Finite Group Databasebeta

← #13162013

Subgroups — #13162013

Order 1728.

756 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C3×Q8×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C2×C3×#3755C212yesmaximal, complemented
3#11723697C211yesmaximal, characteristic
4#11722078C211yesmaximal, characteristic, complemented
5#11722075C212yesmaximal, complemented
6#10591622—31nomaximal
7#10591617—32nomaximal
8C2×C33×Q8C2211yescharacteristic
9C2×C3×C4⋊(C3⋊D3)C2212yes—
10C2×#6295—21no—
11C2×#6295—21no—
12C3×#3755—24no—
13C3×C4⋊(C3⋊Dic3)—21no—
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4×C3⋊Dic3—21no—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2212yes—
18Q8×C3⋊Dic3D311yescharacteristic, complemented
19C2×#3755D312yescomplemented
20C3×Q8×Dic3—32no—
21C3×Q8×Dic3—61no—
22C2×C3×#215—32no—
23C2×C3×#215—32no—
24C2×C3×#215—62no—
25#63914—31no—
26#63911—32no—
27#63572—31no—
28Dic3⋊(C3⋊C8)—32no—
29#63564—32no—
30#63563—32no—
31#63562—32no—
32C22×C3×C3⋊D3—22no—
33C33×Q8—21no—
34C33×Q8D412yescomplemented
35C2×C33×C4—21no—
36C2×C33×C4C2311yescharacteristic
37C2×C3×C3⋊Dic3—21no—
38C2×C3×C3⋊Dic3D411yescharacteristic
39C2×C3×C3⋊Dic3D411yescharacteristic
40C3×C4⋊(C3⋊D3)—22no—
41C3×C4⋊(C3⋊D3)—22no—
42#6295—42no—
43#6295—42no—
44C3×C3⋊(C3⋊C8)—22no—
45#5163—92no—
46#5027—91no—
47C2×C32×Q8—21no—
48C2×C32×Q8—22no—
49C2×C32×Q8—41no—
50C2×C32×Q8C2×D311yescharacteristic
51C2×C32×Q8C2×D312yes—
52C2×C4⋊(C3⋊D3)C2×D312yes—
53C2×C3×D12—32no—
54C2×C3×D12—32no—
55C2×C3×D12—62no—
56C2×#626—62no—
57C2×#626—62no—
58C2×#625—61no—
59C2×#625—61no—
60#3755—24no—
61C4⋊(C3⋊Dic3)—21no—
62C4⋊(C3⋊Dic3)C2×D311yescharacteristic
63C4×C3⋊Dic3—21no—
64C4×C3⋊Dic3C2×D311yescharacteristic
65C2×C3⋊(C3⋊C8)C2×D312yes—
66C3×#215—68no—
67C3×#215—64no—
68C3×#215—64no—
69C3×C4⋊Dic3—32no—
70C3×C4⋊Dic3—62no—
71C3×C4⋊Dic3—61no—
72C3×C4⋊Dic3—62no—
73C3×C4×Dic3—32no—
74C3×C4×Dic3—62no—
75C3×C4×Dic3—61no—
76C3×C4×Dic3—62no—
77C2×C3×C3⋊C8—32no—
78C2×C3×C3⋊C8—32no—
79C2×C3×C3⋊C8—62no—
80C22×C33C2×D411yescharacteristic
81C2×C3×C3⋊D3—42no—
82C2×C3×C3⋊D3—42no—
83C2×C3×C3⋊D3—42no—
84C33×C4—22no—
85C33×C4C2×D411yescharacteristic
86C33×C4#4111yescharacteristic
87C3×C3⋊Dic3—21no—
88C3×C3⋊Dic3—41no—
89C3×C3⋊Dic3SD1612yescomplemented
90C2×C3×SD16—92no—
91C3×C4×Q8—91no—
92Q8×Dic3—32no—
93Q8×Dic3—61no—
94C2×#215—32no—
95C2×#215—32no—
96C2×#215—62no—
97#940—92no—
98#874—91no—
99#832—92no—
100#821—92no—
101Dic3⋊C8—92no—
102#809—92no—
103C4⋊(C3⋊C8)—91no—
104C22×C3⋊D3—22no—
105C22×C3×D3—62no—
106C22×C3×D3—62no—
107C22×C3×D3—122no—
108C32×Q8—21no—
109C32×Q8—22no—
110C32×Q8—22no—
111C32×Q8—22no—
112C32×Q8—24no—
113C32×Q8—42no—
114C32×Q8—42no—
115C32×Q8—42no—
116C32×Q8D1212yescomplemented
117C32×Q8#6714yescomplemented
118C2×C32×C4—21no—
119C2×C32×C4—22no—
120C2×C32×C4—22no—
121C2×C32×C4—22no—
122C2×C32×C4—21no—
123C2×C32×C4—41no—
124C2×C32×C4—42no—
125C2×C32×C4—42no—
126C2×C32×C4C22×D311yescharacteristic
127C2×C32×C4C22×D312yes—
