Finite Group Databasebeta

← #13159901

Subgroups — #13159901

Order 1728.

760 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 142 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#11723536C211yesmaximal, characteristic, complemented
2#11723434C211yesmaximal, characteristic
3#11722185C211yesmaximal, characteristic, complemented
4C3×#63643C211yesmaximal, characteristic, complemented
5C32×#821C211yesmaximal, characteristic
6#11722039C211yesmaximal, characteristic, complemented
7#11722031C211yesmaximal, characteristic
8#10592464—31nomaximal
9#10590866—34nomaximal
10C2×C3×#634C2211yescharacteristic
11C2×C32×D12C2211yescharacteristic
12C2×#6293—21no—
13C2×C33×C8C2211yescharacteristic
14C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18#88806—21no—
19(C2×D3)⋊(C3⋊Dic3)—21no—
20(C2×D3)⋊(C3⋊Dic3)—21no—
21#64145—31no—
22#64097—31no—
23#63919—34no—
24#63863—34no—
25C32×#102—31no—
26#63659—31no—
27#63643D311yescharacteristic, complemented
28#63642—31no—
29#63642—34no—
30#63642—64no—
31#63637—31no—
32C3×#821D314yes—
33C3×#816—34no—
34#63565—34no—
35#63562—34no—
36C22×C32×D3—21no—
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)D411yescharacteristic
39C2×C3⋊(C3⋊Dic3)D411yescharacteristic
40C2×C3×C3⋊Dic3—21no—
41C2×C3×C3⋊Dic3—21no—
42C3×#634—21no—
43C3×#634—21no—
44C2×C32×Dic3—21no—
45C32×D12—21no—
46C32×D12—21no—
47#6293—42no—
48C33×C8—21no—
49C3⋊(C3⋊(C3⋊C8))—21no—
50#4794—91no—
51#4765—94no—
52C2×C32×D4—31no—
53C2×#634—61no—
54C2×#634C2×D311yescharacteristic
55C2×C3×D12C2×D314yes—
56C2×C3×Dic6—34no—
57C2×#627—64no—
58C2×C32×C8—24no—
59C2×C32×C8C2×D314yes—
60C2×C32×C8C2×D311yescharacteristic
61C32×C4⋊C4—31no—
62#3757—61no—
63#3757—61no—
64C4⋊(C3⋊Dic3)C2×D311yescharacteristic
65#3750—61no—
66C4×C3⋊Dic3—31no—
67C4×C3⋊Dic3—34no—
68C4×C3⋊Dic3—64no—
69C2×C3⋊(C3⋊C8)—31no—
70C2×C3⋊(C3⋊C8)—34no—
71C2×C3⋊(C3⋊C8)—64no—
72C3×C4⋊Dic3—34no—
73C3×C4⋊Dic3C2×D314yes—
74#3724—64no—
75(C2×D3)⋊Dic3—64no—
76(C2×D3)⋊Dic3—64no—
77C22×C33C2×D411yescharacteristic
78C2×C32×D3—41no—
79C2×C32×D3—41no—
80C2×C32×D3—41no—
81C33×C4#4111yescharacteristic
82C33×C4#4111yescharacteristic
83C3⋊(C3⋊Dic3)—21no—
84C3⋊(C3⋊Dic3)—21no—
85C3×C3⋊Dic3—41no—
86C3×C3⋊Dic3—41no—
87C32×Dic3—41no—
88#947—91no—
89#936—94no—
90#888—94no—
91#888—91no—
92C3×#103—91no—
93C3×#102—34no—
94#835—91no—
95#832—94no—
96#821C3⋊D311yescharacteristic
97#816—34no—
98#815—94no—
99#815—91no—
100#815—184no—
101#810—94no—
102#809—91no—
103C23×C32—61no—
104C22×C3×D3—24no—
