Finite Group Databasebeta

← #13181135

Subgroups — #13181135

Order 1728.

940 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 940 conjugacy classes of subgroups — too many to draw. Only its 123 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#11724618C212yesmaximal, complemented
2C2×C3×#3787C211yesmaximal, characteristic, complemented
3#11724351C214yesmaximal, complemented
4#11724341C211yesmaximal, characteristic, complemented
5#11724339C212yesmaximal, complemented
6#11724337C214yesmaximal, complemented
7#11724336C211yesmaximal, characteristic, complemented
8#10596500—31nomaximal
9C3×#3828—22no—
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11#89049—22no—
12#89046C2214yescomplemented
13C4⋊#1113C2212yes—
14C4×#1113—22no—
15#89042C2214yescomplemented
16#89041—22no—
17#89040C2212yes—
18C2×C3×#622C2212yescomplemented
19C3×#3787C2211yescharacteristic
20C3×#3787C2211yescharacteristic
21C3×#3787C2212yescomplemented
22#88943C2214yescomplemented
23#88942C2218yescomplemented
24#88941C2214yescomplemented
25#64306—32no—
26D3×#235—62no—
27C2×#3787D311yescharacteristic, complemented
28#64237—34no—
29#64228—31no—
30#64227—32no—
31#64225—31no—
32#64223—34no—
33C22×C3×C3⋊D3D411yescharacteristic
34C2×#1113—22no—
35C2×C33×C4D411yescharacteristic
36C3×#638—24no—
37C2×C3×C3⋊Dic3C2311yescharacteristic
38C3×C4⋊(C3⋊D3)—22no—
39C3×C4×C3⋊D3—24no—
40C3×C4×C3⋊D3D411yescharacteristic
41C3×C4×C3⋊D3D411yescharacteristic
42C3×C4×C3⋊D3C2311yescharacteristic
43C3×C4×C3⋊D3C2311yescharacteristic
44C3×#634—22no—
45#6296C2314yes—
46#6295—24no—
47#6295C2314yes—
48#6294—22no—
49#6294—24no—
50C3×#622C2312yes—
51C3×#622C2312yes—
52C2×#904—91nomaximal
53#3828—22no—
54C2×C4×C3⋊D3C2×D311yescharacteristic
55C3×#235—62no—
56C3×#235—62no—
57C3×#235—122no—
58C2×C3×C4×D3—62no—
59#3811—62no—
60D3×#67—64no—
61#3805—34no—
62#3804—64no—
63#3802—32no—
64D3×D12—62no—
65C4×D32—64no—
66C4×D32—62no—
67#3799—62no—
68#3798—34no—
69#3798—64no—
70#3797—32no—
71#3796—62no—
72#3795—62no—
73D3×Dic6—62no—
74C2×#622C2×D312yescomplemented
75#3787C2×D311yescharacteristic
76#3787C2×D311yescharacteristic
77#3787C2×D312yescomplemented
78#3776—34no—
79#3775—38no—
80#3774—34no—
81C22×C33C2×D411yescharacteristic
82C2×C3×C3⋊D3—22no—
83C2×C3×C3⋊D3—24no—
84C2×C3×C3⋊D3C2×D411yescharacteristic
85C2×C3×C3⋊D3C2×D411yescharacteristic
86#1113—44no—
87C33×C4C2×D411yescharacteristic
88C33×C4C2×D411yescharacteristic
89C3×C3⋊Dic3—24no—
90C3×C3⋊Dic3C2×D411yescharacteristic
91C3×C3⋊Dic3C2411yescharacteristic
92D3×#41—182no—
93C2×#235—92no—
94C22×C3×C8—91no—
95C2×Dic12—91no—
96#904—94no—
97#904—94no—
98C2×D24—91no—
99C2×#204—92no—
100C22×C3⋊D3D1211yescharacteristic
101C22×C3×D3—62no—
102C2×D32—64no—
103C2×D32—62no—
104C2×C32×C4—22no—
105C2×C32×C4—42no—
106C2×C32×C4D1211yescharacteristic
107#638—24no—
108C2×C3⋊Dic3C22×D311yescharacteristic
109C4⋊(C3⋊D3)—22no—
110C4×C3⋊D3—24no—
