Finite Group Databasebeta

← #13155984

Subgroups — #13155984

Order 1728.

708 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 110 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
1Dic6×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C3×#63914C211yesmaximal, characteristic
3C2×C3×#3715C212yesmaximal, complemented
4#11722078C211yesmaximal, characteristic, complemented
5#11722032C212yesmaximal, complemented
6#10591622—31nomaximal
7#10591438—32nomaximal
8C2×C32×D12C2212yes—
9C2×C32×Dic6C2211yescharacteristic
10C2×C3×#625—21no—
11C2×C3×#625—21no—
12C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2212yes—
15Dic3⋊(C3⋊Dic3)—21no—
16Dic3×C3⋊Dic3—21no—
17C3×#3715—24no—
18Q8×C3⋊Dic3—31no—
19C2×C3×#215—32no—
20C3×#940—32no—
21C2×C3×#214—32no—
22#63914D311yescharacteristic
23Dic3×Dic6—32no—
24Dic3×Dic6—61no—
25C2×#3715D312yescomplemented
26#63659—32no—
27#63572—31no—
28Dic3⋊(C3⋊C8)—32no—
29#63563—32no—
30#63563—32no—
31#63563—62no—
32C22×C32×D3—22no—
33C2×C33×C4C2311yescharacteristic
34C2×C3⋊(C3⋊Dic3)—21no—
35C2×C3×C3⋊Dic3D411yescharacteristic
36C2×C3×C3⋊Dic3D411yescharacteristic
37C2×C32×Dic3—21no—
38C32×D12—22no—
39C32×D12—22no—
40C32×Dic6—21no—
41C32×Dic6D412yescomplemented
42C3×#625—42no—
43C3×#625—42no—
44C3×C3⋊(C3⋊C8)—22no—
45#5163—92no—
46#5075—91no—
47C2×C32×Q8—31no—
48C2×C32×D4—32no—
49C2×C3×#67—62no—
50C2×C3×#67—62no—
51C2×C3×D12—22no—
52C2×C3×D12C2×D312yes—
53C2×C3×D12C2×D312yes—
54C2×C3×Dic6—21no—
55C2×C3×Dic6C2×D312yes—
56C2×#625—21no—
57C2×#625—21no—
58C4⋊(C3⋊Dic3)—31no—
59C4⋊(C3⋊Dic3)—32no—
60C4⋊(C3⋊Dic3)—61no—
61C4⋊(C3⋊Dic3)—61no—
62C4⋊(C3⋊Dic3)—62no—
63C4⋊(C3⋊Dic3)—121no—
64C4×C3⋊Dic3—61no—
65C4×C3⋊Dic3C2×D311yescharacteristic
66C2×C3⋊(C3⋊C8)C2×D312yes—
67C3×#215—64no—
68C3×#214—64no—
69C3×C4×Dic3—32no—
70C3×C4×Dic3—61no—
71C2×C3×C3⋊C8—32no—
72C2×C3×C3⋊C8—32no—
73C2×C3×C3⋊C8—62no—
74Dic3⋊Dic3—62no—
75Dic3⋊Dic3—62no—
76Dic32—62no—
77Dic32—62no—
78#3715—24no—
79C22×C33C2×D411yescharacteristic
80C2×C32×D3—42no—
81C2×C32×D3—42no—
82C2×C32×D3—42no—
83C33×C4C2×D411yescharacteristic
84C33×C4#4111yescharacteristic
85C3⋊(C3⋊Dic3)—41no—
86C3×C3⋊Dic3—21no—
87C3×C3⋊Dic3SD1612yescomplemented
88C32×Dic3—22no—
89C2×C3×SD16—92no—
90C3×#127—91no—
91Q8×Dic3—92no—
92Q8×Dic3—181no—
93C2×#215—32no—
94#940—32no—
95C2×#214—32no—
96C4×Dic6—91no—
97#832—92no—
98#832—92no—
99#832—182no—
100#821—92no—
101Dic3⋊C8—92no—
102C4⋊(C3⋊C8)—91no—
103C23×C32—62no—
104C22×C3×D3—22no—
105C22×C3×D3—22no—
106C22×C3×D3—42no—
107C32×Q8—32no—
108C32×Q8—61no—
109C32×D4—62no—
110C32×D4—62no—
111C2×C32×C4—22no—
112C2×C32×C4—21no—
113C2×C32×C4—41no—
114C2×C32×C4—61no—
115C2×C32×C4C22×D311yescharacteristic
116C2×C32×C4C22×D312yes—
117C2×C3⋊Dic3—61no—
118C2×C3⋊Dic3—62no—
119C2×C3⋊Dic3—62no—
120C2×C3⋊Dic3—122no—
121C2×C3⋊Dic3—122no—
122C2×C3⋊Dic3#6711yescharacteristic
123C2×C3⋊Dic3#6711yescharacteristic
124C3×#67—124no—
125C3×#67—124no—