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3#6711yescharacteristic
130C2×C3⋊Dic3#6711yescharacteristic
131C4⋊(C3⋊D3)—22no—
132C4⋊(C3⋊D3)—22no—
133C2×C3×Dic3—32no—
134C2×C3×Dic3—32no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—61no—
137C2×C3×Dic3—62no—
138C2×C3×Dic3—62no—
139C3×D12—64no—
140C3×D12—62no—
141C3×D12—62no—
142C3×D12—62no—
143C3×D12—62no—
144C3×D12—122no—
145#626—124no—
146#626—124no—
147#625—122no—
148#625—122no—
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
157C3×C3⋊D3—82no—
158C2×C3×Q8—21no—
159C2×C3×Q8—22no—
160C2×C3×Q8—41no—
161C2×C3×Q8D3211yescharacteristic
162C2×C3×Q8D3212yes—
163C2×C3×D4—92no—
164C2×#67—182no—
165C2×#67—182no—
166C2×D12—32no—
167C2×D12—32no—
168C2×D12—62no—
169C2×D12—181no—
170C2×D12—181no—
171C3×SD16—184no—
172C2×C3×C8—92no—
173C3×C4⋊C4—91no—
174C3×C4⋊C4—181no—
175C3×C42—91no—
176C3×C42—181no—
177#215—68no—
178#215—64no—
179#215—64no—
180C4⋊Dic3—32no—
181C4⋊Dic3—61no—
182C4⋊Dic3—62no—
183C4⋊Dic3—62no—
184C4×Dic3—32no—
185C4×Dic3—61no—
186C4×Dic3—62no—
187C4×Dic3—62no—
188C2×C3⋊C8—32no—
189C2×C3⋊C8—32no—
190C2×C3⋊C8—62no—
191C22×C32—22no—
192C22×C32—21no—
193C22×C32—41no—
194C22×C32D3×D412yes—
195C22×C32C2×#6711yescharacteristic
196C2×C3⋊D3—42no—
197C2×C3⋊D3—42no—
198C2×C3⋊D3—42no—
199C2×C3×D3—124no—
200C2×C3×D3—122no—
201C2×C3×D3—122no—
202C2×C3×D3—122no—
203C2×C3×D3—122no—
204C2×C3×D3—122no—
205C2×C3×D3—122no—
206C2×C3×D3—242no—
207C2×C3×D3—242no—
208C32×C4—22no—
209C32×C4—24no—
210C32×C4—24no—
211C32×C4—22no—
212C32×C4—21no—
213C32×C4—21no—
214C32×C4—22no—
215C32×C4—41no—
216C32×C4—41no—
217C32×C4—44no—
218C32×C4—44no—
219C32×C4C2×D1211yescharacteristic
220C32×C4#23511yescharacteristic
221C32×C4#23712yes—
222C32×C4C2×#6712yes—
223C3⋊Dic3—21no—
224C3⋊Dic3—41no—
225C3⋊Dic3#21512yescomplemented
226C3×Dic3—34no—
227C3×Dic3—62no—
228C3×Dic3—62no—
229C3×Dic3—121no—
230C3×Dic3—122no—
231C3×Dic3—122no—
232C2×SD16—92no—
233#127—271no—
234C4×Q8—91no—
235C4⋊C8—271no—
236#102—272no—
237C33#46011yescharacteristic, complemented, Sylow
238C23×C3—182no—
239C22×D3—62no—
240C22×D3—62no—
241C22×D3—122no—
242C3×Q8—22no—
243C3×Q8—22no—
244C3×Q8—21no—
245C3×Q8—22no—
246C3×Q8—24no—
247C3×Q8—42no—
248C3×Q8—42no—
249C3×Q8—42no—
250C3×Q8#62512yescomplemented
251C3×Q8#62614yescomplemented
252C3×D4—182no—
253C3×D4—182no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—21no—
257C2×C3×C4—22no—
258C2×C3×C4—21no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—41no—
262C2×C3×C4—91no—
263C2×C3×C4—91no—
264C2×C3×C4—181no—
265C2×C3×C4C2×D3212yes—
266C2×C3×C4C2×D3211yescharacteristic
267#67—364no—
268#67—364no—
269C2×Dic3—32no—
270C2×Dic3—32no—
271C2×Dic3—61no—
272C2×Dic3—61no—
273C2×Dic3—62no—
274C2×Dic3—62no—
275D12—64no—
276D12—62no—
277D12—62no—
278D12—62no—
279D12—62no—
280D12—122no—
281D12—362no—
282D12—362no—
283C3×C8—182no—
284C3⋊C8—64no—
285C3⋊C8—62no—
286C3⋊C8—62no—
287C2×C32—22no—
288C2×C32—22no—
289C2×C32—21no—
290C2×C32—21no—
291C2×C32—22no—
