C2×C3⋊C8—184no—
183ΓL(1,𝔽25)—182no—
184C22×C32—24no—
185C22×C32—61no—
186C22×C32—61no—
187C22×C32—61no—
188C22×C32D3×D411yescharacteristic
189C22×C32C2×#6714yes—
190C2×C3×D3—24no—
191C2×C3×D3—24no—
192C2×C3×D3—24no—
193C32×C4—24no—
194C32×C4—24no—
195C32×C4—48no—
196C32×C4—41no—
197C32×C4—44no—
198C32×C4—61no—
199C32×C4—121no—
200C32×C4#23511yescharacteristic
201C32×C4D3×D414yes—
202C32×C4D3×D411yescharacteristic
203C32×C4#23714yes—
204C3⋊Dic3—41no—
205C3⋊Dic3—128no—
206C3⋊Dic3—121no—
207C3⋊Dic3—124no—
208C3×Dic3—24no—
209C3×Dic3—44no—
210C3×Dic3—124no—
211C2×Q16—271no—
212C2×M16—271no—
213C22⋊Q8—31no—
214C22⋊Q8—271no—
215#106—271no—
216#103—91no—
217#103—271no—
218C33#47011yescharacteristic, complemented, Sylow
219C23×C3—34no—
220C22×D3#63811yescharacteristic
221C3×Q8—28no—
222C3×Q8—24no—
223C3×Q8—24no—
224C3×Q8—21no—
225C3×Q8—21no—
226C3×Q8—44no—
227C2×C3×C4—24no—
228C2×C3×C4—21no—
229C2×C3×C4—24no—
230C2×C3×C4—34no—
231C2×C3×C4—44no—
232C2×C3×C4—64no—
233C2×C3×C4—64no—
234C2×C3×C4—64no—
235C2×C3×C4—181no—
236C2×C3×C4C2×D3214yes—
237C2×C3×C4C22×C3⋊D311yescharacteristic
238C2×Dic3—21no—
239C2×Dic3—64no—
240C2×Dic3—184no—
241C2×Dic3—181no—
242C2×Dic3—364no—
243C2×Dic3#63811yescharacteristic
244C4×D3—21no—
245C4×D3—21no—
246Dic6—188no—
247Dic6—184no—
248Dic6—184no—
249Dic6—181no—
250Dic6—181no—
251Dic6—364no—
252C3×C8—181no—
253C3⋊C8—64no—
254C3⋊C8—98no—
255C3⋊C8—92no—
256C3⋊C8—188no—
257C2×C32—24no—
258C2×C32—24no—
259C2×C32—24no—
260C2×C32—121no—
261C2×C32#88411yescharacteristic
262C2×C32#91511yescharacteristic
263C2×C32#91811yescharacteristic
264C2×C32#93914yes—
265C2×C32#94214yes—
266C2×C32#95114yes—
267C3×D3—44no—
268C2×Q8—271no—
269C2×Q8D3×C3⋊D311yescharacteristic
270C22×C4—31no—
271Q16—542no—
272M16—542no—
273C2×C8—91no—
274C2×C8—271no—
275C4⋊C4—31no—
276C4⋊C4—61no—
277C4⋊C4—91no—
278C4⋊C4—541no—
279C22⋊C4—61no—
280C22⋊C4—541no—
281C22×C3—24no—
282C22×C3—64no—
283C22×C3—64no—
284C22×C3—64no—
285C22×C3D3×#6714yes—
286C22×C3C2×#63811yescharacteristic
287C2×D3—21no—
288C2×D3—21no—
289C2×D3—21no—
290C3×C4—24no—
291C3×C4—24no—
292C3×C4—48no—
293C3×C4—41no—
294C3×C4—44no—
295C3×C4—64no—
296C3×C4—124no—
297C3×C4—361no—
298C3×C4#380414yes—
299C3×C4#381114yes—
300C3×C4D4×C3⋊D311yescharacteristic
301C3×C4#383011yescharacteristic
302Dic3—21no—
303Dic3—41no—
304Dic3—124no—
305Dic3—368no—
306Dic3—361no—
307Dic3—364no—
308C32—24no—
309C32#481611yescharacteristic, complemented
310C32#505214yescomplemented
311C23—31no—
312Q8—21no—
313Q8—21no—
314Q8—541no—
315Q8—541no—
316C2×C4—21no—
317C2×C4—31no—
318C2×C4—61no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—181no—
322C2×C4—541no—
323C2×C4C2×D3×C3⋊D311yescharacteristic
324C8—181no—
325C8—272no—
326C2×C3—24no—
327C2×C3—24no—
328C2×C3—24no—
329C2×C3—124no—
330C2×C3#6394314yes—
331C2×C3#6394614yes—
332C2×C3#6397114yes—
333C2×C3#6414811yescharacteristic
334C2×C3#6415111yescharacteristic
335C2×C3#6416011yescharacteristic
336D3—41no—
337C22—61no—
338C22—61no—
339C22—61no—
340C22D3×#63811yescharacteristic
341C4—41no—
342C4—61no—
343C4—121no—
344C4—361no—
345C4—1081no—
346C4#8902811yescharacteristic
347C4#8903911yescharacteristic
348C3—24no—
349C3#1059159814yescomplemented
350C3#1059275111yescharacteristic, complemented
351C2—121no—
352C2#1172357411yescharacteristic
353C2#1172357911yescharacteristic
354C2#1172361611yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#470
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×#88863C2semidirect
5. C3×#63662C2semidirect
6. #11722053C2semidirect
7. #11722041C2semidirect
32. #63662D3semidirect
218. C33#470semidirect
309. C32#4816semidirect
310. C32#5052semidirect
349. C3#10591598semidirect
350. C3#10592751semidirect

Maximal subgroups

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

subgroupindex
C2×#888632
C32×#9462
#117234262
C2×#887862
C3×#636622
#117220532
#117220412
#105927513
#105915983