Finite Group Databasebeta

← #13159687

Subgroups — #13159687

Order 1728.

848 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 848 conjugacy classes of subgroups — too many to draw. Only its 166 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#11723534C211yesmaximal, characteristic, complemented
2#11723484C211yesmaximal, characteristic
3#11722191C211yesmaximal, characteristic
4C3×#63645C211yesmaximal, characteristic, complemented
5C32×Dic3⋊C8C211yesmaximal, characteristic, complemented
6#11722055C211yesmaximal, characteristic, complemented
7#11722036C211yesmaximal, characteristic
8#10592545—31nomaximal
9#10590764—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×#6293—21no—
12C2×#6289—21no—
13C2×C33×C8C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19Dic3⋊(C3⋊Dic3)—21no—
20#88804—21no—
21C4×C4⋊(C3⋊D3)—31no—
22#64078—31no—
23#63918—34no—
24#63894—34no—
25C32×C4⋊C8—31no—
26#63648—31no—
27#63648—31no—
28#63648—34no—
29#63648—64no—
30#63645—31no—
31#63645D311yescharacteristic, complemented
32C3×#818—34no—
33C3×Dic3⋊C8D314yescomplemented
34#63575—34no—
35#63564—34no—
36C22×C3⋊(C3⋊D3)—21no—
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)—21no—
39C4⋊(C3⋊(C3⋊D3))—21no—
40C4⋊(C3⋊(C3⋊D3))Q812yescomplemented
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3—21no—
43C2×C32×Dic3D411yescharacteristic
44C2×C32×Dic3D411yescharacteristic
45#6293—42no—
46#6289—42no—
47C33×C8—21no—
48C32×C3⋊C8—21no—
49#4890—91no—
50#4846—94no—
51C2×C4⋊(C3⋊D3)—31no—
52C2×C4⋊(C3⋊D3)—34no—
53C2×C4⋊(C3⋊D3)—64no—
54C2×C4×C3⋊D3—61no—
55C2×#634—61no—
56C2×#627—64no—
57C2×#624—64no—
58C2×C32×C8—24no—
59C2×C32×C8—31no—
60C2×C32×C8C2×D314yes—
61C2×C32×C8C2×D311yescharacteristic
62C32×C42—31no—
63#3752—61no—
64C4⋊(C3⋊Dic3)—31no—
65C4⋊(C3⋊Dic3)—34no—
66C4⋊(C3⋊Dic3)—61no—
67C4⋊(C3⋊Dic3)—64no—
68C4⋊(C3⋊Dic3)C2×D311yescharacteristic
69C4⋊(C3⋊Dic3)C2×D311yescharacteristic
70C3×C4⋊Dic3—34no—
71C3×C4⋊Dic3—34no—
72C3×C4×Dic3C2×D314yes—
73C2×C3×C3⋊C8C2×D314yes—
74Dic3⋊Dic3—64no—
75#3722—64no—
76C22×C33C2×D411yescharacteristic
77C2×C3⋊(C3⋊D3)—22no—
78C2×C3⋊(C3⋊D3)—41no—
79C2×C3⋊(C3⋊D3)—41no—
80C33×C4C2×Q811yescharacteristic
81C33×C4#4111yescharacteristic
82C3⋊(C3⋊Dic3)—41no—
83C3×C3⋊Dic3—41no—
84C3×C3⋊Dic3—41no—
85C32×Dic3—21no—
86C32×Dic3D812yes—
87#893—91no—
88#888—91no—
89C4×D12—94no—
90#869—94no—
91C3×#107—91no—
92C3×C4⋊C8—34no—
93#821—94no—
94#821—94no—
95#821—91no—
96#821—184no—
97#818—34no—
98#818—94no—
99Dic3⋊C8C3⋊D311yescharacteristic, complemented
100#809—91no—
101#807—91no—
102C22×C3⋊D3—61no—
103C22×C3⋊D3—64no—
104C22×C3⋊D3—124no—
105C2×C32×C4—24no—
