Finite Group Databasebeta

← #13156511

Subgroups — #13156511

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
1C4⋊#6296C211yesmaximal, characteristic, complemented
2C3×#63914C211yesmaximal, characteristic
3#11722031C212yesmaximal, complemented
4C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
5C3×#63563C212yesmaximal, complemented
6#10591487—32nomaximal
7#10590885—31nomaximal
8C2×C32×D12C2212yes—
9C2×#6296—21no—
10C2×C3×#625—21no—
11C2×C3×#625—21no—
12C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2212yes—
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C2×C32×C3⋊C8C2211yescharacteristic
16#88815—21no—
17C3×#940—32no—
18#63919—31no—
19#63918—32no—
20#63914D311yescharacteristic
21#63659—32no—
22C8×C3⋊Dic3—31no—
23C3×#832—32no—
24C3×#821—32no—
25#63563D312yescomplemented
26#63562—32no—
27#63562—32no—
28#63562—62no—
29Dic3×C3⋊C8—32no—
30Dic3×C3⋊C8—61no—
31C22×C32×D3—22no—
32C2×C33×C4C2311yescharacteristic
33C2×C3×C3⋊Dic3—22no—
34C2×C3×C3⋊Dic3D411yescharacteristic
35C2×C3×C3⋊Dic3D411yescharacteristic
36C32×D12—22no—
37C32×D12—22no—
38#6296—42no—
39C3×#625—42no—
40C3×#625—42no—
41C3⋊(C3⋊(C3⋊C8))—21no—
42C32×C3⋊C8—21no—
43#5070—91no—
44#4760—92no—
45C2×C32×D4—32no—
46C2×C3×#67—62no—
47C2×C3×#67—62no—
48C2×C3×D12—22no—
49C2×C3×D12C2×D312yes—
50C2×C3×D12C2×D312yes—
51C2×#627—62no—
52C2×#627—61no—
53C2×#625—21no—
54C2×#625—21no—
55C2×C32×C8—31no—
56C4⋊(C3⋊Dic3)C2×D312yes—
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—32no—
64C3×C4⋊Dic3—32no—
65C3×C4⋊Dic3—62no—
66C3×C4×Dic3—32no—
67C3×C4×Dic3—61no—
68C2×C3×C3⋊C8—21no—
69C2×C3×C3⋊C8C2×D312yes—
70#3724—61no—
71Dic3⋊Dic3—62no—
72C22×C33C2×D411yescharacteristic
73C2×C32×D3—42no—
74C2×C32×D3—42no—
75C2×C32×D3—42no—
76C33×C4#4111yescharacteristic
77C33×C4#4111yescharacteristic
78C3×C3⋊Dic3—42no—
79C3×C3⋊Dic3D812yes—
80C3×C3⋊Dic3SD1612yes—
81C3×#127—91no—
82#940—32no—
83#888—92no—
84#869—91no—
85C3×#102—92no—
86#832—32no—
87#832—92no—
88#832—92no—
89#832—182no—
90#821—32no—
91C8×Dic3—92no—
92C8×Dic3—181no—
93#809—92no—
94C4×C3⋊C8—91no—
95C23×C32—62no—
96C22×C3×D3—22no—
97C22×C3×D3—22no—
98C22×C3×D3—42no—
99C32×D4—62no—
100C32×D4—62no—
101C2×C32×C4—21no—
102C2×C32×C4—22no—
103C2×C32×C4—41no—
104C2×C32×C4C22×D312yes—
105C2×C32×C4C22×D311yescharacteristic
106C2×C3⋊Dic3—22no—
107C2×C3⋊Dic3#6711yescharacteristic
108C2×C3⋊Dic3#6711yescharacteristic
109C3×#67—124no—
110C3×#67—124no—
111C2×C3×Dic3—32no—
112C2×C3×Dic3—32no—
113C2×C3×Dic3—61no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—62no—
116C2×C3×Dic3—62no—
117C2×C3×Dic3—122no—
118C3×D12—24no—
119C3×D12—22no—
120C3×D12—22no—
121C3×D12—22no—
122C3×D12—22no—
123C3×D12—42no—
124#627—124no—
125#627—122no—
126#625—42no—
127#625—42no—
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
140C32×D3—82no—
141C2×C3×D4—32no—
142C2×C3×D4—32no—
143C2×C3×D4—62no—
144C2×C3×D4—181no—
145C2×C3×D4—181no—
146C2×#67—62no—
147C2×#67—62no—
148C2×D12D3212yes—
149C2×Dic6—182no—
150C2×Dic6—181no—
151C2×C3×C8—32no—
152C2×C3×C8—61no—
153C3×C4⋊C4—92no—
154C3×C42—91no—
155C4⋊Dic3—32no—
156C4⋊Dic3—32no—
157C4⋊Dic3—62no—
158C4⋊Dic3—181no—
159Dic3⋊C4—182no—
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×C32—122no—
172C22×C32—122no—
173C22×C32—122no—
174C22×C32D3×D412yes—
175C22×C32C2×#6711yescharacteristic
176C2×C3×D3—44no—
177C2×C3×D3—42no—
178C2×C3×D3—42no—
179C2×C3×D3—42no—
180C2×C3×D3—42no—
181C2×C3×D3—42no—
182C2×C3×D3—42no—
183C2×C3×D3—82no—
184C2×C3×D3—82no—
185C32×C4—21no—
