Finite Group Databasebeta

← #13162020

Subgroups — #13162020

Order 1728.

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

normal subgroups only

This group has 660 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
2C2×C3×#3756C212yesmaximal
3#11723678C211yesmaximal, characteristic
4#11722078C211yesmaximal, characteristic
5#11722076C212yesmaximal
6#10591633—31nomaximal
7#10591625—32nomaximal
8C2×C33×Q8C2211yescharacteristic
9C2×C3×#634C2212yes—
10C3×#3756—24no—
11C3×C4⋊(C3⋊Dic3)—21no—
12C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
13C3×C4×C3⋊Dic3—21no—
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2212yes—
16#88815—21no—
17#88815—21no—
18Q8×C3⋊Dic3D311yescharacteristic
19C2×#3756D312yes—
20C3×Q8×Dic3—32no—
21C3×Q8×Dic3—61no—
22C2×C3×#216—32no—
23C2×C3×#216—32no—
24C2×C3×#216—62no—
25#63873—31no—
26#63866—32no—
27#63572—31no—
28Dic3⋊(C3⋊C8)—32no—
29#63567—32no—
30#63566—32no—
31#63565—32no—
32C33×Q8—21no—
33C33×Q8D412yes—
34C2×C33×C4—21no—
35C2×C33×C4C2311yescharacteristic
36C2×C3×C3⋊Dic3—21no—
37C2×C3×C3⋊Dic3—22no—
38C2×C3×C3⋊Dic3D411yescharacteristic
39C2×C3×C3⋊Dic3D411yescharacteristic
40C3×#634—22no—
41C3×#634—22no—
42C3×C3⋊(C3⋊C8)—22no—
43#5182—92no—
44#5031—91no—
45C2×C32×Q8—21no—
46C2×C32×Q8—22no—
47C2×C32×Q8—41no—
48C2×C32×Q8C2×D311yescharacteristic
49C2×C32×Q8C2×D312yes—
50C2×#634C2×D312yes—
51C2×C3×Dic6—32no—
52C2×C3×Dic6—32no—
53C2×C3×Dic6—62no—
54#3756—24no—
55C4⋊(C3⋊Dic3)—21no—
56C4⋊(C3⋊Dic3)C2×D311yescharacteristic
57C4×C3⋊Dic3—21no—
58C4×C3⋊Dic3C2×D311yescharacteristic
59C2×C3⋊(C3⋊C8)C2×D312yes—
60C3×#216—68no—
61C3×#216—64no—
62C3×#216—64no—
63C3×C4⋊Dic3—32no—
64C3×C4⋊Dic3—62no—
65C3×C4⋊Dic3—61no—
66C3×C4⋊Dic3—62no—
67C3×C4×Dic3—32no—
68C3×C4×Dic3—62no—
69C3×C4×Dic3—61no—
70C3×C4×Dic3—62no—
71C2×C3×C3⋊C8—32no—
72C2×C3×C3⋊C8—32no—
73C2×C3×C3⋊C8—62no—
74#3724—61no—
75#3724—61no—
76Dic3⋊Dic3—62no—
77Dic3⋊Dic3—62no—
78C22×C33C2×D411yescharacteristic
79C33×C4—22no—
80C33×C4C2×D411yescharacteristic
81C33×C4#4111yescharacteristic
82C3×C3⋊Dic3—21no—
83C3×C3⋊Dic3—42no—
84C3×C3⋊Dic3—41no—
85C3×C3⋊Dic3Q1612yescomplemented
86C2×C3×Q16—92no—
87C3×C4×Q8—91no—
88Q8×Dic3—32no—
89Q8×Dic3—61no—
90#944—92no—
91C2×#216—32no—
92C2×#216—32no—
93C2×#216—62no—
94#869—91no—
95#835—92no—
96#816—92no—
97Dic3⋊C8—92no—
98#810—92no—
99C4⋊(C3⋊C8)—91no—
100C32×Q8—21no—
101C32×Q8—22no—
102C32×Q8—22no—
103C32×Q8—22no—
104C32×Q8—24no—
105C32×Q8—42no—
106C32×Q8—42no—
107C32×Q8—42no—
108C32×Q8D1212yes—
109C32×Q8#6714yes—
110C2×C32×C4—21no—
111C2×C32×C4—22no—
112C2×C32×C4—22no—
113C2×C32×C4—22no—
114C2×C32×C4—21no—
115C2×C32×C4—41no—
116C2×C32×C4—42no—
117C2×C32×C4—42no—