111C4×C3⋊D3D1211yescharacteristic
112C4×C3⋊D3D1211yescharacteristic
113C4×C3⋊D3C22×D311yescharacteristic
114C4×C3⋊D3C22×D311yescharacteristic
115#634—22no—
116C3×#67—64no—
117C3×#67—64no—
118C3×#67—124no—
119C2×C3×Dic3—62no—
120C3×D12—62no—
121C3×D12—62no—
122C3×D12—122no—
123C3×C4×D3—62no—
124C3×C4×D3—62no—
125C3×C4×D3—62no—
126C3×C4×D3—62no—
127C3×C4×D3—64no—
128C3×C4×D3—64no—
129C3×C4×D3—124no—
130C3×Dic6—62no—
131C3×Dic6—62no—
132C3×Dic6—122no—
133#627—34no—
134#627—64no—
135#626—64no—
136#626—64no—
137#626—64no—
138#625—34no—
139#625—64no—
140#624—64no—
141#624—62no—
142D3×Dic3—64no—
143D3×Dic3—64no—
144D3×Dic3—64no—
145#622C22×D312yes—
146#622C22×D312yes—
147C2×#135—271noSylow
148C2×C33—21no—
149C2×C33C22×D411yescharacteristic
150C3×C3⋊D3—21no—
151C3×C3⋊D3—44no—
152C3×C3⋊D3#13512yescomplemented
153C3×#41—182no—
154C22×C3×C4—91no—
155C2×#67—182no—
156#239—182no—
157D3×Q8—182no—
158#237—184no—
159#237—182no—
160D3×D4—182no—
161D3×D4—184no—
162#235—62no—
163#235—62no—
164#235—94no—
165#235—94no—
166#235—122no—
167#235—184no—
168#235—182no—
169C2×D12—92no—
170C2×C4×D3—62no—
171C2×C4×D3—182no—
172C2×C4×D3—184no—
173C2×Dic6—92no—
174C2×C3×C8—91no—
175C2×C3×C8—91no—
176C2×C3×C8—92no—
177C2×C3×C8—92no—
178Dic12—94no—
179D24—94no—
180#204—98no—
181C22×C32—22no—
182C22×C32—42no—
183C22×C32C2×D1211yescharacteristic
184C2×C3⋊D3—24no—
185C2×C3⋊D3—22no—
186C2×C3⋊D3C2×D1211yescharacteristic
187C2×C3⋊D3C2×D1211yescharacteristic
188C2×C3×D3—62no—
189C2×C3×D3—62no—
190C2×C3×D3—64no—
191C2×C3×D3—64no—
192C2×C3×D3—124no—
193C2×C3×D3—124no—
194D32—128no—
195D32—124no—
196C32×C4—22no—
197C32×C4—22no—
198C32×C4—42no—
199C32×C4—42no—
200C32×C4C2×D1211yescharacteristic
201C32×C4C2×D1211yescharacteristic
202C3⋊Dic3—24no—
203C3⋊Dic3C2×D1211yescharacteristic
204C3⋊Dic3C23×D311yescharacteristic
205C3×Dic3—62no—
206C3×Dic3—62no—
207C3×Dic3—64no—
208C3×Dic3—64no—
209C3×Dic3—124no—
210C2×#41—272no—
211#135—274no—
212#135—274no—
213C2×Q16—271no—
214C2×SD16—272no—
215C2×D8—271no—
216C22×C8—91no—
217C33C2×#13511yescharacteristic, complemented, Sylow
218C23×C3—91no—
219C22×D3—62no—
220C22×D3—182no—
221C22×D3—184no—
222C3×Q8—182no—
223C3×D4—182no—
224C3×D4—184no—
225C2×C3×C4—22no—
226C2×C3×C4—42no—
227C2×C3×C4—91no—
228C2×C3×C4—91no—
229C2×C3×C4—91no—
230C2×C3×C4—91no—
231C2×C3×C4—91no—
232C2×C3×C4—184no—
233C2×C3×C4GO+(4,𝔽2)11yescharacteristic
234#67—64no—
235#67—64no—
236#67—124no—
237#67—184no—
238#67—184no—
239#67—184no—
240C2×Dic3—62no—
241C2×Dic3—182no—
242C2×Dic3—184no—
243D12—62no—
244D12—62no—
245D12—94no—
246D12—122no—
247D12—184no—
248C4×D3—62no—
249C4×D3—62no—
250C4×D3—62no—
251C4×D3—62no—
252C4×D3—64no—
253C4×D3—64no—
254C4×D3—124no—
255C4×D3—184no—
256C4×D3—184no—
257C4×D3—184no—
258Dic6—62no—
259Dic6—62no—
260Dic6—94no—
261Dic6—122no—
262Dic6—184no—
263C3×C8—92no—
264C3×C8—92no—
265C2×C32—22no—
266C2×C32—21no—
267C2×C32—42no—
268C2×C32—42no—
