Finite Group Databasebeta

← #13159651

Subgroups — #13159651

Order 1728.

888 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 160 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×#204C211yesmaximal, characteristic
2D12×C3⋊Dic3C211yesmaximal, characteristic
3#11723455C211yesmaximal, characteristic, complemented
4C2×#88794C211yesmaximal, characteristic
5#11722186C211yesmaximal, characteristic, complemented
6C3×#63641C211yesmaximal, characteristic, complemented
7#11722038C211yesmaximal, characteristic, complemented
8#10592861—31nomaximal
9#10590752—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C32×D12C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×D3×C3⋊Dic3—21no—
14C2×C33×C8C2211yescharacteristic
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×#204—22no—
19#88807—21no—
20#88807—21no—
21(C2×D3)⋊(C3⋊Dic3)—21no—
22#88794—22no—
23C2×C32×SD16—31no—
24#64153—31no—
25D4×C3⋊Dic3—31no—
26C2×#3754—31no—
27C2×C3×#204D314yes—
28Dic3×D12—34no—
29#63866—34no—
30C2×#3716—34no—
31#63662—31no—
32#63643—31no—
33#63643—34no—
34#63643—64no—
35#63641D311yescharacteristic, complemented
36C3×Dic3⋊C8—34no—
37#63566—34no—
38C22×C32×D3—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3⋊(C3⋊Dic3)—21no—
42#6308—21no—
43#6308—21no—
44C2×C3×C3⋊Dic3D411yescharacteristic
45C2×C3×C3⋊Dic3D411yescharacteristic
46C2×C32×Dic3—21no—
47C32×D12—21no—
48C32×D12D412yes—
49C32×Dic6—21no—
50C32×Dic6—21no—
51D3×C3⋊Dic3—42no—
52C33×C8—21no—
53C3×C3⋊(C3⋊C8)—21no—
54#5162—94no—
55#4841—91no—
56C2×C32×Q8—31no—
57C2×C32×D4—31no—
58C22×C3⋊Dic3—61no—
59C2×#634—31no—
60C2×#634—34no—
61C2×#634—64no—
62C2×C3×D12C2×D314yes—
63C2×C3×Dic6C2×D314yes—
64C2×D3×Dic3—64no—
65C32×SD16—62no—
66C2×C32×C8—24no—
67C2×C32×C8C2×D314yes—
68C2×C32×C8C2×D311yescharacteristic
69#3757—61no—
70#3754—62no—
71C4⋊(C3⋊Dic3)—31no—
72C4⋊(C3⋊Dic3)—34no—
73C4⋊(C3⋊Dic3)—64no—
74#3750—61no—
75#3750—61no—
76C4×C3⋊Dic3C2×D311yescharacteristic
77C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
78C3×C4×Dic3—34no—
79C2×C3×C3⋊C8—34no—
80C3×#204—28no—
81Dic3⋊Dic3—64no—
82Dic3⋊Dic3—64no—
83(C2×D3)⋊Dic3—64no—
84#3716—68no—
85C22×C33C2×D411yescharacteristic
86C2×C32×D3—22no—
87C2×C32×D3—41no—
88C2×C32×D3—41no—
89C33×C4C2×D411yescharacteristic
90C33×C4#4111yescharacteristic
91C3⋊(C3⋊Dic3)—41no—
92C3⋊(C3⋊Dic3)—41no—
93C3×C3⋊Dic3—21no—
94C3×C3⋊Dic3SD1612yescomplemented
95C32×Dic3—41no—
96C2×C3×SD16—34no—
97#944—94no—
98D4×Dic3—94no—
99C2×#214—94no—
100C2×#204—91no—
101C2×#204C3⋊D311yescharacteristic
102C4×D12—91no—
103#869—91no—
104C3×C4⋊C8—91no—
105#835—94no—
106#816—91no—
107#816—94no—
108#816—91no—
109#816—184no—
110Dic3⋊C8—34no—
111C23×C32—61no—
