Finite Group Databasebeta

← #13156567

Subgroups — #13156567

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#11723697C211yesmaximal, characteristic
2#11723447C211yesmaximal, characteristic, complemented
3#11722034C212yesmaximal, complemented
4C3×#63564C212yesmaximal, complemented
5C3×#63555C211yesmaximal, characteristic, complemented
6#10590882—32nomaximal
7#10590554—31nomaximal
8C2×C3×C4⋊(C3⋊D3)C2212yes—
9C2×#6295—21no—
10C2×#6295—21no—
11C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C32×C4⋊Dic3C2211yescharacteristic
14C2×C32×C3⋊C8C2212yes—
15#88806—21no—
16Dic3⋊(C3⋊Dic3)—21no—
17#64099—31no—
18#63914—31no—
19#63911—32no—
20#63869—32no—
21#63869—61no—
22#63648—32no—
23C3×#821—32no—
24C3×#815—32no—
25C3×#809—32no—
26#63564D312yescomplemented
27#63563—32no—
28#63563—32no—
29#63563—62no—
30#63555D311yescharacteristic, complemented
31C22×C3×C3⋊D3—22no—
32C2×C33×C4C2311yescharacteristic
33C2×C3⋊(C3⋊Dic3)—21no—
34C2×C3×C3⋊Dic3D411yescharacteristic
35C2×C3×C3⋊Dic3D411yescharacteristic
36C3×C4⋊(C3⋊D3)—22no—
37C3×C4⋊(C3⋊D3)—22no—
38C2×C32×Dic3—21no—
39#6295—42no—
40#6295—42no—
41C32×C3⋊C8—22no—
42#4764—92no—
43#4713—91no—
44C2×C4⋊(C3⋊D3)C2×D312yes—
45C2×C3×D12—32no—
46C2×C3×D12—32no—
47C2×C3×D12—62no—
48C2×#626—62no—
49C2×#626—62no—
50C2×#625—61no—
51C2×#625—61no—
52C2×C32×C8—32no—
53C32×C4⋊C4—31no—
54C4⋊(C3⋊Dic3)—31no—
55C4⋊(C3⋊Dic3)—32no—
56C4⋊(C3⋊Dic3)—61no—
57C4⋊(C3⋊Dic3)—61no—
58C4⋊(C3⋊Dic3)—62no—
59C4⋊(C3⋊Dic3)—121no—
60#3750—61no—
61C4×C3⋊Dic3C2×D311yescharacteristic
62C3×C4⋊Dic3—21no—
63C3×C4⋊Dic3C2×D312yes—
64C3×C4×Dic3—32no—
65C3×C4×Dic3—61no—
66C2×C3×C3⋊C8—22no—
67C2×C3×C3⋊C8C2×D312yes—
68C2×C3×C3⋊C8C2×D312yes—
69#3724—62no—
70#3724—62no—
71Dic3⋊Dic3—62no—
72Dic3⋊Dic3—62no—
73C22×C33C2×D411yescharacteristic
74C2×C3×C3⋊D3—42no—
75C2×C3×C3⋊D3—42no—
76C2×C3×C3⋊D3—42no—
77C33×C4#4112yes—
78C3⋊(C3⋊Dic3)—41no—
79C3×C3⋊Dic3—21no—
80C3×C3⋊Dic3—21no—
81C32×Dic3—41no—
82#940—92no—
83#890—92no—
84#890—181no—
85#874—91no—
86#870—91no—
87C3×#102—92no—
88C3×#97—91no—
89#832—92no—
90#821—32no—
91#821—92no—
92#821—92no—
93#821—182no—
94#815—32no—
95#809—32no—
96C22×C3⋊D3—22no—
97C22×C3×D3—62no—
98C22×C3×D3—62no—
99C22×C3×D3—122no—
100C2×C32×C4—22no—
101C2×C32×C4—21no—
102C2×C32×C4—41no—
103C2×C32×C4—61no—
104C2×C32×C4C22×D311yescharacteristic
105C2×C32×C4C22×D312yes—
106C2×C3⋊Dic3—61no—
107C2×C3⋊Dic3—62no—
108C2×C3⋊Dic3—62no—
109C2×C3⋊Dic3—122no—
110C2×C3⋊Dic3—122no—
111C2×C3⋊Dic3D1211yescharacteristic
112C2×C3⋊Dic3D1211yescharacteristic
113C4⋊(C3⋊D3)—22no—
114C4⋊(C3⋊D3)—22no—
115C2×C3×Dic3—22no—
116C2×C3×Dic3—22no—
117C2×C3×Dic3—32no—
118C2×C3×Dic3—32no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—61no—
121C3×D12—64no—
122C3×D12—62no—
123C3×D12—62no—
124C3×D12—62no—
125C3×D12—62no—
126C3×D12—122no—
127#626—124no—
128#626—124no—
129#625—122no—
130#625—122no—
131C32×C8—62no—
132C3×C3⋊C8—22no—
133C3×C3⋊C8—22no—
134C3×C3⋊C8—42no—
135#490—271noSylow
136C2×C33#12411yescharacteristic
137C2×C33AGL(1,ℤ/8)11yescharacteristic
138C2×C33AGL(1,ℤ/8)11yescharacteristic
139C3×C3⋊D3—82no—
140C2×C3×D4—92no—
141C2×#67—182no—
142C2×#67—182no—
143C2×D12—32no—
144C2×D12—32no—
145C2×D12—62no—
146C2×D12—181no—
147C2×D12—181no—
148C2×C3×C8—32no—
149C2×C3×C8—32no—
150C2×C3×C8—62no—
151C3×C4⋊C4—32no—
152C3×C4⋊C4—61no—
153C3×C42—91no—
154C4⋊Dic3—91no—
155C4⋊Dic3—92no—
156C4⋊Dic3—181no—
157C4⋊Dic3—182no—
158C4⋊Dic3—182no—
159C4⋊Dic3—182no—
160C4⋊Dic3—361no—
161C4⋊Dic3D3211yescharacteristic
162Dic3⋊C4—181no—
163Dic3⋊C4—181no—
164Dic3⋊C4—182no—
165Dic3⋊C4—182no—
166C4×Dic3—32no—
167C4×Dic3—61no—
168C2×C3⋊C8D3212yes—
169C22×C32—22no—
170C22×C32—21no—
171C22×C32—41no—
172C22×C32C2×D1211yescharacteristic
