Finite Group Databasebeta

← #13159971

Subgroups — #13159971

Order 1728.

872 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 872 conjugacy classes of subgroups — too many to draw. Only its 156 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
2#11723499C211yesmaximal, characteristic
3#11723451C211yesmaximal, characteristic, complemented
4C2×#88792C211yesmaximal, characteristic
5#11722184C211yesmaximal, characteristic, complemented
6C3×#63643C211yesmaximal, characteristic, complemented
7#11722037C211yesmaximal, characteristic, complemented
8#10592861—31nomaximal
9#10590748—34nomaximal
10C2×C3×#634C2211yescharacteristic
11C2×C32×D12C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×D3×C3⋊Dic3—21no—
14C2×C33×C8C2211yescharacteristic
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C32×#204—22no—
19#88806—21no—
20#88806—21no—
21(C2×D3)⋊(C3⋊Dic3)—21no—
22#88792—22no—
23C2×C32×SD16—31no—
24#64153—31no—
25D4×C3⋊Dic3—31no—
26C2×#3754—31no—
27C2×C3×#204D314yes—
28#63898—34no—
29#63873—34no—
30C2×#3715—34no—
31#63662—31no—
32#63643D311yescharacteristic, complemented
33#63641—31no—
34#63641—34no—
35#63641—64no—
36C3×#816—34no—
37#63565—34no—
38C22×C32×D3—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)D411yescharacteristic
41C2×C3⋊(C3⋊Dic3)D411yescharacteristic
42C2×C3×C3⋊Dic3—21no—
43C2×C3×C3⋊Dic3—21no—
44C3×#634—21no—
45C3×#634—21no—
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#4818—91no—
56C2×C32×Q8—31no—
57C2×C32×D4—31no—
58C22×C3⋊Dic3—61no—
59C2×#634C2×D311yescharacteristic
60C2×C3×D12C2×D314yes—
61C2×C3×Dic6—34no—
62C2×C3×Dic6C2×D314yes—
63C2×D3×Dic3—64no—
64C32×SD16—62no—
65C2×C32×C8—24no—
66C2×C32×C8C2×D314yes—
67C2×C32×C8C2×D311yescharacteristic
68#3757—61no—
69#3754—62no—
70C4⋊(C3⋊Dic3)C2×D311yescharacteristic
71#3750—61no—
72#3750—61no—
73C4×C3⋊Dic3—31no—
74C4×C3⋊Dic3—34no—
75C4×C3⋊Dic3—64no—
76C2×C3⋊(C3⋊C8)—31no—
77C2×C3⋊(C3⋊C8)—34no—
78C2×C3⋊(C3⋊C8)—64no—
79C3×C4⋊Dic3—34no—
80C3×#204—28no—
81#3724—64no—
82#3724—64no—
83(C2×D3)⋊Dic3—64no—
84#3715—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)—21no—
92C3⋊(C3⋊Dic3)SD1612yescomplemented
93C3×C3⋊Dic3—41no—
94C3×C3⋊Dic3—41no—
95C32×Dic3—41no—
96C2×C3×SD16—34no—
97#944—94no—
98#944—91no—
99C2×#215—91no—
100D4×Dic3—94no—
101C2×#214—94no—
102C2×#204C3⋊D311yescharacteristic
103#893—91no—
104C3×#103—91no—
105#835—94no—
106#816—34no—
107Dic3⋊C8—94no—
108Dic3⋊C8—91no—
109Dic3⋊C8—184no—
110#810—91no—
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—21no—