269C2×C32—82no—
270C2×C32C22×D1211yescharacteristic
271C3⋊D3—21no—
272C3⋊D3—44no—
273C3⋊D3#90412yescomplemented
274C3×D3—64no—
275C3×D3—122no—
276C3×D3—124no—
277C3×D3—124no—
278C3×D3—244no—
279#41—182no—
280#41—274no—
281#41—274no—
282#41—542no—
283C2×Q8—272no—
284C2×D4—272no—
285C2×D4—542no—
286C22×C4—91no—
287C22×C4—542no—
288Q16—274no—
289SD16—278no—
290D8—274no—
291C2×C8—91no—
292C2×C8—91no—
293C2×C8—92no—
294C2×C8—92no—
295C22×C3—22no—
296C22×C3—42no—
297C22×C3—91no—
298C22×C3—91no—
299C22×C3—182no—
300C22×C3—184no—
301C22×C3C2×GO+(4,𝔽2)11yescharacteristic
302C2×D3—62no—
303C2×D3—62no—
304C2×D3—64no—
305C2×D3—64no—
306C2×D3—124no—
307C2×D3—124no—
308C2×D3—368no—
309C2×D3—364no—
310C3×C4—22no—
311C3×C4—22no—
312C3×C4—42no—
313C3×C4—42no—
314C3×C4—91no—
315C3×C4—91no—
316C3×C4—184no—
317C3×C4C2×GO+(4,𝔽2)11yescharacteristic
318C3×C4C2×GO+(4,𝔽2)11yescharacteristic
319Dic3—62no—
320Dic3—62no—
321Dic3—64no—
322Dic3—64no—
323Dic3—124no—
324C32—22no—
325C32—42no—
326C32C2×#90411yescharacteristic, complemented
327C23—91no—
328C23—542no—
329Q8—182no—
330Q8—274no—
331D4—184no—
332D4—182no—
333D4—274no—
334D4—544no—
335C2×C4—91no—
336C2×C4—91no—
337C2×C4—91no—
338C2×C4—91no—
339C2×C4—91no—
340C2×C4—184no—
341C2×C4—544no—
342C2×C4—542no—
343C2×C4#157⋊D311yescharacteristic
344C8—92no—
345C8—92no—
346C2×C3—21no—
347C2×C3—22no—
348C2×C3—42no—
349C2×C3—42no—
350C2×C3—82no—
351C2×C3—92no—
352C2×C3—181no—
353C2×C3—364no—
354C2×C3C22×GO+(4,𝔽2)11yescharacteristic
355D3—64no—
356D3—122no—
357D3—124no—
358D3—124no—
359D3—244no—
360C22—91no—
361C22—91no—
362C22—182no—
363C22—184no—
364C22—1084no—
365C22C2×#157⋊D311yescharacteristic
366C4—91no—
367C4—91no—
368C4—184no—
369C4C2×#157⋊D311yescharacteristic
370C4C2×#157⋊D311yescharacteristic
371C3—22no—
372C3—42no—
373C3#1059650011yescharacteristic, complemented
374C2—21no—
375C2—92no—
376C2—181no—
377C2—364no—
378C2C22×#157⋊D311yescharacteristic
subgroupG/H
centerC4C2×#157⋊D3
commutator, cosocleC3×C3⋊Dic3C24
FrattiniC2C22×#157⋊D3
FittingC2×C33×C4D4
socleC2×C33C22×D4
hypercenterC2×C4#157⋊D3
hypocenterC33C2×#135
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11724618C2semidirect
2. C2×C3×#3787C2semidirect
3. #11724351C2semidirect
4. #11724341C2semidirect
5. #11724339C2semidirect
6. #11724337C2semidirect
7. #11724336C2semidirect
12. #89046C22semidirect
15. #89042C22semidirect
18. C2×C3×#622C22semidirect
21. C3×#3787C22semidirect
22. #88943C22semidirect
23. #88942C22semidirect
24. #88941C22semidirect
27. C2×#3787D3semidirect
74. C2×#622C2×D3semidirect
77. #3787C2×D3semidirect
152. C3×C3⋊D3#135semidirect
217. C33C2×#135semidirect
273. C3⋊D3#904semidirect
326. C32C2×#904semidirect
373. C3#10596500semidirect

Maximal subgroups

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

subgroupindex
#117246182
C2×C3×#37872
#117243512
#117243412
#117243392
#117243372
#117243362
#105965003
C2×#9049