Finite Group Databasebeta

← #13156561

Subgroups — #13156561

Order 1728.

622 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 98 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#11723447C211yesmaximal, characteristic, complemented
2C3×#63914C211yesmaximal, characteristic
3#11722071C211yesmaximal, characteristic, complemented
4#11722032C212yesmaximal, complemented
5C3×#63562C212yesmaximal, complemented
6#10591477—32nomaximal
7#10590554—31nomaximal
8C2×C32×D12C2212yes—
9C2×C3×#625—21no—
10C2×C3×#625—21no—
11C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C2×C3×C3⋊(C3⋊C8)C2212yes—
14C32×C4⋊Dic3C2211yescharacteristic
15#88806—21no—
16Dic3⋊(C3⋊Dic3)—21no—
17#64099—31no—
18C3×#940—32no—
19#63914D311yescharacteristic
20#63869—32no—
21#63869—61no—
22#63659—32no—
23C3×#832—32no—
24C3×#809—32no—
25#63563—32no—
26#63563—32no—
27#63563—62no—
28#63562D312yescomplemented
29#63555—31no—
30#63554—32no—
31C22×C32×D3—22no—
32C2×C33×C4C2311yescharacteristic
33C2×C3⋊(C3⋊Dic3)—21no—
34C2×C3×C3⋊Dic3D411yescharacteristic
35C2×C3×C3⋊Dic3D411yescharacteristic
36C2×C32×Dic3—21no—
37C32×D12—22no—
38C32×D12—22no—
39C3×#625—42no—
40C3×#625—42no—
41C3×C3⋊(C3⋊C8)—22no—
42#5071—91no—
43#4764—92no—
44C2×C32×D4—32no—
45C2×C3×#67—62no—
46C2×C3×#67—62no—
47C2×C3×D12—22no—
48C2×C3×D12C2×D312yes—
49C2×C3×D12C2×D312yes—
50C2×#625—21no—
51C2×#625—21no—
52C32×C4⋊C4—31no—
53C4⋊(C3⋊Dic3)—31no—
54C4⋊(C3⋊Dic3)—32no—
55C4⋊(C3⋊Dic3)—61no—
56C4⋊(C3⋊Dic3)—61no—
57C4⋊(C3⋊Dic3)—62no—
58C4⋊(C3⋊Dic3)—121no—
59#3750—61no—
60C4×C3⋊Dic3C2×D311yescharacteristic
61C2×C3⋊(C3⋊C8)C2×D312yes—
62C3×C4⋊Dic3—21no—
63C3×C4⋊Dic3C2×D312yes—
64C3×C4×Dic3—32no—
65C3×C4×Dic3—61no—
66C2×C3×C3⋊C8—32no—
67C2×C3×C3⋊C8—32no—
68C2×C3×C3⋊C8—62no—
69#3724—62no—
70#3724—62no—
71Dic3⋊Dic3—62no—
72Dic3⋊Dic3—62no—
73C22×C33C2×D411yescharacteristic
74C2×C32×D3—42no—
75C2×C32×D3—42no—
76C2×C32×D3—42no—
77C33×C4#4112yes—
78C3⋊(C3⋊Dic3)—41no—
79C3×C3⋊Dic3—21no—
80C3×C3⋊Dic3—21no—
81C32×Dic3—41no—
82C3×#127—91no—
83#940—32no—
84#890—92no—
85#890—181no—
86#870—91no—
87C3×#102—92no—
88#832—32no—
89#832—92no—
90#832—92no—
91#832—182no—
92#821—92no—
93#815—92no—
94#809—32no—
95#803—91no—
96C23×C32—62no—
97C22×C3×D3—22no—
98C22×C3×D3—22no—
99C22×C3×D3—42no—
100C32×D4—62no—
101C32×D4—62no—
102C2×C32×C4—22no—
103C2×C32×C4—21no—
104C2×C32×C4—41no—
105C2×C32×C4—61no—
106C2×C32×C4C22×D312yes—
107C2×C32×C4C22×D311yescharacteristic
108C2×C3⋊Dic3—61no—
109C2×C3⋊Dic3—62no—
110C2×C3⋊Dic3—62no—
111C2×C3⋊Dic3—122no—
112C2×C3⋊Dic3—122no—
113C2×C3⋊Dic3#6711yescharacteristic
114C2×C3⋊Dic3#6711yescharacteristic
115C3×#67—124no—
116C3×#67—124no—
