Finite Group Databasebeta

← #10592927⋊C3 #13143599

Subgroups — #10592927⋊C3 #13143599

Order 1728.

674 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 674 conjugacy classes of subgroups — too many to draw. Only its 122 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
1C22×Q8×H(3,𝔽3)C211yesmaximal, characteristic
2C2×#3756⋊C3C212yesmaximal, complemented
3#64133⋊C3C211yesmaximal, characteristic
4#63662⋊C3C212yesmaximal, complemented
5#63657⋊C3C211yesmaximal, characteristic
6#10592927C311yesmaximal, characteristic, complemented
7C3×#5228—31nomaximal
8C2×Q8×H(3,𝔽3)—24no—
9C2×Q8×H(3,𝔽3)—21no—
10C2×Q8×H(3,𝔽3)C2212yes—
11C22×C4×H(3,𝔽3)—21no—
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C2×#634⋊C3C2211yescharacteristic
14#3757⋊C3—21no—
15#3756⋊C3—24no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17#3750⋊C3—21no—
18C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
19C22×C32×Q8—21no—
20C22×C32×Q8D311yescharacteristic
21C22×C32×Q8C2×C311yescharacteristic
22C2×#3756C2×C312yescomplemented
23#64133C2×C311yescharacteristic
24C2×C3×#216—32no—
25C3×#924—31no—
26#63662C2×C312yescomplemented
27#63657C2×C311yescharacteristic
28C3×#835—32no—
29C3×#830—31no—
30C23×H(3,𝔽3)D411yescharacteristic
31Q8×H(3,𝔽3)—24no—
32Q8×H(3,𝔽3)—22no—
33Q8×H(3,𝔽3)D414yes—
34C2×C4×H(3,𝔽3)—24no—
35C2×C4×H(3,𝔽3)—22no—
36C2×C4×H(3,𝔽3)—22no—
37C2×C4×H(3,𝔽3)D411yescharacteristic
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×(C3⋊Dic3)⋊C3—21no—
40C2×(C3⋊Dic3)⋊C3—21no—
41#634⋊C3—22no—
42(C3⋊(C3⋊C8))⋊C3—22no—
43C3×#451—91no—
44#5228—31no—
45#5228—91no—
46C2×C32×Q8—28no—
47C2×C32×Q8—22no—
48C2×C32×Q8—22no—
49C2×C32×Q8—24no—
50C2×C32×Q8—24no—
51C2×C32×Q8—21no—
52C2×C32×Q8—21no—
53C2×C32×Q8C2×D312yes—
54C2×C32×Q8C22×C312yes—
55C22×C32×C4—22no—
56C22×C32×C4—21no—
57C22×C32×C4—21no—
58C22×C32×C4—21no—
59C22×C32×C4C2×D311yescharacteristic
60C22×C32×C4C22×C311yescharacteristic
61C2×#634C22×C311yescharacteristic
62C2×C3×Dic6—31no—
63#3757—21no—
64#3756—24no—
65C4⋊(C3⋊Dic3)C22×C311yescharacteristic
66#3750—21no—
67C2×C3⋊(C3⋊C8)C22×C312yes—
68C3×#217—61no—
69C3×#216—64no—
70C3×C4⋊Dic3—31no—
71C3×Dic3⋊C4—61no—
72C2×C3×C3⋊C8—32no—
73C22×H(3,𝔽3)—21no—
74C22×H(3,𝔽3)—21no—
75C22×H(3,𝔽3)Q1612yescomplemented
76C22×H(3,𝔽3)C2×D411yescharacteristic
77C4×H(3,𝔽3)—24no—
78C4×H(3,𝔽3)—21no—
79C4×H(3,𝔽3)C2×D412yes—
80(C3⋊Dic3)⋊C3—41no—
81(C3⋊Dic3)⋊C3—41no—
82C22×C3×Q8—31no—
83C22×C3×Q8—31no—
84C22×C3×Q8—61no—
85C22×C3×Q8C3×D311yescharacteristic
86C2×C3×Q16—92no—
87C3×C22⋊Q8—91no—
88C2×#216—32no—
89C2×#216—92no—
90#924—31no—
91#924—91no—
92C3×#103—92no—
93C3×C22⋊C8—91no—
94#835—32no—
95#835—92no—
96#830—31no—
97#830—91no—
98C23×C32—21no—
99C23×C32#6711yescharacteristic
100C23×C32C3×D411yescharacteristic
101C32×Q8—28no—
102C32×Q8—24no—
103C32×Q8—24no—
104C32×Q8—24no—
105C32×Q8—22no—
106C32×Q8—22no—
107C32×Q8—24no—
108C32×Q8#6714yes—
109C32×Q8C3×D414yes—