122C2×C3⋊Dic3—21no—
123C2×C3⋊Dic3—31no—
124C2×C3⋊Dic3—31no—
125C2×C3⋊Dic3—34no—
126C2×C3⋊Dic3—34no—
127C2×C3⋊Dic3—64no—
128C2×C3⋊Dic3—64no—
129C2×C3⋊Dic3—122no—
130#634—21no—
131#634—21no—
132C2×C3×Dic3—24no—
133C2×C3×Dic3—64no—
134C2×C3×Dic3—64no—
135C3×D12—24no—
136C3×D12#6718yes—
137C3×Dic6—24no—
138C3×Dic6—24no—
139C3×Dic6—64no—
140C3×Dic6—64no—
141D3×Dic3—128no—
142C32×C8—21no—
143C32×C8—24no—
144C32×C8—44no—
145C3⋊(C3⋊C8)—61no—
146C3⋊(C3⋊C8)—64no—
147C3⋊(C3⋊C8)—124no—
148#461—271noSylow
149C2×C33#12111yescharacteristic
150C2×C33C2×SD1611yescharacteristic
151C2×C33#13711yescharacteristic
152C32×D3—42no—
153C2×C3×Q8—34no—
154C2×C3×Q8—91no—
155C2×C3×D4—34no—
156C22×Dic3—184no—
157C2×D12C2×C3⋊D311yescharacteristic
158C2×C4×D3—181no—
159C2×Dic6—34no—
160C2×Dic6C2×C3⋊D311yescharacteristic
161C3×SD16—68no—
162C2×C3×C8—24no—
163C2×C3×C8D3214yes—
164C2×C3×C8C2×C3⋊D311yescharacteristic
165C3×C4⋊C4—91no—
166#217—184no—
167#215—182no—
168#214—188no—
169(C2×D3)⋊C4—181no—
170C4⋊Dic3—34no—
171Dic3⋊C4—181no—
172Dic3⋊C4—181no—
173Dic3⋊C4—184no—
174Dic3⋊C4—184no—
175C4×Dic3—94no—
176C4×Dic3—91no—
177C4×Dic3—184no—
178C2×C3⋊C8—94no—
179C2×C3⋊C8—91no—
180C2×C3⋊C8—184no—
181#204—22no—
182C22×C32—24no—
183C22×C32—62no—
184C22×C32—121no—
185C22×C32—121no—
186C22×C32D3×D414yes—
187C22×C32D3×D411yescharacteristic
188C2×C3×D3—28no—
189C2×C3×D3—44no—
190C2×C3×D3—44no—
191C32×C4—24no—
192C32×C4—24no—
193C32×C4—121no—
194C32×C4#23511yescharacteristic
195C32×C4D3×D411yescharacteristic
196C32×C4#23714yes—
197C32×C4C2×#6714yes—
198C3⋊Dic3—32no—
199C3⋊Dic3—38no—
200C3⋊Dic3—41no—
201C3⋊Dic3—41no—
202C3⋊Dic3—68no—
203C3⋊Dic3—61no—
204C3⋊Dic3—64no—
205C3⋊Dic3—124no—
206C3×Dic3—44no—
207C3×Dic3—124no—
208C3×Dic3—124no—
209C2×SD16—31no—
210C2×SD16—271no—
211C4⋊Q8—271no—
212C4×D4—271no—
213C4⋊C8—271no—
214#103—91no—
215#103—271no—
216C33#46111yescharacteristic, complemented, Sylow
217C23×C3—64no—
218C22×D3—21no—
219C3×Q8—64no—
220C3×Q8—64no—
221C3×Q8—181no—
222C3×Q8—181no—
223C3×D4—38no—
224C3×D4—64no—
225C2×C3×C4—24no—
226C2×C3×C4—64no—
227C2×C3×C4—181no—
228C2×C3×C4—181no—
229C2×C3×C4C2×D3214yes—
230C2×C3×C4C22×C3⋊D311yescharacteristic
231C2×Dic3—21no—
232C2×Dic3—64no—
233C2×Dic3—64no—
234C2×Dic3—94no—
235C2×Dic3—91no—
236C2×Dic3—94no—
237C2×Dic3—91no—
238C2×Dic3—184no—
239C2×Dic3—184no—
240C2×Dic3—368no—
241D12—21no—
242D12#63812yes—
243C4×D3—362no—
244Dic6—21no—
245Dic6—21no—
246Dic6—64no—
247Dic6—64no—
248C3×C8—24no—
249C3×C8—21no—
250C3×C8—44no—