173C22×C32D3×D412yes—
174C2×C3⋊D3—42no—
175C2×C3⋊D3—42no—
176C2×C3⋊D3—42no—
177C2×C3×D3—124no—
178C2×C3×D3—122no—
179C2×C3×D3—122no—
180C2×C3×D3—122no—
181C2×C3×D3—122no—
182C2×C3×D3—122no—
183C2×C3×D3—122no—
184C2×C3×D3—242no—
185C2×C3×D3—242no—
186C32×C4—22no—
187C32×C4—22no—
188C32×C4—22no—
189C32×C4—42no—
190C32×C4—121no—
191C32×C4#23512yes—
192C32×C4#23512yes—
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—271no—
208#125—271no—
209#102—92no—
210#102—272no—
211#97—91no—
212C33#49011yescharacteristic, complemented, Sylow
213C23×C3—182no—
214C22×D3—62no—
215C22×D3—62no—
216C22×D3—122no—
217C3×D4—182no—
218C3×D4—182no—
219C2×C3×C4—21no—
220C2×C3×C4—22no—
221C2×C3×C4—41no—
222C2×C3×C4—62no—
223C2×C3×C4—62no—
224C2×C3×C4—91no—
225C2×C3×C4—91no—
226C2×C3×C4C2×D3211yescharacteristic
227C2×C3×C4C2×D3212yes—
228#67—364no—
229#67—364no—
230C2×Dic3—21no—
231C2×Dic3—32no—
232C2×Dic3—32no—
233C2×Dic3—61no—
234C2×Dic3—61no—
235C2×Dic3—182no—
236C2×Dic3—182no—
237C2×Dic3—181no—
238C2×Dic3—362no—
239C2×Dic3—362no—
240D12—64no—
241D12—62no—
242D12—62no—
243D12—62no—
244D12—62no—
245D12—122no—
246D12—362no—
247D12—362no—
248C3×C8—62no—
249C3×C8—62no—
250C3×C8—122no—
251C3⋊C8—22no—
252C2×C32—22no—
253C2×C32—22no—
254C2×C32—22no—
255C2×C32—21no—
256C2×C32—21no—
257C2×C32—21no—
258C2×C32—41no—
259C2×C32—41no—
260C2×C32—41no—
261C2×C32#87511yescharacteristic
262C2×C32#88512yes—
263C2×C32#90811yescharacteristic
264C2×C32#90811yescharacteristic
265C2×C32#91112yes—
266C2×C32#91412yes—
267C3⋊D3—82no—
268C3×D3—244no—
269C3×D3—242no—
270C3×D3—242no—
271C2×D4—92no—
272C2×D4—541no—
273C2×D4—541no—
274C2×C8—32no—
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—362no—
284C22×C3—362no—
285C22×C3—362no—
286C22×C3D3×D1212yes—
287C22×C3#381111yescharacteristic
288C2×D3—124no—
289C2×D3—122no—
290C2×D3—122no—
291C2×D3—122no—
292C2×D3—122no—
293C2×D3—122no—
294C2×D3—122no—
295C2×D3—242no—
296C2×D3—242no—
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#379812yes—
307C3×C4#380412yes—
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#471311yescharacteristic, complemented
322C32#476412yescomplemented
323C23—182no—
324D4—182no—
325D4—182no—
326D4—1082no—
327D4—1082no—
328C2×C4—61no—
329C2×C4—91no—
330C2×C4—91no—
331C2×C4—541no—
332C2×C4D3311yescharacteristic
333C8—62no—
334C2×C3—22no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3—22no—
338C2×C3—21no—
339C2×C3—22no—
340C2×C3—41no—
341C2×C3—41no—
342C2×C3—41no—
343C2×C3—722no—
344C2×C3#6379412yes—
345C2×C3#6379512yes—
346C2×C3#6385212yes—
347C2×C3#6391311yescharacteristic
348C2×C3#6392711yescharacteristic
349C2×C3#6394111yescharacteristic
350D3—244no—
351D3—242no—
352D3—242no—
353C22—362no—
354C22—362no—
355C22—362no—
356C22#8895511yescharacteristic
357C4—121no—
358C4—181no—
359C4—181no—
360C4—1081no—
361C4#8896012yes—
362C3—22no—
363C3—21no—
364C3—41no—
365C3#1059055411yescharacteristic, complemented
366C3#1059088212yescomplemented
367C2—722no—
368C2#1172288011yescharacteristic
369C2#1172288111yescharacteristic
370C2#1172301011yescharacteristic
subgroupG/H
centerC22#88955
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radical#11723697C2
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
2. #11723447C2semidirect
3. #11722034C2semidirect
4. C3×#63564C2semidirect
5. C3×#63555C2semidirect
26. #63564D3semidirect
30. #63555D3semidirect
212. C33#490semidirect
321. C32#4713semidirect
322. C32#4764semidirect
365. C3#10590554semidirect
366. C3#10590882semidirect

Maximal subgroups

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

subgroupindex
#117236972
#117234472
#117220342
C3×#635642
C3×#635552
#105908823
#105905543