110C2×C32×C4—28no—
111C2×C32×C4—24no—
112C2×C32×C4—24no—
113C2×C32×C4—24no—
114C2×C32×C4—24no—
115C2×C32×C4—22no—
116C2×C32×C4—22no—
117C2×C32×C4—22no—
118C2×C32×C4—22no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4#6711yescharacteristic
122C2×C32×C4C3×D411yescharacteristic
123C2×C32×C4C22×D311yescharacteristic
124C2×C32×C4C23×C311yescharacteristic
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3—21no—
127#634—22no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C3×Dic6—62no—
131C3⋊(C3⋊C8)—22no—
132C3×C3⋊C8—62no—
133#451—91noSylow
134C2×H(3,𝔽3)—22no—
135C2×H(3,𝔽3)#12011yescharacteristic
136C2×H(3,𝔽3)C2×Q1611yescharacteristic
137C2×H(3,𝔽3)#13711yescharacteristic
138C2×C3×Q8—24no—
139C2×C3×Q8—21no—
140C2×C3×Q8—32no—
141C2×C3×Q8—32no—
142C2×C3×Q8—61no—
143C2×C3×Q8—61no—
144C2×C3×Q8—64no—
145C2×C3×Q8—64no—
146C2×C3×Q8—68no—
147C2×C3×Q8—62no—
148C2×C3×Q8—62no—
149C2×C3×Q8—91no—
150C2×C3×Q8C2×C3×D312yes—
151C22×C3×C4—21no—
152C22×C3×C4—31no—
153C22×C3×C4—31no—
154C22×C3×C4—61no—
155C22×C3×C4—61no—
156C22×C3×C4—62no—
157C22×C3×C4—61no—
158C22×C3×C4C2×C3×D311yescharacteristic
159C2×Dic6—31no—
160C2×Dic6—91no—
161C3×Q16—184no—
162C2×C3×C8—92no—
163C3×C4⋊C4—91no—
164C3×C4⋊C4—181no—
165C3×C22⋊C4—181no—
166#217—61no—
167#217—181no—
168#216—64no—
169#216—184no—
170C4⋊Dic3—31no—
171C4⋊Dic3—91no—
172Dic3⋊C4—61no—
173Dic3⋊C4—181no—
174C2×C3⋊C8—32no—
175C2×C3⋊C8—92no—
176C22×C32—22no—
177C22×C32—22no—
178C22×C32—21no—
179C22×C32—21no—
180C22×C32—21no—
181C22×C32—21no—
182C22×C32—22no—
183C22×C32—21no—
184C22×C32#21612yescomplemented
185C22×C32C3×Q1612yescomplemented
186C22×C32C2×#6711yescharacteristic
187C22×C32C2×C3×D411yescharacteristic
188C32×C4—28no—
189C32×C4—22no—
190C32×C4—24no—
191C32×C4—21no—
192C32×C4—24no—
193C32×C4—21no—
194C32×C4—22no—
195C32×C4C2×#6712yes—
196C32×C4C2×C3×D412yes—
197C3⋊Dic3—41no—
198C3⋊Dic3—41no—
199C3×Dic3—121no—
200C3×Dic3—121no—
201C22×Q8(C3⋊D3)⋊C311yescharacteristic
202C2×Q16—92no—
203C22⋊Q8—91no—
204#103—92no—
205C22⋊C8—91no—
206H(3,𝔽3)#45111yescharacteristic, complemented, Sylow
207C23×C3—31no—
208C23×C3—31no—
209C23×C3—61no—
210C23×C3C3×#6711yescharacteristic
211C3×Q8—24no—
212C3×Q8—22no—
213C3×Q8—34no—
214C3×Q8—34no—
215C3×Q8—62no—
216C3×Q8—62no—
217C3×Q8—64no—
218C3×Q8—64no—
219C3×Q8—68no—
220C3×Q8—64no—
221C3×Q8—64no—
222C3×Q8—182no—
223C3×Q8C3×#6714yes—
224C2×C3×C4—24no—
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—31no—
228C2×C3×C4—31no—
229C2×C3×C4—31no—
230C2×C3×C4—31no—
231C2×C3×C4—62no—
232C2×C3×C4—62no—
233C2×C3×C4—62no—
234C2×C3×C4—62no—
235C2×C3×C4—64no—
236C2×C3×C4—64no—
237C2×C3×C4—68no—
238C2×C3×C4—64no—
239C2×C3×C4—64no—
240C2×C3×C4—61no—
241C2×C3×C4—61no—
242C2×C3×C4—181no—
243C2×C3×C4—181no—
244C2×C3×C4C3×#6711yescharacteristic
245C2×C3×C4C22×C3×D311yescharacteristic
246C2×Dic3—61no—
247C2×Dic3—61no—
248C2×Dic3—181no—
249C2×Dic3—181no—
250Dic6—62no—