118C2×C32×C4C22×D311yescharacteristic
119C2×C32×C4C22×D312yes—
120C2×C3⋊Dic3—21no—
121C2×C3⋊Dic3—22no—
122C2×C3⋊Dic3#6711yescharacteristic
123C2×C3⋊Dic3#6711yescharacteristic
124#634—22no—
125#634—22no—
126C2×C3×Dic3—32no—
127C2×C3×Dic3—32no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—62no—
131C2×C3×Dic3—62no—
132C2×C3×Dic3—62no—
133C2×C3×Dic3—62no—
134C2×C3×Dic3—122no—
135C3×Dic6—64no—
136C3×Dic6—62no—
137C3×Dic6—62no—
138C3×Dic6—62no—
139C3×Dic6—62no—
140C3×Dic6—122no—
141C3⋊(C3⋊C8)—22no—
142C3×C3⋊C8—64no—
143C3×C3⋊C8—62no—
144C3×C3⋊C8—62no—
145#462—271noSylow
146C2×C33#12111yescharacteristic
147C2×C33C2×Q1611yescharacteristic
148C2×C33#13711yescharacteristic
149C2×C3×Q8—21no—
150C2×C3×Q8—22no—
151C2×C3×Q8—41no—
152C2×C3×Q8—92no—
153C2×C3×Q8D3211yescharacteristic
154C2×C3×Q8D3212yes—
155C2×Dic6—32no—
156C2×Dic6—32no—
157C2×Dic6—62no—
158C3×Q16—184no—
159C2×C3×C8—92no—
160C3×C4⋊C4—91no—
161C3×C4⋊C4—181no—
162C3×C42—91no—
163C3×C42—181no—
164#216—68no—
165#216—64no—
166#216—64no—
167C4⋊Dic3—32no—
168C4⋊Dic3—61no—
169C4⋊Dic3—62no—
170C4⋊Dic3—62no—
171C4⋊Dic3—181no—
172C4⋊Dic3—181no—
173Dic3⋊C4—182no—
174Dic3⋊C4—182no—
175C4×Dic3—32no—
176C4×Dic3—61no—
177C4×Dic3—62no—
178C4×Dic3—62no—
179C2×C3⋊C8—32no—
180C2×C3⋊C8—32no—
181C2×C3⋊C8—62no—
182C22×C32—22no—
183C22×C32—21no—
184C22×C32—41no—
185C22×C32D3×D412yes—
186C22×C32C2×#6711yescharacteristic
187C32×C4—22no—
188C32×C4—24no—
189C32×C4—24no—
190C32×C4—22no—
191C32×C4—21no—
192C32×C4—21no—
193C32×C4—22no—
194C32×C4—41no—
195C32×C4—41no—
196C32×C4—44no—
197C32×C4—44no—
198C32×C4C2×D1211yescharacteristic
199C32×C4#23511yescharacteristic
200C32×C4#23712yes—
201C32×C4C2×#6712yes—
202C3⋊Dic3—21no—
203C3⋊Dic3—42no—
204C3⋊Dic3—41no—
205C3⋊Dic3#21612yescomplemented
206C3×Dic3—34no—
207C3×Dic3—62no—
208C3×Dic3—62no—
209C3×Dic3—121no—
210C3×Dic3—124no—
211C3×Dic3—122no—
212C3×Dic3—122no—
213C3×Dic3—122no—
214C3×Dic3—122no—
215C2×Q16—92no—
216C4⋊Q8—271no—
217C4×Q8—91no—
218C4⋊C8—271no—
219#103—272no—
220C33#46211yescharacteristic, complemented, Sylow
221C3×Q8—22no—
222C3×Q8—22no—
223C3×Q8—21no—
224C3×Q8—22no—
225C3×Q8—24no—
226C3×Q8—42no—
227C3×Q8—42no—
228C3×Q8—42no—
229C3×Q8—182no—
230C3×Q8—182no—
231C3×Q8#62512yes—
232C3×Q8#62614yes—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—21no—
236C2×C3×C4—22no—
237C2×C3×C4—21no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—41no—
241C2×C3×C4—91no—
242C2×C3×C4—91no—
243C2×C3×C4—182no—
244C2×C3×C4—181no—
245C2×C3×C4C2×D3212yes—
246C2×C3×C4C2×D3211yescharacteristic
247C2×Dic3—32no—
248C2×Dic3—32no—
249C2×Dic3—61no—
250C2×Dic3—61no—
251C2×Dic3—62no—
252C2×Dic3—62no—
253C2×Dic3—62no—
