Finite Group Databasebeta

← #13159556

Subgroups — #13159556

Order 1728.

824 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 144 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
1#11723503C211yesmaximal, characteristic
2#11723456C211yesmaximal, characteristic
3#11722190C211yesmaximal, characteristic
4C3×#63645C211yesmaximal, characteristic, complemented
5C32×#816C211yesmaximal, characteristic, complemented
6#11722056C211yesmaximal, characteristic, complemented
7#11722037C211yesmaximal, characteristic, complemented
8#10592510—31nomaximal
9#10590689—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×C32×Dic6C2211yescharacteristic
12C2×#6289—21no—
13C2×C33×C8C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18#88807—21no—
19#88806—21no—
20#88804—21no—
21#64155—31no—
22C4⋊C4×C3⋊D3—31no—
23#63900—34no—
24#63871—34no—
25C32×#103—31no—
26#63662—31no—
27#63647—31no—
28#63647—34no—
29#63647—64no—
30#63645D311yescharacteristic, complemented
31#63634—31no—
32C3×#818—34no—
33C3×#816D314yescomplemented
34#63576—34no—
35#63565—34no—
36C22×C3⋊(C3⋊D3)D411yescharacteristic
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)D411yescharacteristic
39C4×C3⋊(C3⋊D3)—21no—
40C4×C3⋊(C3⋊D3)—21no—
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3—21no—
43C2×C32×Dic3—21no—
44C2×C32×Dic3—21no—
45C32×Dic6—21no—
46C32×Dic6—21no—
47#6289—42no—
48C33×C8—21no—
49C3⋊(C3⋊(C3⋊C8))—21no—
50#4884—91no—
51#4811—94no—
52C2×C32×Q8—31no—
53C2×C4×C3⋊D3—31no—
54C2×C4×C3⋊D3—34no—
55C2×C4×C3⋊D3—61no—
56C2×C4×C3⋊D3—64no—
57C2×C3×Dic6C2×D314yes—
58C2×#624—64no—
59C2×C32×C8—24no—
60C2×C32×C8C2×D314yes—
61C2×C32×C8C2×D311yescharacteristic
62C32×C4⋊C4—31no—
63#3752—61no—
64C4⋊(C3⋊Dic3)C2×D311yescharacteristic
65C4⋊(C3⋊Dic3)C2×D311yescharacteristic
66#3750—61no—
67#3750—61no—
68C2×C3⋊(C3⋊C8)—31no—
69C2×C3⋊(C3⋊C8)—34no—
70C2×C3⋊(C3⋊C8)—64no—
71C3×C4⋊Dic3—34no—
72C3×C4⋊Dic3—34no—
73C3×C4⋊Dic3C2×D314yes—
74#3724—64no—
75Dic3⋊Dic3—64no—
76#3722—64no—
77C22×C33C2×D411yescharacteristic
78C2×C3⋊(C3⋊D3)—21no—
79C2×C3⋊(C3⋊D3)—21no—
80C2×C3⋊(C3⋊D3)Q1612yes—
81C33×C4#4111yescharacteristic
82C33×C4#4111yescharacteristic
83C3⋊(C3⋊Dic3)—21no—
84C3×C3⋊Dic3—41no—
85C3×C3⋊Dic3—41no—
86C32×Dic3—41no—
87C32×Dic3—41no—
88#946—94no—
89#897—91no—
90D3×C4⋊C4—94no—
91D3×C4⋊C4—91no—
92C3×#107—91no—
93C3×#103—34no—
94#835—94no—
95(C2×D3)⋊C8—94no—
96(C2×D3)⋊C8—91no—
97(C2×D3)⋊C8—184no—
98#818—34no—
99#816C3⋊D311yescharacteristic, complemented
100#810—91no—
101#807—94no—
102#807—91no—
103C22×C3⋊D3—31no—
104C22×C3⋊D3—34no—
105C22×C3⋊D3—64no—
106C32×Q8—61no—
107C32×Q8—61no—