105C32×D4—61no—
106C32×D4—61no—
107C2×C32×C4—24no—
108C2×C32×C4—61no—
109C2×C32×C4C22×D314yes—
110C2×C32×C4C22×D311yescharacteristic
111C2×C3⋊Dic3—21no—
112C2×C3⋊Dic3—21no—
113C2×C3⋊Dic3—31no—
114C2×C3⋊Dic3—31no—
115C2×C3⋊Dic3—34no—
116C2×C3⋊Dic3—34no—
117C2×C3⋊Dic3—64no—
118C2×C3⋊Dic3—64no—
119#634—21no—
120#634—21no—
121#634—122no—
122C2×C3×Dic3—24no—
123C2×C3×Dic3—64no—
124C2×C3×Dic3—64no—
125C3×D12—24no—
126C3×D12—24no—
127C3×Dic6—64no—
128C3×Dic6—64no—
129#627—128no—
130C32×C8—28no—
131C32×C8—24no—
132C32×C8—21no—
133C3⋊(C3⋊C8)—68no—
134C3⋊(C3⋊C8)—64no—
135C3⋊(C3⋊C8)—61no—
136#489—271noSylow
137C2×C33#12411yescharacteristic
138C2×C33AGL(1,ℤ/8)11yescharacteristic
139C2×C33#13711yescharacteristic
140C32×D3—81no—
141C2×C3×Q8—91no—
142C2×C3×D4—34no—
143C2×D12C2×C3⋊D311yescharacteristic
144C2×Dic6—34no—
145C2×Dic6—181no—
146C2×Dic6—184no—
147C2×C3×C8—24no—
148C2×C3×C8D3214yes—
149C2×C3×C8C2×C3⋊D311yescharacteristic
150C3×C4⋊C4—34no—
151C3×C4⋊C4—91no—
152#217—184no—
153#217—184no—
154(C2×D3)⋊C4—181no—
155(C2×D3)⋊C4—181no—
156C4⋊Dic3—34no—
157C4⋊Dic3C2×C3⋊D311yescharacteristic
158Dic3⋊C4—181no—
159Dic3⋊C4—184no—
160C4×Dic3—94no—
161C4×Dic3—91no—
162C4×Dic3—184no—
163C2×C3⋊C8—94no—
164C2×C3⋊C8—91no—
165C2×C3⋊C8—184no—
166C22×C32—24no—
167C22×C32—121no—
168C22×C32—121no—
169C22×C32—121no—
170C22×C32D3×D414yes—
171C22×C32D3×D411yescharacteristic
172C2×C3×D3—44no—
173C2×C3×D3—44no—
174C2×C3×D3—44no—
175C32×C4—24no—
176C32×C4—24no—
177C32×C4—121no—
178C32×C4#23514yes—
179C32×C4#23511yescharacteristic
180C32×C4#23714yes—
181C32×C4#23711yescharacteristic
182C3⋊Dic3—41no—
183C3⋊Dic3—41no—
184C3⋊Dic3—61no—
185C3⋊Dic3—61no—
186C3⋊Dic3—64no—
187C3⋊Dic3—64no—
188C3⋊Dic3—124no—
189C3⋊Dic3—124no—
190C3×Dic3—44no—
191C3×Dic3—124no—
192C3×Dic3—124no—
193C4⋊Q8—271no—
194#124—271no—
195#103—91no—
196#103—271no—
197#102—31no—
198#102—271no—
199#97—271no—
200C33#48911yescharacteristic, complemented, Sylow
201C23×C3—64no—
202C22×D3—21no—
203C3×Q8—181no—
204C3×Q8—181no—
205C3×D4—64no—
206C3×D4—64no—
207C2×C3×C4—24no—
208C2×C3×C4—64no—
209C2×C3×C4—181no—
210C2×C3×C4—181no—
211C2×C3×C4C2×D3214yes—
212C2×C3×C4C22×C3⋊D311yescharacteristic
213C2×Dic3—21no—
214C2×Dic3—64no—
215C2×Dic3—64no—
216C2×Dic3—94no—
217C2×Dic3—94no—
218C2×Dic3—91no—
219C2×Dic3—91no—
220C2×Dic3—184no—
221C2×Dic3—184no—
222D12—21no—
223D12—21no—
224Dic6—64no—
225Dic6—64no—
226Dic6—362no—
227Dic6—368no—
228C3×C8—28no—
229C3×C8—24no—
230C3×C8—21no—
231C3⋊C8—188no—
