Finite Group Databasebeta

← #13156509

Subgroups — #13156509

Order 1728.

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

normal subgroups only

This group has 666 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#11723535C211yesmaximal, characteristic, complemented
2C3×#63914C211yesmaximal, characteristic
3#11722070C211yesmaximal, characteristic, complemented
4#11722032C212yesmaximal, complemented
5C3×#63562C212yesmaximal, complemented
6#10591487—32nomaximal
7#10590552—31nomaximal
8C2×C32×D12C2212yes—
9C2×#6293—21no—
10C2×C3×#625—21no—
11C2×C3×#625—21no—
12C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2212yes—
15C32×C4⋊Dic3C2211yescharacteristic
16#88807—21no—
17#64097—31no—
18C3×#940—32no—
19#63918—32no—
20#63918—61no—
21#63914D311yescharacteristic
22#63659—32no—
23C3×#832—32no—
24C3×#809—32no—
25#63563—32no—
26#63563—32no—
27#63563—62no—
28#63562D312yescomplemented
29#63553—31no—
30Dic3×C3⋊C8—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—
39#6293—42no—
40C3×#625—42no—
41C3×#625—42no—
42C3×C3⋊(C3⋊C8)—22no—
43#5070—91no—
44#4760—92no—
45C2×C32×D4—32no—
46C2×#634—61no—
47C2×C3×#67—62no—
48C2×C3×#67—62no—
49C2×C3×D12—22no—
50C2×C3×D12C2×D312yes—
51C2×C3×D12C2×D312yes—
52C2×#627—62no—
53C2×#627—62no—
54C2×#625—21no—
55C2×#625—21no—
56C32×C4⋊C4—31no—
57C4⋊(C3⋊Dic3)—31no—
58C4⋊(C3⋊Dic3)—32no—
59C4⋊(C3⋊Dic3)—61no—
60C4⋊(C3⋊Dic3)—62no—
61C4⋊(C3⋊Dic3)—121no—
62#3750—61no—
63C4×C3⋊Dic3C2×D311yescharacteristic
64C2×C3⋊(C3⋊C8)C2×D312yes—
65C3×C4⋊Dic3—21no—
66C3×C4⋊Dic3C2×D312yes—
67C3×C4×Dic3—32no—
68C3×C4×Dic3—61no—
69C2×C3×C3⋊C8—32no—
70C2×C3×C3⋊C8—32no—
71C2×C3×C3⋊C8—62no—
72Dic3⋊Dic3—62no—
73Dic3⋊Dic3—62no—
74C22×C33C2×D411yescharacteristic
75C2×C32×D3—42no—
76C2×C32×D3—42no—
77C2×C32×D3—42no—
78C33×C4#4112yes—
79C3⋊(C3⋊Dic3)—41no—
80C3×C3⋊Dic3D812yes—
81C3×C3⋊Dic3SD1612yes—
82C32×Dic3—41no—
83C3×#127—91no—
84#940—32no—
85#888—92no—
86#888—181no—
87#869—91no—
88C3×#102—92no—
89#832—32no—
90#832—92no—
91#832—92no—
92#832—182no—
93#821—92no—
94C8×Dic3—92no—
95#809—32no—
96C4×C3⋊C8—91no—
97C23×C32—62no—
98C22×C3×D3—22no—
99C22×C3×D3—22no—
100C22×C3×D3—42no—
101C32×D4—62no—
102C32×D4—62no—
103C2×C32×C4—22no—
104C2×C32×C4—21no—
105C2×C32×C4—41no—
106C2×C32×C4—61no—
107C2×C32×C4C22×D312yes—
108C2×C32×C4C22×D311yescharacteristic
109C2×C3⋊Dic3—61no—
110C2×C3⋊Dic3—62no—
111C2×C3⋊Dic3—62no—
112C2×C3⋊Dic3—122no—
113C2×C3⋊Dic3—122no—
114C2×C3⋊Dic3#6711yescharacteristic
115C2×C3⋊Dic3#6711yescharacteristic
116#634—122no—
117C3×#67—124no—
118C3×#67—124no—
119C2×C3×Dic3—22no—
120C2×C3×Dic3—22no—
121C2×C3×Dic3—32no—