112C22×C3×D3—24no—
113C32×Q8—61no—
114C32×Q8—61no—
115C32×D4—32no—
116C32×D4—61no—
117C2×C32×C4—24no—
118C2×C32×C4—61no—
119C2×C32×C4C22×D314yes—
120C2×C32×C4C22×D311yescharacteristic
121C2×C3⋊Dic3—61no—
122C2×C3⋊Dic3—61no—
123C2×C3⋊Dic3—64no—
124C2×C3⋊Dic3—64no—
125C2×C3⋊Dic3—124no—
126C2×C3⋊Dic3—124no—
127C2×C3⋊Dic3—122no—
128C2×C3⋊Dic3D1211yescharacteristic
129C2×C3⋊Dic3D1211yescharacteristic
130#634—68no—
131#634—61no—
132#634—64no—
133#634—61no—
134#634—64no—
135#634—124no—
136C2×C3×Dic3—24no—
137C2×C3×Dic3—34no—
138C2×C3×Dic3—34no—
139C3×D12—24no—
140C3×D12#6718yes—
141C3×Dic6—24no—
142C3×Dic6—24no—
143D3×Dic3—128no—
144C32×C8—21no—
145C32×C8—24no—
146C32×C8—44no—
147C3⋊(C3⋊C8)—21no—
148C3×C3⋊C8—64no—
149#461—271noSylow
150C2×C33#12111yescharacteristic
151C2×C33C2×SD1611yescharacteristic
152C2×C33#13711yescharacteristic
153C32×D3—42no—
154C2×C3×Q8—34no—
155C2×C3×D4—34no—
156C22×Dic3—184no—
157C2×D12C2×C3⋊D311yescharacteristic
158C2×C4×D3—181no—
159C2×Dic6—94no—
160C2×Dic6—91no—
161C2×Dic6—184no—
162C2×Dic6C2×C3⋊D311yescharacteristic
163C3×SD16—68no—
164C2×C3×C8—24no—
165C2×C3×C8—91no—
166C2×C3×C8D3214yes—
167C2×C3×C8C2×C3⋊D311yescharacteristic
168C3×C42—91no—
169#217—184no—
170#214—188no—
171(C2×D3)⋊C4—181no—
172C4⋊Dic3—94no—
173C4⋊Dic3—91no—
174C4⋊Dic3—184no—
175C4⋊Dic3—181no—
176C4⋊Dic3—181no—
177Dic3⋊C4—184no—
178Dic3⋊C4—184no—
179C4×Dic3—34no—
180C2×C3⋊C8—34no—
181#204—22no—
182#204—182no—
183C22×C32—24no—
184C22×C32—62no—
185C22×C32—121no—
186C22×C32—121no—
187C22×C32C2×D1211yescharacteristic
188C22×C32D3×D414yes—
189C2×C3×D3—28no—
190C2×C3×D3—44no—
191C2×C3×D3—44no—
192C32×C4—24no—
193C32×C4—24no—
194C32×C4—121no—
195C32×C4D3×D411yescharacteristic
196C32×C4#23714yes—
197C32×C4#23911yescharacteristic
198C32×C4C2×#6714yes—
199C3⋊Dic3—21no—
200C3⋊Dic3—128no—
201C3⋊Dic3—121no—
202C3⋊Dic3—121no—
203C3⋊Dic3—124no—
204C3⋊Dic3—124no—
205C3⋊Dic3—244no—
206C3⋊Dic3#20412yescomplemented
207C3×Dic3—38no—
208C3×Dic3—44no—
209C3×Dic3—64no—
210C2×SD16—31no—
211C2×SD16—271no—
212C4⋊Q8—271no—
213C4×D4—271no—
214C4⋊C8—91no—
215#103—271no—
216#103—271no—
217C33#46111yescharacteristic, complemented, Sylow
218C23×C3—64no—
219C22×D3—21no—
220C3×Q8—64no—
221C3×Q8—64no—
222C3×D4—38no—
223C3×D4—64no—
224C2×C3×C4—24no—
225C2×C3×C4—64no—
226C2×C3×C4—91no—
227C2×C3×C4—91no—
228C2×C3×C4C2×D3214yes—
229C2×C3×C4C22×C3⋊D311yescharacteristic
230C2×Dic3—21no—
231C2×Dic3—34no—
232C2×Dic3—34no—
233C2×Dic3—184no—
234C2×Dic3—184no—
235C2×Dic3—181no—
236C2×Dic3—181no—
237C2×Dic3—364no—
238C2×Dic3—364no—
239C2×Dic3—368no—
240D12—21no—
241D12#63812yes—
242C4×D3—362no—
243Dic6—21no—
244Dic6—21no—
245Dic6—188no—
246Dic6—184no—
247Dic6—184no—
