Finite Group Databasebeta

← #13156512

Subgroups — #13156512

Order 1728.

638 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 106 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
2C3×#63919C211yesmaximal, characteristic, complemented
3#11722033C212yesmaximal, complemented
4C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
5C3×#63564C212yesmaximal, complemented
6#10590885—31nomaximal
7#10590548—32nomaximal
8C2×C3×C4⋊(C3⋊D3)C2212yes—
9C2×#6295—21no—
10C2×#6295—21no—
11C2×C3×#627—21no—
12C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C32×C4⋊Dic3C2212yes—
15C2×C32×C3⋊C8C2211yescharacteristic
16C3×#3724—21no—
17C3×#888—32no—
18#63919D311yescharacteristic, complemented
19#63914—31no—
20#63911—32no—
21C8×C3⋊Dic3—31no—
22#63636—32no—
23C3×#821—32no—
24C3×#809—32no—
25#63564D312yescomplemented
26#63562—32no—
27#63562—32no—
28#63562—62no—
29Dic3×C3⋊C8—32no—
30Dic3×C3⋊C8—61no—
31C22×C3×C3⋊D3—22no—
32C2×C33×C4C2311yescharacteristic
33C2×C3×C3⋊Dic3D411yescharacteristic
34C2×C3×C3⋊Dic3D411yescharacteristic
35C3×C4⋊(C3⋊D3)—22no—
36C3×C4⋊(C3⋊D3)—22no—
37C2×C32×Dic3—22no—
38#6295—42no—
39#6295—42no—
40C3×#627—42no—
41C3⋊(C3⋊(C3⋊C8))—21no—
42C32×C3⋊C8—21no—
43#5088—91no—
44#4760—92no—
45C2×C4⋊(C3⋊D3)C2×D312yes—
46C2×C3×D12—32no—
47C2×C3×D12—32no—
48C2×C3×D12—62no—
49C2×C3×Dic6—62no—
50C2×#627—21no—
51C2×#626—62no—
52C2×#626—62no—
53C2×#625—61no—
54C2×#625—61no—
55C2×C32×C8—31no—
56C32×C4⋊C4—32no—
57C4×C3⋊Dic3C2×D311yescharacteristic
58C2×C3⋊(C3⋊C8)—32no—
59C2×C3⋊(C3⋊C8)—31no—
60C2×C3⋊(C3⋊C8)—61no—
61C2×C3⋊(C3⋊C8)—62no—
62C2×C3⋊(C3⋊C8)—121no—
63C3×C4⋊Dic3—22no—
64C3×C4⋊Dic3C2×D312yes—
65C3×C4⋊Dic3C2×D312yes—
66C3×Dic3⋊C4—62no—
67C3×C4×Dic3—32no—
68C3×C4×Dic3—61no—
69C2×C3×C3⋊C8—21no—
70C2×C3×C3⋊C8C2×D312yes—
71#3724—21no—
72C22×C33C2×D411yescharacteristic
73C2×C3×C3⋊D3—42no—
74C2×C3×C3⋊D3—42no—
75C2×C3×C3⋊D3—42no—
76C33×C4#4111yescharacteristic
77C33×C4#4111yescharacteristic
78C3×C3⋊Dic3D812yes—
79C3×C3⋊Dic3SD1612yes—
80C32×Dic3—42no—
81C3×C4⋊Q8—91no—
82#940—92no—
83#888—32no—
84#874—91no—
85C3×#102—92no—
86#832—92no—
87#821—32no—
88C8×Dic3—92no—
89C8×Dic3—181no—
90#809—32no—
91#809—92no—
92#809—92no—
93#809—182no—
94C4×C3⋊C8—91no—
95C22×C3⋊D3—22no—
96C22×C3×D3—62no—
97C22×C3×D3—62no—
98C22×C3×D3—122no—
99C2×C32×C4—21no—
100C2×C32×C4—22no—
101C2×C32×C4—41no—
102C2×C32×C4—62no—
103C2×C32×C4C22×D312yes—
104C2×C32×C4C22×D311yescharacteristic
105C2×C3⋊Dic3#6711yescharacteristic
106C2×C3⋊Dic3#6711yescharacteristic
107C4⋊(C3⋊D3)—22no—
108C4⋊(C3⋊D3)—22no—
109C2×C3×Dic3—22no—
110C2×C3×Dic3—22no—
111C2×C3×Dic3—32no—
112C2×C3×Dic3—32no—