108C2×C32×C4—24no—
109C2×C32×C4—61no—
110C2×C32×C4—61no—
111C2×C32×C4C22×D314yes—
112C2×C32×C4C22×D311yescharacteristic
113C2×C3⋊Dic3—21no—
114C2×C3⋊Dic3—21no—
115C2×C3⋊Dic3—31no—
116C2×C3⋊Dic3—34no—
117C2×C3⋊Dic3—64no—
118C4×C3⋊D3—61no—
119C4×C3⋊D3—61no—
120C4×C3⋊D3—64no—
121C4×C3⋊D3—64no—
122C4×C3⋊D3—124no—
123C4×C3⋊D3—124no—
124C4×C3⋊D3—122no—
125C2×C3×Dic3—24no—
126C2×C3×Dic3—24no—
127C2×C3×Dic3—64no—
128C2×C3×Dic3—64no—
129C3×Dic6—24no—
130C3×Dic6—24no—
131#624—128no—
132C32×C8—21no—
133C32×C8—24no—
134C32×C8—44no—
135C3⋊(C3⋊C8)—61no—
136C3⋊(C3⋊C8)—64no—
137C3⋊(C3⋊C8)—124no—
138#484—271noSylow
139C2×C33#12311yescharacteristic
140C2×C33C2×Q1611yescharacteristic
141C2×C33AGL(1,ℤ/8)11yescharacteristic
142C3⋊(C3⋊D3)—22no—
143C2×C3×Q8—34no—
144C2×C4×D3—94no—
145C2×C4×D3—91no—
146C2×C4×D3—184no—
147C2×C4×D3—181no—
148C2×C4×D3—184no—
149C2×Dic6C2×C3⋊D311yescharacteristic
150C2×C3×C8—24no—
151C2×C3×C8D3214yes—
152C2×C3×C8C2×C3⋊D311yescharacteristic
153C3×C4⋊C4—34no—
154C3×C4⋊C4—91no—
155C3×C4⋊C4—91no—
156(C2×D3)⋊C4—181no—
157(C2×D3)⋊C4—184no—
158C4⋊Dic3—34no—
159C4⋊Dic3—34no—
160C4⋊Dic3—181no—
161C4⋊Dic3C2×C3⋊D311yescharacteristic
162Dic3⋊C4—181no—
163Dic3⋊C4—184no—
164Dic3⋊C4—184no—
165C2×C3⋊C8—94no—
166C2×C3⋊C8—91no—
167C2×C3⋊C8—184no—
168C22×C32—24no—
169C22×C32D3×D414yes—
170C22×C32D3×D411yescharacteristic
171C2×C3⋊D3—32no—
172C2×C3⋊D3—38no—
173C2×C3⋊D3—68no—
174C2×C3⋊D3—61no—
175C2×C3⋊D3—61no—
176C2×C3⋊D3—64no—
177C2×C3⋊D3—64no—
178C2×C3⋊D3—124no—
179C2×C3⋊D3—124no—
180C32×C4—24no—
181C32×C4—24no—
182C32×C4—121no—
183C32×C4—121no—
184C32×C4#23514yes—
185C32×C4#23511yescharacteristic
186C32×C4#23714yes—
187C32×C4#23911yescharacteristic
188C3⋊Dic3—41no—
189C3⋊Dic3—41no—
190C3⋊Dic3—61no—
191C3⋊Dic3—64no—
192C3⋊Dic3—124no—
193C3×Dic3—44no—
194C3×Dic3—44no—
195C3×Dic3—124no—
196C3×Dic3—124no—
197C22⋊Q8—271no—
198C2×C4⋊C4—271no—
199#107—91no—
200#107—271no—
201#103—31no—
202#103—271no—
203C22⋊C8—271no—
204C33#48411yescharacteristic, complemented, Sylow
205C22×D3—94no—
206C22×D3—91no—
207C22×D3—184no—
208C3×Q8—64no—
209C3×Q8—64no—
210C2×C3×C4—24no—
211C2×C3×C4—64no—
212C2×C3×C4—64no—
213C2×C3×C4—181no—
214C2×C3×C4—181no—
215C2×C3×C4C2×D3214yes—
216C2×C3×C4C22×C3⋊D311yescharacteristic
217C2×Dic3—21no—
218C2×Dic3—21no—
219C2×Dic3—64no—
220C2×Dic3—64no—
221C2×Dic3—94no—
222C2×Dic3—91no—
223C2×Dic3—184no—
224C4×D3—184no—
225C4×D3—184no—
226C4×D3—181no—
227C4×D3—181no—
228C4×D3—364no—
229C4×D3—364no—
230C4×D3—362no—
231C4×D3—368no—
232Dic6—21no—
233Dic6—21no—
234C3×C8—24no—
235C3×C8—21no—
236C3×C8—44no—
237C3⋊C8—184no—
238C3⋊C8—181no—
239C3⋊C8—364no—