254C2×Dic3—62no—
255C2×Dic3—122no—
256Dic6—64no—
257Dic6—62no—
258Dic6—62no—
259Dic6—62no—
260Dic6—62no—
261Dic6—122no—
262C3×C8—182no—
263C3⋊C8—64no—
264C3⋊C8—62no—
265C3⋊C8—62no—
266C2×C32—22no—
267C2×C32—22no—
268C2×C32—21no—
269C2×C32—21no—
270C2×C32—22no—
271C2×C32—21no—
272C2×C32—41no—
273C2×C32—41no—
274C2×C32—41no—
275C2×C32D3×Q1612yes—
276C2×C32#91812yes—
277C2×C32#93011yescharacteristic
278C2×C32#93912yes—
279C2×C32C2×#21611yescharacteristic
280C2×C32#95111yescharacteristic
281C2×Q8—92no—
282C2×Q8#111311yescharacteristic
283Q16—184no—
284C2×C8—92no—
285C4⋊C4—91no—
286C4⋊C4—181no—
287C4⋊C4—541no—
288C4⋊C4—541no—
289C42—91no—
290C42—181no—
291C22×C3—21no—
292C22×C3—22no—
293C22×C3—41no—
294C22×C3D3×#6712yes—
295C22×C3#381111yescharacteristic
296C3×C4—24no—
297C3×C4—24no—
298C3×C4—22no—
299C3×C4—22no—
300C3×C4—21no—
301C3×C4—22no—
302C3×C4—21no—
303C3×C4—44no—
304C3×C4—44no—
305C3×C4—41no—
306C3×C4—41no—
307C3×C4—92no—
308C3×C4—181no—
309C3×C4—361no—
310C3×C4—362no—
311C3×C4#380412yes—
312C3×C4#380511yescharacteristic
313C3×C4C2×#62511yescharacteristic
314C3×C4C2×#62612yes—
315Dic3—34no—
316Dic3—62no—
317Dic3—62no—
318Dic3—121no—
319Dic3—124no—
320Dic3—122no—
321Dic3—122no—
322Dic3—122no—
323Dic3—122no—
324C32—22no—
325C32—21no—
326C32—41no—
327C32#503111yescharacteristic, complemented
328C32#518212yescomplemented
329Q8—21no—
330Q8—182no—
331Q8—182no—
332Q8#629512yes—
333C2×C4—21no—
334C2×C4—91no—
335C2×C4—91no—
336C2×C4—181no—
337C2×C4—182no—
338C2×C4C2×#111311yescharacteristic
339C8—182no—
340C2×C3—21no—
341C2×C3—22no—
342C2×C3—21no—
343C2×C3—22no—
344C2×C3—22no—
345C2×C3—21no—
346C2×C3—41no—
347C2×C3—41no—
348C2×C3—41no—
349C2×C3D3×#21612yes—
350C2×C3#6394312yes—
351C2×C3#6394411yescharacteristic
352C2×C3#6394511yescharacteristic
353C2×C3#6397812yes—
354C2×C3#6398011yescharacteristic
355C22#8904911yescharacteristic
356C4—22no—
357C4—92no—
358C4—181no—
359C4—361no—
360C4—362no—
361C4#8904611yescharacteristic
362C4C2×#629511yescharacteristic
363C3—22no—
364C3—21no—
365C3—41no—
366C3#1059162512yescomplemented
367C3#1059163311yescharacteristic, complemented
368C2#1172371011yescharacteristic
369C2#1172371111yescharacteristic
370C2#1172371711yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×Q8#1113
hypocenterC33#462
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
85. C3×C3⋊Dic3Q16semidirect
205. C3⋊Dic3#216semidirect
220. C33#462semidirect
327. C32#5031semidirect
328. C32#5182semidirect
366. C3#10591625semidirect
367. C3#10591633semidirect

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×#37562
#117236782
#117220782
#117220762
#105916333
#105916253