126C2×C3×Dic3—22no—
127C2×C3×Dic3—22no—
128C2×C3×Dic3—32no—
129C2×C3×Dic3—32no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C3×D12—24no—
133C3×D12—22no—
134C3×D12—22no—
135C3×D12—22no—
136C3×D12—22no—
137C3×D12—42no—
138C3×Dic6—22no—
139C3×Dic6—22no—
140C3×Dic6—22no—
141C3×Dic6#6714yescomplemented
142#625—42no—
143#625—42no—
144C3⋊(C3⋊C8)—22no—
145C3×C3⋊C8—62no—
146C3×C3⋊C8—62no—
147C3×C3⋊C8—122no—
148#460—271noSylow
149C2×C33#12111yescharacteristic
150C2×C33C2×SD1611yescharacteristic
151C2×C33AGL(1,ℤ/8)11yescharacteristic
152C32×D3—82no—
153C2×C3×Q8—32no—
154C2×C3×Q8—61no—
155C2×C3×D4—32no—
156C2×C3×D4—32no—
157C2×C3×D4—62no—
158C2×C3×D4—181no—
159C2×C3×D4—181no—
160C2×#67—62no—
161C2×#67—62no—
162C2×D12D3212yes—
163C2×Dic6D3211yescharacteristic
164C3×SD16—184no—
165C2×C3×C8—92no—
166C3×C42—91no—
167#215—64no—
168#214—64no—
169C4⋊Dic3—92no—
170C4⋊Dic3—91no—
171C4⋊Dic3—181no—
172C4⋊Dic3—182no—
173C4⋊Dic3—182no—
174C4⋊Dic3—182no—
175C4⋊Dic3—361no—
176Dic3⋊C4—181no—
177C4×Dic3—32no—
178C4×Dic3—61no—
179C4×Dic3—181no—
180C4×Dic3—182no—
181C4×Dic3—182no—
182C2×C3⋊C8—32no—
183C2×C3⋊C8—32no—
184C2×C3⋊C8—62no—
185C22×C32—22no—
186C22×C32—21no—
187C22×C32—41no—
188C22×C32—122no—
189C22×C32—122no—
190C22×C32—122no—
191C22×C32D3×D412yes—
192C22×C32C2×#6711yescharacteristic
193C2×C3×D3—44no—
194C2×C3×D3—42no—
195C2×C3×D3—42no—
196C2×C3×D3—42no—
197C2×C3×D3—42no—
198C2×C3×D3—42no—
199C2×C3×D3—42no—
200C2×C3×D3—82no—
201C2×C3×D3—82no—
202C32×C4—21no—
203C32×C4—22no—
204C32×C4—21no—
205C32×C4—22no—
206C32×C4—41no—
207C32×C4—41no—
208C32×C4—62no—
209C32×C4D3×D411yescharacteristic
210C32×C4#23711yescharacteristic
211C32×C4#23712yes—
212C32×C4C2×#6712yes—
213C3⋊Dic3—21no—
214C3⋊Dic3—121no—
215C3⋊Dic3—122no—
216C3⋊Dic3—122no—
217C3⋊Dic3—242no—
218C3⋊Dic3—242no—
219C3⋊Dic3#21412yescomplemented
220C3×Dic3—24no—
221C3×Dic3—24no—
222C3×Dic3—34no—
223C3×Dic3—62no—
224C3×Dic3—62no—
225C3×Dic3—121no—
226C2×SD16—92no—
227#127—91no—
228C4×Q8—271no—
229C4⋊C8—271no—
230#102—272no—
231C33#46011yescharacteristic, complemented, Sylow
232C23×C3—62no—
233C23×C3—62no—
234C23×C3—122no—
235C22×D3—22no—
236C3×Q8—34no—
237C3×Q8—62no—
238C3×Q8—62no—
239C3×Q8—62no—
240C3×D4—64no—
241C3×D4—62no—
242C3×D4—62no—
243C3×D4—62no—
244C3×D4—62no—
245C3×D4—122no—
246C3×D4—362no—
247C3×D4—362no—
248C2×C3×C4—22no—
249C2×C3×C4—21no—
250C2×C3×C4—41no—
251C2×C3×C4—62no—
252C2×C3×C4—62no—
253C2×C3×C4—91no—
254C2×C3×C4—91no—
255C2×C3×C4C2×D3211yescharacteristic
256C2×C3×C4C2×D3212yes—
257#67—124no—
258#67—124no—
259C2×Dic3—21no—
260C2×Dic3—32no—
261C2×Dic3—32no—
262C2×Dic3—61no—
263C2×Dic3—61no—
264C2×Dic3—182no—
265C2×Dic3—182no—
266C2×Dic3—181no—
267C2×Dic3—362no—
268C2×Dic3—362no—
269D12—22no—
270D12—22no—
271Dic6—21no—
272Dic6#62512yescomplemented
273C3×C8—182no—
274C3⋊C8—62no—
275C3⋊C8—62no—
276C3⋊C8—122no—
277C2×C32—22no—
278C2×C32—22no—
279C2×C32—21no—