117C2×C3×Dic3—22no—
118C2×C3×Dic3—22no—
119C2×C3×Dic3—32no—
120C2×C3×Dic3—32no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C3×D12—24no—
124C3×D12—22no—
125C3×D12—22no—
126C3×D12—22no—
127C3×D12—22no—
128C3×D12—42no—
129#625—42no—
130#625—42no—
131C3⋊(C3⋊C8)—22no—
132C3×C3⋊C8—62no—
133C3×C3⋊C8—62no—
134C3×C3⋊C8—122no—
135#490—271noSylow
136C2×C33#12411yescharacteristic
137C2×C33AGL(1,ℤ/8)11yescharacteristic
138C2×C33AGL(1,ℤ/8)11yescharacteristic
139C32×D3—82no—
140C2×C3×D4—32no—
141C2×C3×D4—32no—
142C2×C3×D4—62no—
143C2×C3×D4—181no—
144C2×C3×D4—181no—
145C2×#67—62no—
146C2×#67—62no—
147C2×D12D3212yes—
148C2×C3×C8—92no—
149C3×C4⋊C4—32no—
150C3×C4⋊C4—61no—
151C3×C42—91no—
152C4⋊Dic3—92no—
153C4⋊Dic3—91no—
154C4⋊Dic3—181no—
155C4⋊Dic3—182no—
156C4⋊Dic3—182no—
157C4⋊Dic3—182no—
158C4⋊Dic3—361no—
159C4⋊Dic3D3211yescharacteristic
160Dic3⋊C4—181no—
161Dic3⋊C4—181no—
162Dic3⋊C4—182no—
163Dic3⋊C4—182no—
164C4×Dic3—32no—
165C4×Dic3—61no—
166C2×C3⋊C8—32no—
167C2×C3⋊C8—32no—
168C2×C3⋊C8—62no—
169C22×C32—22no—
170C22×C32—21no—
171C22×C32—41no—
172C22×C32—122no—
173C22×C32—122no—
174C22×C32—122no—
175C22×C32D3×D412yes—
176C22×C32C2×#6711yescharacteristic
177C2×C3×D3—44no—
178C2×C3×D3—42no—
179C2×C3×D3—42no—
180C2×C3×D3—42no—
181C2×C3×D3—42no—
182C2×C3×D3—42no—
183C2×C3×D3—42no—
184C2×C3×D3—82no—
185C2×C3×D3—82no—
186C32×C4—22no—
187C32×C4—22no—
188C32×C4—22no—
189C32×C4—42no—
190C32×C4—121no—
191C32×C4#23512yes—
192C32×C4#23712yes—
193C32×C4#23712yes—
194C3⋊Dic3—21no—
195C3⋊Dic3—21no—
196C3⋊Dic3—121no—
197C3⋊Dic3—122no—
198C3⋊Dic3—122no—
199C3⋊Dic3—242no—
200C3⋊Dic3—242no—
201C3×Dic3—42no—
202C3×Dic3—42no—
203C3×Dic3—62no—
204C3×Dic3—62no—
205C3×Dic3—121no—
206C3×Dic3—121no—
207#127—91no—
208#125—271no—
209#102—92no—
210#102—272no—
211#97—271no—
212C33#49011yescharacteristic, complemented, Sylow
213C23×C3—62no—
214C23×C3—62no—
215C23×C3—122no—
216C22×D3—22no—
217C3×D4—64no—
218C3×D4—62no—
219C3×D4—62no—
220C3×D4—62no—
221C3×D4—62no—
222C3×D4—122no—
223C3×D4—362no—
224C3×D4—362no—
225C2×C3×C4—22no—
226C2×C3×C4—21no—
227C2×C3×C4—41no—
228C2×C3×C4—62no—
229C2×C3×C4—62no—
230C2×C3×C4—91no—
231C2×C3×C4—91no—
232C2×C3×C4C2×D3211yescharacteristic
233C2×C3×C4C2×D3212yes—
234#67—124no—
235#67—124no—
236C2×Dic3—21no—
237C2×Dic3—32no—
238C2×Dic3—32no—
239C2×Dic3—61no—
240C2×Dic3—61no—
241C2×Dic3—182no—
242C2×Dic3—182no—
243C2×Dic3—181no—
244C2×Dic3—362no—
245C2×Dic3—362no—
246D12—22no—
247D12—22no—
248C3×C8—182no—
249C3⋊C8—62no—
250C3⋊C8—62no—
251C3⋊C8—122no—
252C2×C32—21no—
253C2×C32—21no—
254C2×C32—22no—
255C2×C32—22no—
256C2×C32—21no—
257C2×C32—22no—
258C2×C32—41no—