292C2×C32—21no—
293C2×C32—41no—
294C2×C32—41no—
295C2×C32—41no—
296C2×C32D3×SD1612yes—
297C2×C32#91412yes—
298C2×C32#93011yescharacteristic
299C2×C32#93912yes—
300C2×C32C2×#21511yescharacteristic
301C2×C32#94911yescharacteristic
302C3⋊D3—82no—
303C3×D3—244no—
304C3×D3—242no—
305C3×D3—242no—
306C2×Q8#111311yescharacteristic
307C2×D4—92no—
308C2×D4—541no—
309C2×D4—541no—
310SD16—184no—
311C2×C8—92no—
312C4⋊C4—91no—
313C4⋊C4—181no—
314C42—91no—
315C42—181no—
316C22×C3—21no—
317C22×C3—22no—
318C22×C3—41no—
319C22×C3—362no—
320C22×C3—362no—
321C22×C3—362no—
322C22×C3D3×#6712yes—
323C22×C3#381111yescharacteristic
324C2×D3—124no—
325C2×D3—122no—
326C2×D3—122no—
327C2×D3—122no—
328C2×D3—122no—
329C2×D3—122no—
330C2×D3—122no—
331C2×D3—242no—
332C2×D3—242no—
333C3×C4—24no—
334C3×C4—24no—
335C3×C4—22no—
336C3×C4—22no—
337C3×C4—21no—
338C3×C4—22no—
339C3×C4—21no—
340C3×C4—44no—
341C3×C4—44no—
342C3×C4—41no—
343C3×C4—41no—
344C3×C4—92no—
345C3×C4—181no—
346C3×C4—361no—
347C3×C4#380412yes—
348C3×C4#380511yescharacteristic
349C3×C4C2×#62511yescharacteristic
350C3×C4C2×#62612yes—
351Dic3—34no—
352Dic3—62no—
353Dic3—62no—
354Dic3—121no—
355Dic3—122no—
356Dic3—122no—
357C32—22no—
358C32—21no—
359C32—41no—
360C32#502711yescharacteristic, complemented
361C32#516312yescomplemented
362C23—182no—
363Q8—21no—
364Q8#629512yescomplemented
365D4—182no—
366D4—182no—
367D4—1082no—
368D4—1082no—
369C2×C4—21no—
370C2×C4—91no—
371C2×C4—91no—
372C2×C4—181no—
373C2×C4C2×#111311yescharacteristic
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—722no—
385C2×C3D3×#21512yes—
386C2×C3#6393912yes—
387C2×C3#6394011yescharacteristic
388C2×C3#6394111yescharacteristic
389C2×C3#6397812yes—
390C2×C3#6398011yescharacteristic
391D3—244no—
392D3—242no—
393D3—242no—
394C22—362no—
395C22—362no—
396C22—362no—
397C22#8904911yescharacteristic
398C4—22no—
399C4—92no—
400C4—181no—
401C4—361no—
402C4#8904611yescharacteristic
403C4C2×#629511yescharacteristic
404C3—22no—
405C3—21no—
406C3—41no—
407C3#1059161712yescomplemented
408C3#1059162211yescharacteristic, complemented
409C2—722no—
410C2#1172370811yescharacteristic
411C2#1172370911yescharacteristic
412C2#1172371711yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radical#11723697C2
hypercenterC2×Q8#1113
hypocenterC33#460
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C3×Q8×C3⋊Dic3C2semidirect
2. C2×C3×#3755C2semidirect
4. #11722078C2semidirect
5. #11722075C2semidirect
18. Q8×C3⋊Dic3D3semidirect
19. C2×#3755D3semidirect
34. C33×Q8D4semidirect
89. C3×C3⋊Dic3SD16semidirect
116. C32×Q8D12semidirect
117. C32×Q8#67semidirect
225. C3⋊Dic3#215semidirect
237. C33#460semidirect
250. C3×Q8#625semidirect
251. C3×Q8#626semidirect
360. C32#5027semidirect
361. C32#5163semidirect
364. Q8#6295semidirect
407. C3#10591617semidirect
408. C3#10591622semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×C3⋊Dic32
C2×C3×#37552
#117236972
#117220782
#117220752
#105916223
#105916173