251C3⋊C8—184no—
252C3⋊C8—181no—
253C3⋊C8—364no—
254C2×C32—24no—
255C2×C32—24no—
256C2×C32—24no—
257C2×C32—122no—
258C2×C32#88411yescharacteristic
259C2×C32D3×SD1614yes—
260C2×C32D3×SD1611yescharacteristic
261C2×C32#91514yes—
262C2×C32#91811yescharacteristic
263C2×C32#93914yes—
264C3×D3—48no—
265C2×Q8—31no—
266C2×Q8—91no—
267C2×D4—31no—
268C22×C4—541no—
269SD16—62no—
270SD16—542no—
271C2×C8—271no—
272C2×C8D3×C3⋊D311yescharacteristic
273C4⋊C4—91no—
274C4⋊C4—541no—
275C4⋊C4—541no—
276C22⋊C4—541no—
277C42—271no—
278C22×C3—24no—
279C22×C3—68no—
280C22×C3—124no—
281C22×C3—124no—
282C22×C3#380214yes—
283C22×C3D4×C3⋊D311yescharacteristic
284C2×D3—22no—
285C2×D3—41no—
286C2×D3—41no—
287C3×C4—24no—
288C3×C4—24no—
289C3×C4—124no—
290C3×C4—361no—
291C3×C4—361no—
292C3×C4#379514yes—
293C3×C4D3×#6714yes—
294C3×C4#383011yescharacteristic
295C3×C4C2×#63811yescharacteristic
296Dic3—41no—
297Dic3—98no—
298Dic3—92no—
299Dic3—124no—
300Dic3—124no—
301Dic3—188no—
302Dic3—184no—
303Dic3—181no—
304Dic3—364no—
305C32—24no—
306C32#481811yescharacteristic, complemented
307C32#516214yescomplemented
308C23—61no—
309Q8—61no—
310Q8—61no—
311Q8—181no—
312Q8—181no—
313D4—32no—
314D4—61no—
315C2×C4—61no—
316C2×C4—181no—
317C2×C4—181no—
318C2×C4—271no—
319C2×C4—271no—
320C2×C4—1082no—
321C2×C4C2×D3×C3⋊D311yescharacteristic
322C8—21no—
323C8—541no—
324C2×C3—24no—
325C2×C3—24no—
326C2×C3—24no—
327C2×C3—128no—
328C2×C3#6379614yes—
329C2×C3#6381114yes—
330C2×C3#6388514yes—
331C2×C3SD16×C3⋊D311yescharacteristic
332C2×C3#6412411yescharacteristic
333C2×C3#6414811yescharacteristic
334D3—42no—
335C22—62no—
336C22—121no—
337C22—121no—
338C22#8902611yescharacteristic
339C4—121no—
340C4—272no—
341C4—361no—
342C4—361no—
343C4—541no—
344C4#8901411yescharacteristic
345C4D3×#63811yescharacteristic
346C3—24no—
347C3#1059074814yescomplemented
348C3#1059286111yescharacteristic, complemented
349C2—122no—
350C2#1172332111yescharacteristic
351C2#1172334411yescharacteristic
352C2#1172347411yescharacteristic
subgroupG/H
centerC22#89026
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. #11723451C2semidirect
5. #11722184C2semidirect
6. C3×#63643C2semidirect
7. #11722037C2semidirect
32. #63643D3semidirect
92. C3⋊(C3⋊Dic3)SD16semidirect
216. C33#461semidirect
306. C32#4818semidirect
307. C32#5162semidirect
347. C3#10590748semidirect
348. 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
#117234992
#117234512
C2×#887922
#117221842
C3×#636432
#117220372
#105928613
#105907483