Finite Group Databasebeta

← #13155993

Subgroups — #13155993

Order 1728.

756 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 104 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#11723406C211yesmaximal, characteristic, complemented
2C2×#88794C212yesmaximal, complemented
3C3×#63914C211yesmaximal, characteristic
4#11722078C211yesmaximal, characteristic, complemented
5C3×#63562C212yesmaximal, complemented
6#10591439—32nomaximal
7#10590553—31nomaximal
8C2×#6308C2211yescharacteristic
9C2×C32×D12C2212yes—
10C2×C3×#625—21no—
11C2×C3×#625—21no—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C2×C3×C3⋊(C3⋊C8)C2212yes—
14C32×C4⋊Dic3C2211yescharacteristic
15#88806—21no—
16Dic3×C3⋊Dic3—21no—
17#88794—24no—
18C2×#3754—32no—
19#64096—31no—
20C3×#940—32no—
21#63914D311yescharacteristic
22#63844—32no—
23#63844—61no—
24C2×#3716—32no—
25C2×#3716—32no—
26C2×#3716—62no—
27C3×#832—32no—
28C3×#809—32no—
29#63572—31no—
30Dic3⋊(C3⋊C8)—32no—
31#63562D312yescomplemented
32C22×C32×D3—22no—
33C2×C33×C4C2311yescharacteristic
34C2×C3⋊(C3⋊Dic3)—21no—
35#6308—21no—
36#6308D412yescomplemented
37C2×C3×C3⋊Dic3D411yescharacteristic
38C2×C3×C3⋊Dic3D411yescharacteristic
39C2×C32×Dic3—21no—
40C32×D12—22no—
41C32×D12—22no—
42C3×#625—42no—
43C3×#625—42no—
44C3×C3⋊(C3⋊C8)—22no—
45#5075—91no—
46#4762—92no—
47C2×C32×D4—32no—
48C2×#634—31no—
49C2×#634—32no—
50C2×#634—61no—
51C2×#634—62no—
52C2×#634—121no—
53C2×C3×#67—62no—
54C2×C3×#67—62no—
55C2×C3×D12—22no—
56C2×C3×D12C2×D312yes—
57C2×C3×D12C2×D312yes—
58C2×#625—21no—
59C2×#625—21no—
60C32×C4⋊C4—31no—
61#3754—64no—
62#3750—61no—
63C4×C3⋊Dic3—61no—
64C4×C3⋊Dic3C2×D311yescharacteristic
65C2×C3⋊(C3⋊C8)C2×D312yes—
66C3×C4⋊Dic3—21no—
67C3×C4⋊Dic3C2×D312yes—
68C3×C4×Dic3—32no—
69C3×C4×Dic3—61no—
70C2×C3×C3⋊C8—32no—
71C2×C3×C3⋊C8—32no—
72C2×C3×C3⋊C8—62no—
73#3724—62no—
74#3724—62no—
75Dic32—62no—
76Dic32—62no—
77#3716—64no—
78#3716—64no—
79#3716—68no—
80C22×C33C2×D411yescharacteristic
81C2×C32×D3—42no—
82C2×C32×D3—42no—
83C2×C32×D3—42no—
84C33×C4C2×D411yescharacteristic
85C33×C4#4111yescharacteristic
86C3⋊(C3⋊Dic3)—22no—
87C3×C3⋊Dic3—21no—
88C3×C3⋊Dic3SD1612yescomplemented
89C32×Dic3—41no—
90C3×#127—91no—
91#940—32no—
92C2×#214—92no—
93C2×#214—92no—
94C2×#214—182no—
95C2×#204—92no—
96#887—92no—
97#887—181no—
98C4×Dic6—91no—
99C3×#102—92no—
100#832—32no—
101Dic3⋊C8—92no—
102#809—32no—
103C4⋊(C3⋊C8)—91no—
104C23×C32—62no—
105C22×C3×D3—22no—
106C22×C3×D3—22no—
107C22×C3×D3—42no—
108C32×D4—62no—
109C32×D4—62no—
110C2×C32×C4—22no—
111C2×C32×C4—21no—
112C2×C32×C4—41no—
113C2×C32×C4—61no—
114C2×C32×C4C22×D312yes—
115C2×C32×C4C22×D311yescharacteristic
116C2×C3⋊Dic3—61no—
117C2×C3⋊Dic3—62no—
118C2×C3⋊Dic3—62no—
119C2×C3⋊Dic3—122no—
120C2×C3⋊Dic3—122no—
121C2×C3⋊Dic3#6711yescharacteristic
122C2×C3⋊Dic3#6711yescharacteristic
123#634—32no—
124#634—34no—
125#634—62no—
126#634—64no—
127#634—61no—
128#634—62no—
129#634—62no—
130#634—122no—
131#634—122no—