186C32×C4—22no—
187C32×C4—21no—
188C32×C4—22no—
189C32×C4—41no—
190C32×C4—41no—
191C32×C4#23512yes—
192C32×C4#23711yescharacteristic
193C32×C4#23712yes—
194C32×C4#23711yescharacteristic
195C3⋊Dic3—42no—
196C3⋊Dic3#21312yes—
197C3⋊Dic3#21412yes—
198C3×Dic3—34no—
199C3×Dic3—34no—
200C3×Dic3—62no—
201C3×Dic3—62no—
202C3×Dic3—122no—
203C3×Dic3—122no—
204C3×Dic3—242no—
205C4⋊Q8—271no—
206#127—91no—
207#102—92no—
208#102—272no—
209C4×C8—271no—
210C33#48611yescharacteristic, complemented, Sylow
211C23×C3—62no—
212C23×C3—62no—
213C23×C3—122no—
214C22×D3—22no—
215C3×D4—64no—
216C3×D4—62no—
217C3×D4—62no—
218C3×D4—62no—
219C3×D4—62no—
220C3×D4—122no—
221C3×D4—362no—
222C3×D4—362no—
223C2×C3×C4—21no—
224C2×C3×C4—22no—
225C2×C3×C4—41no—
226C2×C3×C4—91no—
227C2×C3×C4—91no—
228C2×C3×C4—182no—
229C2×C3×C4C2×D3211yescharacteristic
230C2×C3×C4C2×D3212yes—
231#67—124no—
232#67—124no—
233C2×Dic3—32no—
234C2×Dic3—32no—
235C2×Dic3—61no—
236C2×Dic3—61no—
237C2×Dic3—62no—
238C2×Dic3—62no—
239C2×Dic3—122no—
240D12—22no—
241D12—22no—
242Dic6—364no—
243Dic6—362no—
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—21no—
255C2×C32—22no—
256C2×C32—22no—
257C2×C32—22no—
258C2×C32—41no—
259C2×C32—41no—
260C2×C32—41no—
261C2×C32—242no—
262C2×C32#88512yes—
263C2×C32D3×D812yes—
264C2×C32D3×SD1612yes—
265C2×C32C2×#21311yescharacteristic
266C2×C32C2×#21411yescharacteristic
267C2×C32#93611yescharacteristic
268C3×D3—84no—
269C3×D3—82no—
270C3×D3—82no—
271C2×Q8—541no—
272C2×D4—32no—
273C2×D4—181no—
274C2×D4—181no—
275C2×C8—31no—
276C2×C8—271no—
277C4⋊C4—92no—
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×C3#380211yescharacteristic
293C22×C3D3×#6712yes—
294C2×D3—42no—
295C2×D3—42no—
296C2×D3—42no—
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—362no—
306C3×C4#379811yescharacteristic
307C3×C4#380412yes—
308C3×C4#380511yescharacteristic
309C3×C4#380512yes—
310Dic3—34no—
311Dic3—34no—
312Dic3—62no—
313Dic3—62no—
314Dic3—122no—
315Dic3—122no—
316Dic3—242no—
317C32—21no—
318C32—22no—
319C32—41no—
320C32#476012yescomplemented
321C32#507011yescharacteristic, complemented
322C23—62no—
323Q8—1082no—
324D4—62no—
325D4—62no—
326D4—362no—
327D4—362no—
328C2×C4—91no—
329C2×C4—91no—
330C2×C4—182no—
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—244no—
344C2×C3—242no—
345C2×C3—242no—
346C2×C3#6379611yescharacteristic
347C2×C3#6379711yescharacteristic
348C2×C3#6391511yescharacteristic
349C2×C3D3×#21312yes—
350C2×C3D3×#21412yes—
351C2×C3#6395912yes—
352D3—82no—
353C22—122no—
354C22—122no—
355C22—122no—
356C22#8895311yescharacteristic
357C4—92no—
358C4—92no—
359C4—362no—
360C4#8895911yescharacteristic
361C4#8896311yescharacteristic
362C3—22no—
363C3—21no—
364C3—41no—
365C3#1059088511yescharacteristic, complemented
366C3#1059148712yescomplemented
367C2—242no—
368C2#1172283111yescharacteristic
369C2#1172283911yescharacteristic
370C2#1172300611yescharacteristic
subgroupG/H
centerC22#88953
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC3×#63914C2
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
1. C4⋊#6296C2semidirect
3. #11722031C2semidirect
4. C3⋊C8×C3⋊Dic3C2semidirect
5. C3×#63563C2semidirect
25. #63563D3semidirect
210. C33#486semidirect
320. C32#4760semidirect
321. C32#5070semidirect
365. C3#10590885semidirect
366. C3#10591487semidirect

Maximal subgroups

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

subgroupindex
C4⋊#62962
C3×#639142
#117220312
C3⋊C8×C3⋊Dic32
C3×#635632
#105914873
#105908853