122C2×C3×Dic3—32no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C3×D12—24no—
126C3×D12—22no—
127C3×D12—22no—
128C3×D12—22no—
129C3×D12—22no—
130C3×D12—42no—
131#627—124no—
132#627—124no—
133#625—42no—
134#625—42no—
135C3⋊(C3⋊C8)—22no—
136C3×C3⋊C8—62no—
137C3×C3⋊C8—62no—
138C3×C3⋊C8—122no—
139#486—271noSylow
140C2×C33#12411yescharacteristic
141C2×C33C2×D811yescharacteristic
142C2×C33C2×SD1611yescharacteristic
143C32×D3—82no—
144C2×C3×D4—32no—
145C2×C3×D4—32no—
146C2×C3×D4—62no—
147C2×C3×D4—181no—
148C2×C3×D4—181no—
149C2×#67—62no—
150C2×#67—62no—
151C2×D12D3212yes—
152C2×Dic6—181no—
153C2×Dic6—182no—
154C2×Dic6—182no—
155C2×C3×C8—92no—
156C3×C4⋊C4—32no—
157C3×C4⋊C4—61no—
158C3×C42—91no—
159C4⋊Dic3—92no—
160C4⋊Dic3—91no—
161C4⋊Dic3—181no—
162C4⋊Dic3—182no—
163C4⋊Dic3—181no—
164C4⋊Dic3—361no—
165C4⋊Dic3D3211yescharacteristic
166Dic3⋊C4—182no—
167Dic3⋊C4—182no—
168C4×Dic3—32no—
169C4×Dic3—61no—
170C2×C3⋊C8—32no—
171C2×C3⋊C8—32no—
172C2×C3⋊C8—62no—
173C22×C32—22no—
174C22×C32—21no—
175C22×C32—41no—
176C22×C32—122no—
177C22×C32—122no—
178C22×C32—122no—
179C22×C32D3×D412yes—
180C22×C32C2×#6711yescharacteristic
181C2×C3×D3—44no—
182C2×C3×D3—42no—
183C2×C3×D3—42no—
184C2×C3×D3—42no—
185C2×C3×D3—42no—
186C2×C3×D3—42no—
187C2×C3×D3—42no—
188C2×C3×D3—82no—
189C2×C3×D3—82no—
190C32×C4—22no—
191C32×C4—22no—
192C32×C4—22no—
193C32×C4—42no—
194C32×C4—121no—
195C32×C4#23512yes—
196C32×C4#23712yes—
197C32×C4#23712yes—
198C3⋊Dic3—121no—
199C3⋊Dic3—122no—
200C3⋊Dic3—122no—
201C3⋊Dic3—242no—
202C3⋊Dic3—242no—
203C3⋊Dic3#21312yes—
204C3⋊Dic3#21412yes—
205C3×Dic3—34no—
206C3×Dic3—34no—
207C3×Dic3—42no—
208C3×Dic3—42no—
209C3×Dic3—62no—
210C3×Dic3—62no—
211C4⋊Q8—271no—
212#127—91no—
213#102—92no—
214#102—272no—
215C4×C8—271no—
216C33#48611yescharacteristic, complemented, Sylow
217C23×C3—62no—
218C23×C3—62no—
219C23×C3—122no—
220C22×D3—22no—
221C3×D4—64no—
222C3×D4—62no—
223C3×D4—62no—
224C3×D4—62no—
225C3×D4—62no—
226C3×D4—122no—
227C3×D4—362no—
228C3×D4—362no—
229C2×C3×C4—22no—
230C2×C3×C4—21no—
231C2×C3×C4—41no—
232C2×C3×C4—62no—
233C2×C3×C4—62no—
234C2×C3×C4—91no—
235C2×C3×C4—91no—
236C2×C3×C4C2×D3211yescharacteristic
237C2×C3×C4C2×D3212yes—
238#67—124no—
239#67—124no—
240C2×Dic3—21no—
241C2×Dic3—32no—
242C2×Dic3—32no—
243C2×Dic3—61no—
244C2×Dic3—61no—
245C2×Dic3—182no—
246C2×Dic3—182no—
247C2×Dic3—181no—
248C2×Dic3—362no—
249C2×Dic3—362no—
250D12—22no—
251D12—22no—
252Dic6—364no—
253Dic6—362no—
254Dic6—364no—
255C3×C8—182no—
256C3⋊C8—62no—
257C3⋊C8—62no—
258C3⋊C8—122no—
259C2×C32—21no—
260C2×C32—22no—
261C2×C32—21no—
262C2×C32—22no—
263C2×C32—21no—