106C2×C32×C4—31no—
107C2×C32×C4—31no—
108C2×C32×C4C22×D314yes—
109C2×C32×C4C22×D311yescharacteristic
110C2×C3⋊Dic3—21no—
111C2×C3⋊Dic3—21no—
112C2×C3⋊Dic3—61no—
113C2×C3⋊Dic3—64no—
114C2×C3⋊Dic3—124no—
115C4⋊(C3⋊D3)—32no—
116C4⋊(C3⋊D3)—38no—
117C4⋊(C3⋊D3)—68no—
118C4⋊(C3⋊D3)—61no—
119C4⋊(C3⋊D3)—64no—
120C4⋊(C3⋊D3)—124no—
121C4×C3⋊D3—122no—
122#634—122no—
123C2×C3×Dic3—64no—
124C2×C3×Dic3—64no—
125C2×C3×Dic3D1214yes—
126C2×C3×Dic3D1214yes—
127#627—128no—
128#624—128no—
129C32×C8—21no—
130C32×C8—24no—
131C32×C8—44no—
132C32×C8—61no—
133C3×C3⋊C8—24no—
134#474—271noSylow
135C2×C33C22⋊Q811yescharacteristic
136C2×C33C2×D811yescharacteristic
137C2×C33#13711yescharacteristic
138C3⋊(C3⋊D3)—42no—
139C2×D12—94no—
140C2×D12—91no—
141C2×D12—184no—
142C2×C4×D3—184no—
143C2×C4×D3—181no—
144C2×Dic6—184no—
145C2×Dic6—181no—
146C2×C3×C8—24no—
147C2×C3×C8—34no—
148C2×C3×C8D3214yes—
149C2×C3×C8C2×C3⋊D311yescharacteristic
150C3×C4⋊C4—91no—
151C3×C4⋊C4—91no—
152C3×C42—34no—
153(C2×D3)⋊C4—184no—
154(C2×D3)⋊C4—181no—
155C4⋊Dic3—34no—
156C4⋊Dic3—34no—
157C4⋊Dic3—94no—
158C4⋊Dic3—91no—
159C4⋊Dic3—184no—
160C4⋊Dic3—184no—
161Dic3⋊C4—181no—
162C4×Dic3C2×C3⋊D311yescharacteristic
163C2×C3⋊C8C2×C3⋊D311yescharacteristic
164C22×C32—24no—
165C22×C32C2×D1214yes—
166C22×C32D3×D411yescharacteristic
167C2×C3⋊D3—62no—
168C2×C3⋊D3—68no—
169C2×C3⋊D3—128no—
170C2×C3⋊D3—121no—
171C2×C3⋊D3—121no—
172C2×C3⋊D3—124no—
173C2×C3⋊D3—124no—
174C2×C3⋊D3—244no—
175C2×C3⋊D3—244no—
176C32×C4—24no—
177C32×C4—24no—
178C32×C4—32no—
179C32×C4—61no—
180C32×C4#23511yescharacteristic
181C32×C4#23714yes—
182C32×C4D3×Q814yes—
183C32×C4D3×Q811yescharacteristic
184C3⋊Dic3—41no—
185C3⋊Dic3—41no—
186C3⋊Dic3—121no—
187C3⋊Dic3—124no—
188C3⋊Dic3—244no—
189C3×Dic3—24no—
190C3×Dic3—124no—
191C3×Dic3—124no—
192C3×Dic3D2418yes—
193C4⋊Q8—271no—
194C4×D4—271no—
195#107—91no—
196#107—271no—
197C4⋊C8—31no—
198#102—271no—
199#102—271no—
200C33#47411yescharacteristic, complemented, Sylow
201C22×D3—184no—
202C22×D3—181no—
203C22×D3—364no—
204C2×C3×C4—24no—
205C2×C3×C4—34no—
206C2×C3×C4—34no—
207C2×C3×C4—181no—
208C2×C3×C4—181no—
209C2×C3×C4C2×D3214yes—
210C2×C3×C4C22×C3⋊D311yescharacteristic
211C2×Dic3—64no—
212C2×Dic3—64no—
213C2×Dic3—184no—
214C2×Dic3—181no—
215C2×Dic3—364no—
216C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
217C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
218D12—98no—
219D12—92no—
220D12—188no—
221D12—184no—
222D12—181no—
223D12—364no—
224C4×D3—368no—
225C4×D3—362no—
226Dic6—368no—
227Dic6—362no—
228C3×C8—24no—
229C3×C8—21no—
230C3×C8—44no—
231C3×C8—64no—
232C3⋊C8—21no—