232C3⋊C8—184no—
233C3⋊C8—181no—
234C2×C32—24no—
235C2×C32—24no—
236C2×C32—24no—
237C2×C32—241no—
238C2×C32#88514yes—
239C2×C32#88511yescharacteristic
240C2×C32#91114yes—
241C2×C32#91411yescharacteristic
242C2×C32#91514yes—
243C2×C32#91811yescharacteristic
244C3×D3—84no—
245C2×Q8—91no—
246C2×Q8—541no—
247C2×D4—31no—
248C2×C8—271no—
249C2×C8D3×C3⋊D311yescharacteristic
250C4⋊C4—31no—
251C4⋊C4—91no—
252C4⋊C4—541no—
253C22⋊C4—541no—
254C22⋊C4—541no—
255C42—271no—
256C22×C3—24no—
257C22×C3—124no—
258C22×C3—124no—
259C22×C3—124no—
260C22×C3#380214yes—
261C22×C3D4×C3⋊D311yescharacteristic
262C2×D3—41no—
263C2×D3—41no—
264C2×D3—41no—
265C3×C4—24no—
266C3×C4—24no—
267C3×C4—124no—
268C3×C4—361no—
269C3×C4—361no—
270C3×C4#379814yes—
271C3×C4#380514yes—
272C3×C4#382811yescharacteristic
273C3×C4#383011yescharacteristic
274Dic3—41no—
275Dic3—124no—
276Dic3—124no—
277Dic3—184no—
278Dic3—184no—
279Dic3—181no—
280Dic3—181no—
281Dic3—364no—
282Dic3—364no—
283C32—24no—
284C32#476514yescomplemented
285C32#479411yescharacteristic, complemented
286C23—61no—
287Q8—181no—
288Q8—181no—
289Q8—1082no—
290D4—61no—
291D4—61no—
292C2×C4—61no—
293C2×C4—181no—
294C2×C4—181no—
295C2×C4—271no—
296C2×C4—271no—
297C2×C4C2×D3×C3⋊D311yescharacteristic
298C8—21no—
299C8—541no—
300C2×C3—24no—
301C2×C3—24no—
302C2×C3—24no—
303C2×C3—244no—
304C2×C3#6379814yes—
305C2×C3#6381114yes—
306C2×C3#6391514yes—
307C2×C3#6409411yescharacteristic
308C2×C3#6412011yescharacteristic
309C2×C3#6412411yescharacteristic
310D3—81no—
311C22—121no—
312C22—121no—
313C22—121no—
314C22#8902611yescharacteristic
315C4—121no—
316C4—361no—
317C4—361no—
318C4—541no—
319C4—541no—
320C4#8902011yescharacteristic
321C4#8902911yescharacteristic
322C3—24no—
323C3#1059086614yescomplemented
324C3#1059246411yescharacteristic, complemented
325C2—241no—
326C2#1172332411yescharacteristic
327C2#1172334411yescharacteristic
328C2#1172353011yescharacteristic
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#489
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
1. #11723536C2semidirect
3. #11722185C2semidirect
4. C3×#63643C2semidirect
6. #11722039C2semidirect
27. #63643D3semidirect
200. C33#489semidirect
284. C32#4765semidirect
285. C32#4794semidirect
323. C3#10590866semidirect
324. C3#10592464semidirect

Maximal subgroups

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

subgroupindex
#117235362
#117234342
#117221852
C3×#636432
C32×#8212
#117220392
#117220312
#105924643
#105908663