Finite Group Databasebeta

← C3×#10591633 #13158635

Subgroups — C3×#10591633 #13158635

Order 1728.

660 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 136 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×C32×#216C212yesmaximal
3C3×#63873C211yesmaximal, characteristic
4C3×#63572C211yesmaximal, characteristic
5C3×#63566C212yesmaximal
6C3×#5182—32nomaximal
7#10591633C311yesmaximal, characteristic, direct product factor
8C2×C33×Q8C2211yescharacteristic
9C2×C32×Dic6C2212yes—
10C3×C4⋊(C3⋊Dic3)—21no—
11C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
12C3×C4×C3⋊Dic3—21no—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C32×#216—24no—
15C2×C32×C3⋊C8C2212yes—
16C3×#3724—21no—
17C3×#3724—21no—
18C2×C32×Q16—32no—
19Q8×C3⋊Dic3C2×C311yescharacteristic
20C3×Q8×Dic3—32no—
21C3×Q8×Dic3—61no—
22C3×#944—32no—
23C2×C3×#216—22no—
24C2×C3×#216D312yes—
25C2×C3×#216C2×C312yes—
26#63873C2×C311yescharacteristic
27C3×#835—32no—
28C3×#816—32no—
29C3×Dic3⋊C8—32no—
30#63572C2×C311yescharacteristic
31#63566C2×C312yes—
32C33×Q8—21no—
33C33×Q8D412yes—
34C2×C33×C4—21no—
35C2×C33×C4C2311yescharacteristic
36C2×C3×C3⋊Dic3—21no—
37C2×C3×C3⋊Dic3D411yescharacteristic
38C2×C3×C3⋊Dic3D411yescharacteristic
39C2×C32×Dic3—22no—
40C32×Dic6—22no—
41C32×Dic6—22no—
42C32×C3⋊C8—22no—
43C3×#462—91no—
44#5182—32no—
45C2×C32×Q8—21no—
46C2×C32×Q8—22no—
47C2×C32×Q8—32no—
48C2×C32×Q8—41no—
49C2×C32×Q8C2×D312yes—
50C2×C32×Q8C22×C311yescharacteristic
51C2×C3×Dic6—22no—
52C2×C3×Dic6C2×D312yes—
53C2×C3×Dic6C22×C312yes—
54C32×Q16—64no—
55C2×C32×C8—32no—
56C4⋊(C3⋊Dic3)—21no—
57C4⋊(C3⋊Dic3)C22×C311yescharacteristic
58C4×C3⋊Dic3—21no—
59C4×C3⋊Dic3C22×C311yescharacteristic
60C3×#216—28no—
61C3×#216—24no—
62C3×#216—24no—
63C3×C4⋊Dic3—32no—
64C3×C4⋊Dic3—62no—
65C3×C4⋊Dic3—61no—
66C3×C4⋊Dic3—62no—
67C3×Dic3⋊C4—62no—
68C3×Dic3⋊C4—62no—
69C3×C4×Dic3—32no—
70C3×C4×Dic3—62no—
71C3×C4×Dic3—61no—
72C3×C4×Dic3—62no—
73C2×C3×C3⋊C8—22no—
74C2×C3×C3⋊C8C2×D312yes—
75C2×C3×C3⋊C8C22×C312yes—
76#3724—21no—
77#3724—21no—
78C22×C33C2×D411yescharacteristic
79C33×C4—22no—
80C33×C4C2×D411yescharacteristic
81C33×C4#4111yescharacteristic
82C3×C3⋊Dic3—21no—
83C3×C3⋊Dic3—41no—
84C3×C3⋊Dic3Q1612yescomplemented
85C32×Dic3—42no—
86C2×C3×Q16—32no—
87C2×C3×Q16—32no—
88C2×C3×Q16—62no—
89C3×C4⋊Q8—91no—
90C3×C4×Q8—91no—
91Q8×Dic3—32no—
92Q8×Dic3—61no—
93#944—32no—
94C2×#216C3×D312yes—
95C3×C4⋊C8—91no—
96C3×#103—92no—
97#835—32no—
98#816—32no—
99Dic3⋊C8—32no—
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×Q8—62no—
109C32×Q8—62no—
110C32×Q8#6714yes—
111C32×Q8C3×D412yes—
112C2×C32×C4—21no—
113C2×C32×C4—22no—
114C2×C32×C4—22no—
115C2×C32×C4—21no—
116C2×C32×C4—22no—
117C2×C32×C4—41no—
118C2×C32×C4—42no—
119C2×C32×C4—42no—