248Dic6—181no—
249Dic6—181no—
250Dic6—364no—
251C3×C8—24no—
252C3×C8—21no—
253C3×C8—44no—
254C3×C8—181no—
255C3⋊C8—64no—
256C2×C32—24no—
257C2×C32—24no—
258C2×C32—24no—
259C2×C32—122no—
260C2×C32#89511yescharacteristic
261C2×C32C2×#20411yescharacteristic
262C2×C32#90911yescharacteristic
263C2×C32D3×SD1614yes—
264C2×C32#91514yes—
265C2×C32#93914yes—
266C3×D3—48no—
267C2×Q8—31no—
268C2×Q8—271no—
269C2×D4—31no—
270C22×C4—541no—
271SD16—62no—
272SD16—542no—
273C2×C8—91no—
274C2×C8D3×C3⋊D311yescharacteristic
275C4⋊C4—271no—
276C4⋊C4—541no—
277C4⋊C4—541no—
278C22⋊C4—541no—
279C42—91no—
280C22×C3—24no—
281C22×C3—68no—
282C22×C3—124no—
283C22×C3—124no—
284C22×C3D3×D1214yes—
285C22×C3D4×C3⋊D311yescharacteristic
286C2×D3—22no—
287C2×D3—41no—
288C2×D3—41no—
289C3×C4—24no—
290C3×C4—24no—
291C3×C4—92no—
292C3×C4—124no—
293C3×C4—181no—
294C3×C4#379614yes—
295C3×C4D3×#6714yes—
296C3×C4#383011yescharacteristic
297C3×C4C2×#63811yescharacteristic
298Dic3—38no—
299Dic3—41no—
300Dic3—64no—
301Dic3—368no—
302Dic3—364no—
303Dic3—361no—
304Dic3—364no—
305Dic3—361no—
306Dic3—724no—
307C32—24no—
308C32#484111yescharacteristic, complemented
309C32#516214yescomplemented
310C23—61no—
311Q8—61no—
312Q8—61no—
313Q8—541no—
314Q8—541no—
315D4—32no—
316D4—61no—
317C2×C4—61no—
318C2×C4—91no—
319C2×C4—91no—
320C2×C4—541no—
321C2×C4—541no—
322C2×C4—1082no—
323C2×C4C2×D3×C3⋊D311yescharacteristic
324C8—21no—
325C8—181no—
326C2×C3—24no—
327C2×C3—24no—
328C2×C3—24no—
329C2×C3—128no—
330C2×C3D3×#20414yes—
331C2×C3#6380014yes—
332C2×C3#6388614yes—
333C2×C3SD16×C3⋊D311yescharacteristic
334C2×C3#6412411yescharacteristic
335C2×C3#6414811yescharacteristic
336D3—42no—
337C22—62no—
338C22—121no—
339C22—121no—
340C22C3⋊D3×D1211yescharacteristic
341C4—92no—
342C4—121no—
343C4—181no—
344C4—1081no—
345C4—1081no—
346C4#8901511yescharacteristic
347C4D3×#63811yescharacteristic
348C3—24no—
349C3#1059075214yescomplemented
350C3#1059286111yescharacteristic, complemented
351C2—122no—
352C2C3⋊D3×#20411yescharacteristic
353C2#1172334011yescharacteristic
354C2#1172347511yescharacteristic
subgroupG/H
centerC22C3⋊D3×D12
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#461
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
3. #11723455C2semidirect
5. #11722186C2semidirect
6. C3×#63641C2semidirect
7. #11722038C2semidirect
35. #63641D3semidirect
94. C3×C3⋊Dic3SD16semidirect
206. C3⋊Dic3#204semidirect
217. C33#461semidirect
308. C32#4841semidirect
309. C32#5162semidirect
349. C3#10590752semidirect
350. C3#10592861semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2042
D12×C3⋊Dic32
#117234552
C2×#887942
#117221862
C3×#636412
#117220382
#105928613
#105907523