264C2×C32—22no—
265C2×C32—41no—
266C2×C32—41no—
267C2×C32—41no—
268C2×C32—242no—
269C2×C32#88512yes—
270C2×C32D3×D812yes—
271C2×C32D3×SD1612yes—
272C2×C32C2×#21311yescharacteristic
273C2×C32C2×#21411yescharacteristic
274C2×C32#93611yescharacteristic
275C3×D3—84no—
276C3×D3—82no—
277C3×D3—82no—
278C2×Q8—541no—
279C2×D4—32no—
280C2×D4—181no—
281C2×D4—181no—
282C2×C8—92no—
283C4⋊C4—31no—
284C4⋊C4—271no—
285C4⋊C4—541no—
286C42—91no—
287C22×C3—21no—
288C22×C3—22no—
289C22×C3—41no—
290C22×C3—124no—
291C22×C3—122no—
292C22×C3—122no—
293C22×C3—122no—
294C22×C3—122no—
295C22×C3—122no—
296C22×C3—122no—
297C22×C3—242no—
298C22×C3—242no—
299C22×C3D3×#6712yes—
300C22×C3#381111yescharacteristic
301C2×D3—42no—
302C2×D3—42no—
303C2×D3—42no—
304C3×C4—22no—
305C3×C4—22no—
306C3×C4—22no—
307C3×C4—42no—
308C3×C4—92no—
309C3×C4—92no—
310C3×C4—122no—
311C3×C4—122no—
312C3×C4#379512yes—
313C3×C4#380412yes—
314C3×C4#380512yes—
315Dic3—34no—
316Dic3—34no—
317Dic3—41no—
318Dic3—62no—
319Dic3—62no—
320Dic3—362no—
321Dic3—362no—
322Dic3—361no—
323Dic3—722no—
324Dic3—722no—
325C32—21no—
326C32—22no—
327C32—41no—
328C32#476012yescomplemented
329C32#507011yescharacteristic, complemented
330C23—62no—
331Q8—1082no—
332D4—62no—
333D4—62no—
334D4—362no—
335D4—362no—
336C2×C4—61no—
337C2×C4—91no—
338C2×C4—91no—
339C2×C4—541no—
340C2×C4D3311yescharacteristic
341C8—182no—
342C2×C3—21no—
343C2×C3—22no—
344C2×C3—21no—
345C2×C3—22no—
346C2×C3—22no—
347C2×C3—21no—
348C2×C3—41no—
349C2×C3—41no—
350C2×C3—41no—
351C2×C3—244no—
352C2×C3—242no—
353C2×C3—242no—
354C2×C3#6391311yescharacteristic
355C2×C3D3×#21312yes—
356C2×C3#6392611yescharacteristic
357C2×C3D3×#21412yes—
358C2×C3#6394011yescharacteristic
359C2×C3#6395912yes—
360D3—82no—
361C22—122no—
362C22—122no—
363C22—122no—
364C22#8895211yescharacteristic
365C4—92no—
366C4—92no—
367C4—121no—
368C4—1081no—
369C4#8896112yes—
370C3—22no—
371C3—21no—
372C3—41no—
373C3#1059055211yescharacteristic, complemented
374C3#1059148712yescomplemented
375C2—242no—
376C2#1172283011yescharacteristic
377C2#1172283711yescharacteristic
378C2#1172300411yescharacteristic
subgroupG/H
centerC22#88952
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. #11723535C2semidirect
3. #11722070C2semidirect
4. #11722032C2semidirect
5. C3×#63562C2semidirect
28. #63562D3semidirect
216. C33#486semidirect
328. C32#4760semidirect
329. C32#5070semidirect
373. C3#10590552semidirect
374. C3#10591487semidirect

Maximal subgroups

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

subgroupindex
#117235352
C3×#639142
#117220702
#117220322
C3×#635622
#105914873
#105905523