280C2×C32—21no—
281C2×C32—22no—
282C2×C32—21no—
283C2×C32—41no—
284C2×C32—41no—
285C2×C32—41no—
286C2×C32—242no—
287C2×C32D3×SD1612yes—
288C2×C32#91412yes—
289C2×C32#93211yescharacteristic
290C2×C32C2×#21411yescharacteristic
291C2×C32#93811yescharacteristic
292C2×C32#93912yes—
293C3×D3—84no—
294C3×D3—82no—
295C3×D3—82no—
296C2×Q8—31no—
297C2×D4—32no—
298C2×D4—181no—
299C2×D4—181no—
300SD16—184no—
301C2×C8—92no—
302C4⋊C4—271no—
303C4⋊C4—541no—
304C42—91no—
305C42—541no—
306C22×C3—21no—
307C22×C3—22no—
308C22×C3—41no—
309C22×C3—124no—
310C22×C3—122no—
311C22×C3—122no—
312C22×C3—122no—
313C22×C3—122no—
314C22×C3—122no—
315C22×C3—122no—
316C22×C3—242no—
317C22×C3—242no—
318C22×C3D3×#6712yes—
319C22×C3#381111yescharacteristic
320C2×D3—42no—
321C2×D3—42no—
322C2×D3—42no—
323C3×C4—22no—
324C3×C4—21no—
325C3×C4—22no—
326C3×C4—21no—
327C3×C4—41no—
328C3×C4—41no—
329C3×C4—64no—
330C3×C4—64no—
331C3×C4—92no—
332C3×C4—181no—
333C3×C4#380512yes—
334C3×C4#380511yescharacteristic
335C3×C4C2×#62511yescharacteristic
336C3×C4D3×#6712yes—
337Dic3—22no—
338Dic3—34no—
339Dic3—62no—
340Dic3—62no—
341Dic3—121no—
342Dic3—362no—
343Dic3—362no—
344Dic3—361no—
345Dic3—722no—
346Dic3—722no—
347C32—21no—
348C32—22no—
349C32—41no—
350C32#507511yescharacteristic, complemented
351C32#516312yescomplemented
352C23—62no—
353Q8—32no—
354Q8—61no—
355D4—62no—
356D4—62no—
357D4—362no—
358D4—362no—
359C2×C4—61no—
360C2×C4—91no—
361C2×C4—91no—
362C2×C4—541no—
363C2×C4D3311yescharacteristic
364C8—182no—
365C2×C3—21no—
366C2×C3—22no—
367C2×C3—21no—
368C2×C3—22no—
369C2×C3—21no—
370C2×C3—22no—
371C2×C3—41no—
372C2×C3—41no—
373C2×C3—41no—
374C2×C3—244no—
375C2×C3—242no—
376C2×C3—242no—
377C2×C3D3×#21412yes—
378C2×C3#6393412yes—
379C2×C3#6394011yescharacteristic
380C2×C3#6394111yescharacteristic
381C2×C3#6397012yes—
382C2×C3#6398011yescharacteristic
383D3—82no—
384C22—122no—
385C22—122no—
386C22—122no—
387C22#8895211yescharacteristic
388C4—62no—
389C4—92no—
390C4—181no—
391C4—1081no—
392C4D3×#62511yescharacteristic
393C4#8896311yescharacteristic
394C3—22no—
395C3—21no—
396C3—41no—
397C3#1059143812yescomplemented
398C3#1059162211yescharacteristic, complemented
399C2—242no—
400C2#1172283711yescharacteristic
401C2#1172287311yescharacteristic
402C2#1172295311yescharacteristic
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. Dic6×C3⋊Dic3C2semidirect
3. C2×C3×#3715C2semidirect
4. #11722078C2semidirect
5. #11722032C2semidirect
25. C2×#3715D3semidirect
41. C32×Dic6D4semidirect
87. C3×C3⋊Dic3SD16semidirect
141. C3×Dic6#67semidirect
219. C3⋊Dic3#214semidirect
231. C33#460semidirect
272. Dic6#625semidirect
350. C32#5075semidirect
351. C32#5163semidirect
397. C3#10591438semidirect
398. C3#10591622semidirect

Maximal subgroups

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

subgroupindex
Dic6×C3⋊Dic32
C3×#639142
C2×C3×#37152
#117220782
#117220322
#105916223
#105914383