132#634—122no—
133C3×#67—124no—
134C3×#67—124no—
135C2×C3×Dic3—22no—
136C2×C3×Dic3—22no—
137C2×C3×Dic3—32no—
138C2×C3×Dic3—32no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C3×D12—24no—
142C3×D12—22no—
143C3×D12—22no—
144C3×D12—22no—
145C3×D12—22no—
146C3×D12—42no—
147#625—42no—
148#625—42no—
149C3⋊(C3⋊C8)—22no—
150C3×C3⋊C8—64no—
151C3×C3⋊C8—62no—
152C3×C3⋊C8—62no—
153#460—271noSylow
154C2×C33#12111yescharacteristic
155C2×C33C2×SD1611yescharacteristic
156C2×C33AGL(1,ℤ/8)11yescharacteristic
157C32×D3—82no—
158C2×C3×D4—32no—
159C2×C3×D4—32no—
160C2×C3×D4—62no—
161C2×C3×D4—181no—
162C2×C3×D4—181no—
163C2×#67—62no—
164C2×#67—62no—
165C2×D12D3212yes—
166C2×Dic6—91no—
167C2×Dic6—92no—
168C2×Dic6—181no—
169C2×Dic6—182no—
170C2×Dic6—361no—
171C2×C3×C8—92no—
172C3×C4⋊C4—32no—
173C3×C4⋊C4—61no—
174C3×C42—91no—
175#214—184no—
176#214—184no—
177#214—188no—
178C4⋊Dic3D3211yescharacteristic
179Dic3⋊C4—181no—
180Dic3⋊C4—182no—
181Dic3⋊C4—182no—
182C4×Dic3—32no—
183C4×Dic3—61no—
184C4×Dic3—181no—
185C4×Dic3—182no—
186C4×Dic3—182no—
187C2×C3⋊C8—32no—
188C2×C3⋊C8—32no—
189C2×C3⋊C8—62no—
190#204—184no—
191C22×C32—22no—
192C22×C32—21no—
193C22×C32—41no—
194C22×C32—122no—
195C22×C32—122no—
196C22×C32—122no—
197C22×C32D3×D412yes—
198C22×C32C2×#6711yescharacteristic
199C2×C3×D3—44no—
200C2×C3×D3—42no—
201C2×C3×D3—42no—
202C2×C3×D3—42no—
203C2×C3×D3—42no—
204C2×C3×D3—42no—
205C2×C3×D3—42no—
206C2×C3×D3—82no—
207C2×C3×D3—82no—
208C32×C4—21no—
209C32×C4—21no—
210C32×C4—22no—
211C32×C4—22no—
212C32×C4—41no—
213C32×C4—41no—
214C32×C4—121no—
215C32×C4#23512yes—
216C32×C4D3×D412yes—
217C32×C4D3×D411yescharacteristic
218C32×C4#23711yescharacteristic
219C3⋊Dic3—21no—
220C3⋊Dic3—62no—
221C3⋊Dic3—64no—
222C3⋊Dic3—64no—
223C3⋊Dic3—124no—
224C3⋊Dic3—124no—
225C3⋊Dic3#21412yescomplemented
226C3×Dic3—34no—
227C3×Dic3—42no—
228C3×Dic3—42no—
229C3×Dic3—62no—
230C3×Dic3—62no—
231C3×Dic3—121no—
232C2×SD16—272no—
233#127—91no—
234C4×Q8—271no—
235C4⋊C8—271no—
236#102—92no—
237C33#46011yescharacteristic, complemented, Sylow
238C23×C3—62no—
239C23×C3—62no—
240C23×C3—122no—
241C22×D3—22no—
242C3×D4—64no—
243C3×D4—62no—
244C3×D4—62no—
245C3×D4—62no—
246C3×D4—62no—
247C3×D4—122no—
248C3×D4—362no—
249C3×D4—362no—
250C2×C3×C4—22no—
251C2×C3×C4—21no—
252C2×C3×C4—41no—
253C2×C3×C4—62no—
254C2×C3×C4—62no—
255C2×C3×C4—91no—
256C2×C3×C4—91no—
257C2×C3×C4C2×D3211yescharacteristic
258C2×C3×C4C2×D3212yes—
259#67—124no—
260#67—124no—
261C2×Dic3—21no—
262C2×Dic3—32no—
263C2×Dic3—32no—
264C2×Dic3—61no—
265C2×Dic3—61no—
266C2×Dic3—182no—
267C2×Dic3—182no—
268C2×Dic3—181no—
269C2×Dic3—362no—
270C2×Dic3—362no—
271D12—22no—
272D12—22no—
273Dic6—92no—
274Dic6—94no—
275Dic6—182no—
276Dic6—184no—
277Dic6—182no—
278Dic6—181no—
279Dic6—182no—
280Dic6—362no—
281Dic6—362no—
282Dic6—362no—
283C3×C8—182no—
284C3⋊C8—64no—
285C3⋊C8—62no—
286C3⋊C8—62no—
287C2×C32—21no—
288C2×C32—21no—
289C2×C32—22no—
290C2×C32—22no—
291C2×C32—21no—