113C2×C3×Dic3—42no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—61no—
116C3×D12—64no—
117C3×D12—62no—
118C3×D12—62no—
119C3×D12—62no—
120C3×D12—62no—
121C3×D12—122no—
122C3×Dic6—124no—
123#627—42no—
124#626—124no—
125#626—124no—
126#625—122no—
127#625—122no—
128C32×C8—61no—
129C3⋊(C3⋊C8)—61no—
130C3⋊(C3⋊C8)—62no—
131C3⋊(C3⋊C8)—62no—
132C3⋊(C3⋊C8)—122no—
133C3⋊(C3⋊C8)—122no—
134C3×C3⋊C8—22no—
135C3×C3⋊C8—22no—
136#486—271noSylow
137C2×C33#12411yescharacteristic
138C2×C33C2×D811yescharacteristic
139C2×C33C2×SD1611yescharacteristic
140C3×C3⋊D3—82no—
141C2×C3×Q8—181no—
142C2×C3×D4—92no—
143C2×#67—182no—
144C2×#67—182no—
145C2×D12—32no—
146C2×D12—32no—
147C2×D12—62no—
148C2×D12—181no—
149C2×D12—181no—
150C2×Dic6—62no—
151C2×C3×C8—32no—
152C2×C3×C8—61no—
153C3×C4⋊C4—32no—
154C3×C4⋊C4—32no—
155C3×C4⋊C4—62no—
156C3×C4⋊C4—181no—
157C3×C42—91no—
158C4⋊Dic3D3212yes—
159Dic3⋊C4—62no—
160C4×Dic3—32no—
161C4×Dic3—61no—
162C2×C3⋊C8—91no—
163C2×C3⋊C8—92no—
164C2×C3⋊C8—181no—
165C2×C3⋊C8—182no—
166C2×C3⋊C8—361no—
167C2×C3⋊C8D3211yescharacteristic
168C22×C32—21no—
169C22×C32—22no—
170C22×C32—41no—
171C22×C32D3×D412yes—
172C22×C32C2×#6711yescharacteristic
173C2×C3⋊D3—42no—
174C2×C3⋊D3—42no—
175C2×C3⋊D3—42no—
176C2×C3×D3—124no—
177C2×C3×D3—122no—
178C2×C3×D3—122no—
179C2×C3×D3—122no—
180C2×C3×D3—122no—
181C2×C3×D3—122no—
182C2×C3×D3—122no—
183C2×C3×D3—242no—
184C2×C3×D3—242no—
185C32×C4—22no—
186C32×C4—21no—
187C32×C4—22no—
188C32×C4—21no—
189C32×C4—41no—
190C32×C4—41no—
191C32×C4—122no—
192C32×C4#23512yes—
193C32×C4#23712yes—
194C32×C4#23911yescharacteristic
195C32×C4#23911yescharacteristic
196C3⋊Dic3#21312yes—
197C3⋊Dic3#21512yes—
198C3×Dic3—34no—
199C3×Dic3—34no—
200C3×Dic3—42no—
201C3×Dic3—42no—
202C3×Dic3—62no—
203C3×Dic3—62no—
204C3×Dic3—82no—
205C4⋊Q8—91no—
206#127—271no—
207#102—92no—
208#102—272no—
209C4×C8—271no—
210C33#48611yescharacteristic, complemented, Sylow
211C23×C3—182no—
212C22×D3—62no—
213C22×D3—62no—
214C22×D3—122no—
215C3×Q8—362no—
216C3×D4—182no—
217C3×D4—182no—
218C2×C3×C4—21no—
219C2×C3×C4—22no—
220C2×C3×C4—41no—
221C2×C3×C4—62no—
222C2×C3×C4—62no—
223C2×C3×C4—91no—
224C2×C3×C4—91no—
225C2×C3×C4—122no—
226C2×C3×C4C2×D3212yes—
227C2×C3×C4C2×D3211yescharacteristic
228#67—364no—
229#67—364no—
230C2×Dic3—22no—
231C2×Dic3—32no—
232C2×Dic3—32no—
233C2×Dic3—61no—
234C2×Dic3—61no—
235D12—64no—
236D12—62no—
237D12—62no—
238D12—62no—
239D12—62no—
240D12—122no—
241D12—362no—
242D12—362no—
243Dic6—124no—
244C3×C8—62no—
245C3×C8—62no—
246C3⋊C8—21no—
247C3⋊C8—182no—
248C3⋊C8—181no—
249C3⋊C8—182no—
250C3⋊C8—362no—
251C3⋊C8—362no—
252C2×C32—21no—
253C2×C32—21no—
254C2×C32—22no—