120C2×C32×C4—62no—
121C2×C32×C4C22×D312yes—
122C2×C32×C4C23×C311yescharacteristic
123C2×C3⋊Dic3—21no—
124C2×C3⋊Dic3C3×D411yescharacteristic
125C2×C3⋊Dic3C3×D411yescharacteristic
126C2×C3×Dic3—22no—
127C2×C3×Dic3—22no—
128C2×C3×Dic3—32no—
129C2×C3×Dic3—32no—
130C2×C3×Dic3—42no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—62no—
134C2×C3×Dic3—62no—
135C3×Dic6—24no—
136C3×Dic6—22no—
137C3×Dic6—22no—
138C3×Dic6—22no—
139C3×Dic6—22no—
140C3×Dic6—42no—
141C32×C8—62no—
142C3×C3⋊C8—24no—
143C3×C3⋊C8—22no—
144C3×C3⋊C8—22no—
145#462—91noSylow
146C2×C33#12111yescharacteristic
147C2×C33C2×Q1611yescharacteristic
148C2×C33#13711yescharacteristic
149C2×C3×Q8—21no—
150C2×C3×Q8—22no—
151C2×C3×Q8—32no—
152C2×C3×Q8—32no—
153C2×C3×Q8—41no—
154C2×C3×Q8—62no—
155C2×C3×Q8D3211yescharacteristic
156C2×C3×Q8C2×C3×D312yes—
157C2×Dic6C2×C3×D312yes—
158C3×Q16—68no—
159C3×Q16—64no—
160C3×Q16—64no—
161C2×C3×C8—32no—
162C2×C3×C8—32no—
163C2×C3×C8—62no—
164C3×C4⋊C4—91no—
165C3×C4⋊C4—181no—
166C3×C4⋊C4—181no—
167C3×C4⋊C4—181no—
168C3×C42—91no—
169C3×C42—181no—
170#216—24no—
171C4⋊Dic3—32no—
172C4⋊Dic3—61no—
173C4⋊Dic3—62no—
174C4⋊Dic3—62no—
175Dic3⋊C4—62no—
176Dic3⋊C4—62no—
177C4×Dic3—32no—
178C4×Dic3—61no—
179C4×Dic3—62no—
180C4×Dic3—62no—
181C2×C3⋊C8C2×C3×D312yes—
182C22×C32—21no—
183C22×C32—22no—
184C22×C32—41no—
185C22×C32D3×D412yes—
186C22×C32C2×C3×D411yescharacteristic
187C32×C4—22no—
188C32×C4—24no—
189C32×C4—24no—
190C32×C4—21no—
191C32×C4—21no—
192C32×C4—22no—
193C32×C4—22no—
194C32×C4—41no—
195C32×C4—41no—
196C32×C4—44no—
197C32×C4—44no—
198C32×C4—122no—
199C32×C4#23712yes—
200C32×C4C2×#6712yes—
201C32×C4C2×C3×D411yescharacteristic
202C32×C4C3×#4111yescharacteristic
203C3⋊Dic3—21no—
204C3⋊Dic3—41no—
205C3⋊Dic3C3×Q1612yescomplemented
206C3×Dic3—34no—
207C3×Dic3—44no—
208C3×Dic3—42no—
209C3×Dic3—42no—
210C3×Dic3—62no—
211C3×Dic3—62no—
212C3×Dic3—121no—
213C3×Dic3—122no—
214C3×Dic3—122no—
215C2×Q16—32no—
216C4⋊Q8—91no—
217C4×Q8—91no—
218C4⋊C8—91no—
219#103—92no—
220C33#46211yescharacteristic, complemented, Sylow
221C3×Q8—22no—
222C3×Q8—21no—
223C3×Q8—22no—
224C3×Q8—22no—
225C3×Q8—24no—
226C3×Q8—42no—
227C3×Q8—42no—
228C3×Q8—42no—
229C3×Q8—64no—
230C3×Q8—62no—
231C3×Q8—62no—
232C3×Q8—62no—
233C3×Q8—62no—
234C3×Q8—122no—
235C3×Q8#62512yes—
236C3×Q8C3×#6714yes—
237C2×C3×C4—22no—
238C2×C3×C4—21no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—21no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—41no—
245C2×C3×C4—62no—
246C2×C3×C4—62no—
247C2×C3×C4—91no—
248C2×C3×C4—91no—
249C2×C3×C4—122no—
250C2×C3×C4—181no—
251C2×C3×C4C2×D3211yescharacteristic
252C2×C3×C4C22×C3×D312yes—
253C2×Dic3—22no—
254C2×Dic3—32no—
255C2×Dic3—32no—
256C2×Dic3—61no—
257C2×Dic3—61no—