233C2×C32—24no—
234C2×C32—24no—
235C2×C32—24no—
236C2×C32#89611yescharacteristic
237C2×C32#89714yes—
238C2×C32C2×D2414yes—
239C2×C32#90914yes—
240C2×C32D3×D811yescharacteristic
241C2×C32#91811yescharacteristic
242C3⋊D3—122no—
243C3⋊D3—128no—
244C3⋊D3—248no—
245C2×Q8—541no—
246C2×D4—271no—
247C22×C4—541no—
248C2×C8—31no—
249C2×C8D3×C3⋊D311yescharacteristic
250C4⋊C4—91no—
251C4⋊C4—91no—
252C4⋊C4—271no—
253C4⋊C4—541no—
254C22⋊C4—541no—
255C42—31no—
256C22×C3—24no—
257C22×C3D3×D1214yes—
258C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
259C2×D3—182no—
260C2×D3—188no—
261C2×D3—368no—
262C2×D3—364no—
263C2×D3—364no—
264C2×D3—361no—
265C2×D3—361no—
266C2×D3—724no—
267C2×D3—724no—
268C3×C4—24no—
269C3×C4—24no—
270C3×C4—38no—
271C3×C4—64no—
272C3×C4—361no—
273C3×C4—361no—
274C3×C4#379714yes—
275C3×C4#380414yes—
276C3×C4#383011yescharacteristic
277C3×C4Q8×C3⋊D311yescharacteristic
278Dic3—21no—
279Dic3—124no—
280Dic3—124no—
281Dic3—364no—
282Dic3—361no—
283Dic3—724no—
284Dic3#374512yes—
285C32—24no—
286C32#484614yescomplemented
287C32#489011yescharacteristic, complemented
288C23—541no—
289Q8—1082no—
290D4—272no—
291D4—541no—
292C2×C4—31no—
293C2×C4—31no—
294C2×C4—181no—
295C2×C4—181no—
296C2×C4—541no—
297C2×C4—1082no—
298C2×C4C2×D3×C3⋊D311yescharacteristic
299C8—21no—
300C8—61no—
301C2×C3—24no—
302C2×C3—24no—
303C2×C3—24no—
304C2×C3D3×D2414yes—
305C2×C3#6380114yes—
306C2×C3#6389514yes—
307C2×C3#6410611yescharacteristic
308C2×C3C2×#374511yescharacteristic
309C2×C3#6411811yescharacteristic
310D3—362no—
311D3—368no—
312D3—728no—
313C22—542no—
314C22—1081no—
315C22—1081no—
316C22D3×C4⋊(C3⋊D3)11yescharacteristic
317C4—32no—
318C4—61no—
319C4—361no—
320C4—361no—
321C4—1081no—
322C4#8901911yescharacteristic
323C4#8902811yescharacteristic
324C3—24no—
325C3#1059076414yescomplemented
326C3#1059254511yescharacteristic, complemented
327C2—1082no—
328C2D3×#374511yescharacteristic
329C2#1172332711yescharacteristic
330C2#1172348511yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#474
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. #11723534C2semidirect
4. C3×#63645C2semidirect
5. C32×Dic3⋊C8C2semidirect
6. #11722055C2semidirect
31. #63645D3semidirect
33. C3×Dic3⋊C8D3semidirect
40. C4⋊(C3⋊(C3⋊D3))Q8semidirect
99. Dic3⋊C8C3⋊D3semidirect
200. C33#474semidirect
286. C32#4846semidirect
287. C32#4890semidirect
325. C3#10590764semidirect
326. C3#10592545semidirect

Maximal subgroups

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

subgroupindex
#117235342
#117234842
#117221912
C3×#636452
C32×Dic3⋊C82
#117220552
#117220362
#105925453
#105907643