255C2×C32—21no—
256C2×C32—22no—
257C2×C32—22no—
258C2×C32—41no—
259C2×C32—41no—
260C2×C32—41no—
261C2×C32#88512yes—
262C2×C32D3×D812yes—
263C2×C32D3×SD1612yes—
264C2×C32C2×#21311yescharacteristic
265C2×C32C2×#21511yescharacteristic
266C2×C32#94711yescharacteristic
267C3⋊D3—82no—
268C3×D3—244no—
269C3×D3—242no—
270C3×D3—242no—
271C2×Q8—181no—
272C2×D4—92no—
273C2×D4—541no—
274C2×D4—541no—
275C2×C8—31no—
276C2×C8—271no—
277C4⋊C4—32no—
278C4⋊C4—181no—
279C42—91no—
280C22×C3—21no—
281C22×C3—22no—
282C22×C3—41no—
283C22×C3—362no—
284C22×C3—362no—
285C22×C3—362no—
286C22×C3#380211yescharacteristic
287C22×C3D3×#6712yes—
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—21no—
298C3×C4—22no—
299C3×C4—21no—
300C3×C4—22no—
301C3×C4—41no—
302C3×C4—41no—
303C3×C4—92no—
304C3×C4—92no—
305C3×C4—122no—
306C3×C4—122no—
307C3×C4—242no—
308C3×C4#379612yes—
309C3×C4#379811yescharacteristic
310C3×C4#379912yes—
311C3×C4#380511yescharacteristic
312Dic3—34no—
313Dic3—34no—
314Dic3—42no—
315Dic3—62no—
316Dic3—62no—
317C32—22no—
318C32—21no—
319C32—41no—
320C32#476012yescomplemented
321C32#508811yescharacteristic, complemented
322C23—182no—
323Q8—362no—
324D4—182no—
325D4—182no—
326D4—1082no—
327D4—1082no—
328C2×C4—62no—
329C2×C4—91no—
330C2×C4—91no—
331C2×C4D3311yescharacteristic
332C8—61no—
333C8—541no—
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—722no—
344C2×C3#6379611yescharacteristic
345C2×C3#6379711yescharacteristic
346C2×C3#6390712yes—
347C2×C3#6391511yescharacteristic
348C2×C3D3×#21312yes—
349C2×C3D3×#21512yes—
350D3—244no—
351D3—242no—
352D3—242no—
353C22—362no—
354C22—362no—
355C22—362no—
356C22#8895311yescharacteristic
357C4—92no—
358C4—92no—
359C4—122no—
360C4#8896411yescharacteristic
361C4#8896511yescharacteristic
362C3—22no—
363C3—21no—
364C3—41no—
365C3#1059054812yescomplemented
366C3#1059088511yescharacteristic, complemented
367C2—722no—
368C2#1172283111yescharacteristic
369C2#1172284011yescharacteristic
370C2#1172300711yescharacteristic
subgroupG/H
centerC22#88953
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radical#11723697C2
hypocenterC33#486
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. C3×#63919C2semidirect
3. #11722033C2semidirect
4. C3⋊C8×C3⋊Dic3C2semidirect
5. C3×#63564C2semidirect
18. #63919D3semidirect
25. #63564D3semidirect
210. C33#486semidirect
320. C32#4760semidirect
321. C32#5088semidirect
365. C3#10590548semidirect
366. C3#10590885semidirect

Maximal subgroups

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

subgroupindex
#117236972
C3×#639192
#117220332
C3⋊C8×C3⋊Dic32
C3×#635642
#105908853
#105905483