258C2×Dic3—62no—
259C2×Dic3—62no—
260Dic6—22no—
261Dic6—22no—
262C3×C8—64no—
263C3×C8—62no—
264C3×C8—62no—
265C3⋊C8—22no—
266C2×C32—22no—
267C2×C32—22no—
268C2×C32—22no—
269C2×C32—21no—
270C2×C32—21no—
271C2×C32—21no—
272C2×C32—41no—
273C2×C32—41no—
274C2×C32—41no—
275C2×C32D3×Q1612yes—
276C2×C32#91812yes—
277C2×C32#93912yes—
278C2×C32C3×#12111yescharacteristic
279C2×C32C2×C3×Q1611yescharacteristic
280C2×C32C3×#13711yescharacteristic
281C2×Q8—32no—
282C2×Q8C3×D3211yescharacteristic
283Q16—64no—
284C2×C8—32no—
285C4⋊C4—91no—
286C4⋊C4—181no—
287C4⋊C4—181no—
288C4⋊C4—181no—
289C42—91no—
290C42—181no—
291C22×C3—21no—
292C22×C3—22no—
293C22×C3—41no—
294C22×C3#381111yescharacteristic
295C22×C3C3×D3×D412yes—
296C3×C4—24no—
297C3×C4—24no—
298C3×C4—22no—
299C3×C4—21no—
300C3×C4—22no—
301C3×C4—22no—
302C3×C4—21no—
303C3×C4—44no—
304C3×C4—44no—
305C3×C4—41no—
306C3×C4—41no—
307C3×C4—92no—
308C3×C4—124no—
309C3×C4—122no—
310C3×C4—122no—
311C3×C4—181no—
312C3×C4—361no—
313C3×C4#380511yescharacteristic
314C3×C4C2×#62511yescharacteristic
315C3×C4C3×#23712yes—
316C3×C4C2×C3×#6712yes—
317Dic3—34no—
318Dic3—42no—
319Dic3—62no—
320Dic3—62no—
321Dic3—121no—
322Dic3—122no—
323Dic3—122no—
324C32—22no—
325C32—21no—
326C32—41no—
327C32#518212yescomplemented
328C32C3×#46211yescharacteristic, complemented
329Q8—21no—
330Q8—62no—
331Q8—62no—
332Q8C3×#62512yes—
333C2×C4—21no—
334C2×C4—62no—
335C2×C4—91no—
336C2×C4—91no—
337C2×C4—181no—
338C2×C4C2×C3×D3211yescharacteristic
339C8—62no—
340C2×C3—21no—
341C2×C3—22no—
342C2×C3—21no—
343C2×C3—22no—
344C2×C3—21no—
345C2×C3—22no—
346C2×C3—41no—
347C2×C3—41no—
348C2×C3—41no—
349C2×C3#6394411yescharacteristic
350C2×C3#6394511yescharacteristic
351C2×C3#6398011yescharacteristic
352C2×C3C3×D3×Q1612yes—
353C2×C3C3×#91812yes—
354C2×C3C3×#93912yes—
355C22C3×#381111yescharacteristic
356C4—22no—
357C4—92no—
358C4—122no—
359C4—181no—
360C4—361no—
361C4C3×#380511yescharacteristic
362C4C2×C3×#62511yescharacteristic
363C3—22no—
364C3—21no—
365C3—41no—
366C3#1059163311yescharacteristic, direct product factor
367C3C3×#518212yescomplemented
368C2C3×#6394411yescharacteristic
369C2C3×#6394511yescharacteristic
370C2C3×#6398011yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×Q8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×C3×Q8D32
hypocenterC32C3×#462
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
7. #10591633C3direct (N × H)
84. C3×C3⋊Dic3Q16semidirect
205. C3⋊Dic3C3×Q16semidirect
220. C33#462semidirect
327. C32#5182semidirect
328. C32C3×#462semidirect
366. C3#10591633direct (N × H)
367. C3C3×#5182semidirect

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×C32×#2162
C3×#638732
C3×#635722
C3×#635662
C3×#51823
#105916333