240C2×C32—24no—
241C2×C32—24no—
242C2×C32—24no—
243C2×C32#88314yes—
244C2×C32#89411yescharacteristic
245C2×C32#91111yescharacteristic
246C2×C32#91414yes—
247C2×C32D3×Q1614yes—
248C2×C32D3×Q1611yescharacteristic
249C3⋊D3—62no—
250C3⋊D3—68no—
251C3⋊D3—128no—
252C2×Q8—31no—
253C22×C4—271no—
254C22×C4—541no—
255C2×C8—271no—
256C2×C8D3×C3⋊D311yescharacteristic
257C4⋊C4—31no—
258C4⋊C4—91no—
259C4⋊C4—91no—
260C4⋊C4—541no—
261C4⋊C4—541no—
262C22⋊C4—541no—
263C22×C3—24no—
264C22×C3#380214yes—
265C22×C3D4×C3⋊D311yescharacteristic
266C2×D3—98no—
267C2×D3—92no—
268C2×D3—188no—
269C2×D3—184no—
270C2×D3—181no—
271C2×D3—184no—
272C2×D3—181no—
273C2×D3—364no—
274C2×D3—364no—
275C3×C4—24no—
276C3×C4—24no—
277C3×C4—124no—
278C3×C4—124no—
279C3×C4—361no—
280C3×C4—361no—
281C3×C4#379914yes—
282C3×C4#380414yes—
283C3×C4#382811yescharacteristic
284C3×C4#383011yescharacteristic
285Dic3—41no—
286Dic3—41no—
287Dic3—124no—
288Dic3—124no—
289Dic3—181no—
290Dic3—184no—
291Dic3—364no—
292C32—24no—
293C32#481114yescomplemented
294C32#488411yescharacteristic, complemented
295C23—271no—
296Q8—61no—
297Q8—61no—
298C2×C4—61no—
299C2×C4—61no—
300C2×C4—181no—
301C2×C4—181no—
302C2×C4—271no—
303C2×C4—541no—
304C2×C4—541no—
305C2×C4—1082no—
306C2×C4C2×D3×C3⋊D311yescharacteristic
307C8—21no—
308C8—541no—
309C2×C3—24no—
310C2×C3—24no—
311C2×C3—24no—
312C2×C3#6379814yes—
313C2×C3#6380214yes—
314C2×C3#6385414yes—
315C2×C3#6409211yescharacteristic
316C2×C3#6412311yescharacteristic
317C2×C3Q16×C3⋊D311yescharacteristic
318D3—188no—
319D3—182no—
320D3—368no—
321C22—272no—
322C22—541no—
323C22—541no—
324C22#8902611yescharacteristic
325C4—121no—
326C4—121no—
327C4—361no—
328C4—361no—
329C4—541no—
330C4#8902211yescharacteristic
331C4#8902811yescharacteristic
332C3—24no—
333C3#1059068914yescomplemented
334C3#1059251011yescharacteristic, complemented
335C2—542no—
336C2#1172332311yescharacteristic
337C2#1172333011yescharacteristic
338C2#1172342211yescharacteristic
subgroupG/H
centerC22#89026
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#484
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
4. C3×#63645C2semidirect
5. C32×#816C2semidirect
6. #11722056C2semidirect
7. #11722037C2semidirect
30. #63645D3semidirect
33. C3×#816D3semidirect
99. #816C3⋊D3semidirect
204. C33#484semidirect
293. C32#4811semidirect
294. C32#4884semidirect
333. C3#10590689semidirect
334. C3#10592510semidirect

Maximal subgroups

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

subgroupindex
#117235032
#117234562
#117221902
C3×#636452
C32×#8162
#117220562
#117220372
#105925103
#105906893