259C2×C32—41no—
260C2×C32—41no—
261C2×C32—242no—
262C2×C32#88512yes—
263C2×C32#91112yes—
264C2×C32#91412yes—
265C2×C32#93211yescharacteristic
266C2×C32#93211yescharacteristic
267C2×C32#93611yescharacteristic
268C3×D3—84no—
269C3×D3—82no—
270C3×D3—82no—
271C2×D4—32no—
272C2×D4—181no—
273C2×D4—181no—
274C2×C8—92no—
275C4⋊C4—31no—
276C4⋊C4—271no—
277C4⋊C4—541no—
278C4⋊C4—541no—
279C42—91no—
280C22×C3—21no—
281C22×C3—22no—
282C22×C3—41no—
283C22×C3—124no—
284C22×C3—122no—
285C22×C3—122no—
286C22×C3—122no—
287C22×C3—122no—
288C22×C3—122no—
289C22×C3—122no—
290C22×C3—242no—
291C22×C3—242no—
292C22×C3D3×#6712yes—
293C22×C3#381111yescharacteristic
294C2×D3—42no—
295C2×D3—42no—
296C2×D3—42no—
297C3×C4—22no—
298C3×C4—22no—
299C3×C4—22no—
300C3×C4—42no—
301C3×C4—122no—
302C3×C4—122no—
303C3×C4—181no—
304C3×C4—181no—
305C3×C4#379512yes—
306C3×C4#380412yes—
307C3×C4#380512yes—
308Dic3—41no—
309Dic3—62no—
310Dic3—62no—
311Dic3—121no—
312Dic3—121no—
313Dic3—362no—
314Dic3—362no—
315Dic3—361no—
316Dic3—722no—
317Dic3—722no—
318C32—21no—
319C32—22no—
320C32—41no—
321C32#476412yescomplemented
322C32#507111yescharacteristic, complemented
323C23—62no—
324D4—62no—
325D4—62no—
326D4—362no—
327D4—362no—
328C2×C4—61no—
329C2×C4—91no—
330C2×C4—91no—
331C2×C4—541no—
332C2×C4D3311yescharacteristic
333C8—182no—
334C2×C3—21no—
335C2×C3—22no—
336C2×C3—21no—
337C2×C3—22no—
338C2×C3—22no—
339C2×C3—21no—
340C2×C3—41no—
341C2×C3—41no—
342C2×C3—41no—
343C2×C3—244no—
344C2×C3—242no—
345C2×C3—242no—
346C2×C3#6391311yescharacteristic
347C2×C3#6392711yescharacteristic
348C2×C3#6393212yes—
349C2×C3#6393412yes—
350C2×C3#6394111yescharacteristic
351C2×C3#6395912yes—
352D3—82no—
353C22—122no—
354C22—122no—
355C22—122no—
356C22#8895211yescharacteristic
357C4—121no—
358C4—181no—
359C4—181no—
360C4—1081no—
361C4#8896112yes—
362C3—22no—
363C3—21no—
364C3—41no—
365C3#1059055411yescharacteristic, complemented
366C3#1059147712yescomplemented
367C2—242no—
368C2#1172287211yescharacteristic
369C2#1172287311yescharacteristic
370C2#1172300411yescharacteristic
subgroupG/H
centerC22#88952
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC3×#63914C2
hypocenterC33#490
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723447C2semidirect
3. #11722071C2semidirect
4. #11722032C2semidirect
5. C3×#63562C2semidirect
28. #63562D3semidirect
212. C33#490semidirect
321. C32#4764semidirect
322. C32#5071semidirect
365. C3#10590554semidirect
366. C3#10591477semidirect

Maximal subgroups

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

subgroupindex
#117234472
C3×#639142
#117220712
#117220322
C3×#635622
#105914773
#105905543