251Dic6—182no—
252C3×C8—182no—
253C3⋊C8—62no—
254C3⋊C8—182no—
255C2×C32—24no—
256C2×C32—22no—
257C2×C32—22no—
258C2×C32—21no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32#94211yescharacteristic
262C2×C32C2×#21611yescharacteristic
263C2×C32#95311yescharacteristic
264C2×C32C3×#12011yescharacteristic
265C2×C32C2×C3×Q1611yescharacteristic
266C2×C32C3×#13711yescharacteristic
267C2×Q8—24no—
268C2×Q8—21no—
269C2×Q8—91no—
270C2×Q8C2×(C3⋊D3)⋊C312yes—
271C22×C4—21no—
272C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
273Q16—184no—
274C2×C8—92no—
275C4⋊C4—91no—
276C4⋊C4—181no—
277C22⋊C4—181no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—31no—
281C22×C3—31no—
282C22×C3—32no—
283C22×C3—32no—
284C22×C3—61no—
285C22×C3—61no—
286C22×C3—61no—
287C22×C3—61no—
288C22×C3—62no—
289C22×C3—62no—
290C22×C3—61no—
291C22×C3—62no—
292C22×C3C3×#21612yes—
293C22×C3C2×C3×#6711yescharacteristic
294C3×C4—24no—
295C3×C4—21no—
296C3×C4—32no—
297C3×C4—32no—
298C3×C4—61no—
299C3×C4—61no—
300C3×C4—64no—
301C3×C4—64no—
302C3×C4—68no—
303C3×C4—62no—
304C3×C4—62no—
305C3×C4—361no—
306C3×C4—361no—
307C3×C4C2×C3×#6712yes—
308Dic3—121no—
309Dic3—121no—
310Dic3—361no—
311Dic3—361no—
312C32—21no—
313C32#522811yescharacteristic, complemented
314C32C3×#45111yescharacteristic, complemented
315C23#638⋊C311yescharacteristic
316Q8—24no—
317Q8—22no—
318Q8—182no—
319Q8#638⋊C314yes—
320C2×C4—24no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—181no—
324C2×C4—181no—
325C2×C4#638⋊C311yescharacteristic
326C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
327C8—182no—
328C2×C3—22no—
329C2×C3—31no—
330C2×C3—31no—
331C2×C3—31no—
332C2×C3—31no—
333C2×C3—31no—
334C2×C3—31no—
335C2×C3—62no—
336C2×C3—62no—
337C2×C3—64no—
338C2×C3—61no—
339C2×C3—61no—
340C2×C3—61no—
341C2×C3C3×#94211yescharacteristic
342C2×C3C2×C3×#21611yescharacteristic
343C2×C3C3×#95311yescharacteristic
344C22—21no—
345C22—21no—
346C22#3756⋊C312yescomplemented
347C22C2×#638⋊C311yescharacteristic
348C4—24no—
349C4—21no—
350C4—361no—
351C4—361no—
352C4C2×#638⋊C312yes—
353C3—31no—
354C3—31no—
355C3—61no—
356C3C3×#522811yescharacteristic
357C2—22no—
358C2C2×#3756⋊C311yescharacteristic
359C2#64151⋊C311yescharacteristic
360C2#1172170211yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC22×Q8×H(3,𝔽3)C2
socleC22×C3C2×C3×#67
prime radicalC23×H(3,𝔽3)D4
hypercenterC22×Q8(C3⋊D3)⋊C3
hypocenterC32C3×#451
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#3756⋊C3C2semidirect
4. #63662⋊C3C2semidirect
6. #10592927C3semidirect
22. C2×#3756C2×C3semidirect
26. #63662C2×C3semidirect
75. C22×H(3,𝔽3)Q16semidirect
184. C22×C32#216semidirect
185. C22×C32C3×Q16semidirect
206. H(3,𝔽3)#451semidirect
313. C32#5228semidirect
314. C32C3×#451semidirect
346. C22#3756⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×H(3,𝔽3)2
C2×#3756⋊C32
#64133⋊C32
#63662⋊C32
#63657⋊C32
#105929273
C3×#52283