292C2×C32—22no—
293C2×C32—41no—
294C2×C32—41no—
295C2×C32—41no—
296C2×C32—242no—
297C2×C32#88412yes—
298C2×C32#91112yes—
299C2×C32D3×SD1612yes—
300C2×C32#93211yescharacteristic
301C2×C32C2×#21411yescharacteristic
302C2×C32#93811yescharacteristic
303C3×D3—84no—
304C3×D3—82no—
305C3×D3—82no—
306C2×Q8—271no—
307C2×D4—32no—
308C2×D4—181no—
309C2×D4—181no—
310SD16—544no—
311C2×C8—92no—
312C4⋊C4—31no—
313C4⋊C4—541no—
314C42—91no—
315C42—541no—
316C22×C3—21no—
317C22×C3—22no—
318C22×C3—41no—
319C22×C3—124no—
320C22×C3—122no—
321C22×C3—122no—
322C22×C3—122no—
323C22×C3—122no—
324C22×C3—122no—
325C22×C3—122no—
326C22×C3—242no—
327C22×C3—242no—
328C22×C3D3×#6712yes—
329C22×C3#381111yescharacteristic
330C2×D3—42no—
331C2×D3—42no—
332C2×D3—42no—
333C3×C4—22no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—22no—
337C3×C4—41no—
338C3×C4—41no—
339C3×C4—92no—
340C3×C4—122no—
341C3×C4—122no—
342C3×C4—181no—
343C3×C4#379811yescharacteristic
344C3×C4#380211yescharacteristic
345C3×C4#380412yes—
346C3×C4#381112yes—
347Dic3—34no—
348Dic3—41no—
349Dic3—62no—
350Dic3—62no—
351Dic3—121no—
352Dic3—184no—
353Dic3—182no—
354Dic3—184no—
355Dic3—364no—
356Dic3—364no—
357C32—21no—
358C32—22no—
359C32—41no—
360C32#476212yescomplemented
361C32#507511yescharacteristic, complemented
362C23—62no—
363Q8—272no—
364Q8—541no—
365D4—62no—
366D4—62no—
367D4—362no—
368D4—362no—
369C2×C4—61no—
370C2×C4—91no—
371C2×C4—91no—
372C2×C4—541no—
373C2×C4D3311yescharacteristic
374C8—182no—
375C2×C3—21no—
376C2×C3—22no—
377C2×C3—21no—
378C2×C3—22no—
379C2×C3—21no—
380C2×C3—22no—
381C2×C3—41no—
382C2×C3—41no—
383C2×C3—41no—
384C2×C3—244no—
385C2×C3—242no—
386C2×C3—242no—
387C2×C3#6391211yescharacteristic
388C2×C3#6392711yescharacteristic
389C2×C3D3×#21412yes—
390C2×C3#6393212yes—
391C2×C3#6394011yescharacteristic
392C2×C3#6395812yes—
393D3—82no—
394C22—122no—
395C22—122no—
396C22—122no—
397C22#8895211yescharacteristic
398C4—92no—
399C4—121no—
400C4—181no—
401C4—542no—
402C4#8895411yescharacteristic
403C4#8895911yescharacteristic
404C3—22no—
405C3—21no—
406C3—41no—
407C3#1059055311yescharacteristic, complemented
408C3#1059143912yescomplemented
409C2—242no—
410C2#1172283711yescharacteristic
411C2#1172287211yescharacteristic
412C2#1172296411yescharacteristic
subgroupG/H
centerC22#88952
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC3×#63914C2
hypocenterC33#460
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. #11723406C2semidirect
2. C2×#88794C2semidirect
4. #11722078C2semidirect
5. C3×#63562C2semidirect
31. #63562D3semidirect
36. #6308D4semidirect
88. C3×C3⋊Dic3SD16semidirect
225. C3⋊Dic3#214semidirect
237. C33#460semidirect
360. C32#4762semidirect
361. C32#5075semidirect
407. C3#10590553semidirect
408. C3#10591439semidirect

Maximal subgroups

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

subgroupindex
#117234062
C2×#887942
C3×#639142
#117220782
C3×#635622
#105914393
#105905533