Finite Group Databasebeta

← C2×#11721306 #13152053

Subgroups — C2×#11721306 #13152053

Order 1728.

1329 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 1329 conjugacy classes of subgroups — too many to draw. Only its 219 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×#638⋊C3C211yesmaximal, characteristic
2C23×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
3C2×#88552C211yesmaximal, characteristic, complemented
4#11721306C218yesmaximal, direct product factor
5C2×#88450C212yesmaximal, complemented
6C2×#88449C212yesmaximal, complemented
7C2×#63966—31nomaximal
8C2×#63962—31nomaximal
9C24×H(3,𝔽3)C2211yescharacteristic
10C23×(C3⋊D3)⋊C3C2211yescharacteristic
11C22×#1084C2212yes—
12C2×#638⋊C3—24no—
13C2×#638⋊C3C2214yes—
14C22×(C3⋊Dic3)⋊C3—24no—
15C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2212yes—
17C22×(C3⋊Dic3)⋊C3C2214yescomplemented
18#88552C2214yescomplemented
19#88450C2218yescomplemented
20#88449C2218yescomplemented
21C22×#638D311yescharacteristic
22C23×C3⋊Dic3D311yescharacteristic, complemented
23C22×C3×#67—31no—
24C23×C3×Dic3—31no—
25C2×C3×#217—31no—
26C2×C3×#217—31no—
27C2×C3×#217—61no—
28#63966—38no—
29C2×Dic3⋊Dic3—32no—
30C2×Dic3⋊Dic3—32no—
31C2×#3722—32no—
32#63962—38no—
33C2×(C2×D3)⋊Dic3—32no—
34C23×H(3,𝔽3)—24no—
35C23×H(3,𝔽3)D414yes—
36C23×H(3,𝔽3)C2312yes—
37C23×H(3,𝔽3)C2311yescharacteristic
38C22×(C3⋊D3)⋊C3—22no—
39C22×(C3⋊D3)⋊C3—24no—
40C22×(C3⋊D3)⋊C3C2312yes—
41C2×#1084—24no—
42C2×#1084C2314yes—
43#638⋊C3—28no—
44C2×(C3⋊Dic3)⋊C3—24no—
45C2×(C3⋊Dic3)⋊C3—28no—
46C2×(C3⋊Dic3)⋊C3—22no—
47C2×(C3⋊Dic3)⋊C3C2312yes—
48C2×(C3⋊Dic3)⋊C3C2314yes—
49C2×#935—91no—
50C2×#929—91no—
51C2×#886—91no—
52C24×C32—21no—
53C24×C32C2×D311yescharacteristic
54C24×C32C2×D311yescharacteristic
55C23×C3⋊D3C2×D311yescharacteristic
56C23×C3×D3—31no—
57C2×#638—24no—
58C2×#638C2×D314yes—
59C22×C3⋊Dic3—24no—
60C22×C3⋊Dic3C2×D312yes—
61C22×C3⋊Dic3C2×D311yescharacteristic
62C22×C3⋊Dic3C2×D314yescomplemented
63C2×C3×#67—34no—
64C2×C3×#67—64no—
65C22×C3×Dic3—32no—
66C22×C3×Dic3—32no—
67C22×C3×Dic3—32no—
68C22×C3×Dic3—34no—
69C22×C3×Dic3—31no—
70C22×C3×Dic3—64no—
71C22×C3×Dic3—62no—
72C3×#217—34no—
73C3×#217—34no—
74C3×#217—64no—
75Dic3⋊Dic3—38no—
76Dic3⋊Dic3—38no—
77#3722—38no—
78(C2×D3)⋊Dic3—38no—
79C22×H(3,𝔽3)—28no—
80C22×H(3,𝔽3)—24no—
81C22×H(3,𝔽3)C2×D414yes—
82C22×H(3,𝔽3)C2×D412yes—
83C22×H(3,𝔽3)#4114yes—
84C22×H(3,𝔽3)C2411yescharacteristic
85C2×(C3⋊D3)⋊C3—22no—
86C2×(C3⋊D3)⋊C3—24no—
87C2×(C3⋊D3)⋊C3—44no—
88#1084—24no—
89(C3⋊Dic3)⋊C3—24no—
90(C3⋊Dic3)⋊C3—22no—
91C22×C3×D4—91no—
92C23×C3×C4—91no—
93C22×#67—31no—
94C22×#67—91no—
95C23×Dic3—31no—
96C23×Dic3—91no—
97C2×C3×C22⋊C4—91no—
98C2×#217—31no—
99C2×#217—31no—
100C2×#217—61no—
101C2×#217—92no—
102#935—98no—
103#929—98no—
104C2×(C2×D3)⋊C4—92no—
105C2×(C2×D3)⋊C4—92no—
106C2×C4⋊Dic3—92no—
107C2×Dic3⋊C4—92no—
108C2×Dic3⋊C4—92no—
109#886—98no—
110C23×C32—24no—
111C23×C32—24no—
112C23×C32—24no—
113C23×C32—21no—
114C23×C32—22no—
115C23×C32—44no—
116C23×C32#6714yes—
117C23×C32#6714yes—
118C23×C32C22×D312yes—
119C23×C32C22×D312yes—
120C23×C32C22×D311yescharacteristic
121C23×C32C22×D311yescharacteristic
122C22×C3⋊D3—24no—
123C22×C3⋊D3—22no—
124C22×C3⋊D3C22×D312yes—
125C22×C3×D3—32no—
126C22×C3×D3—64no—
127C22×C3×D3—62no—
128#638—28no—
129C2×C3⋊Dic3—24no—
130C2×C3⋊Dic3—28no—
131C2×C3⋊Dic3—22no—
132C2×C3⋊Dic3C22×D314yes—
133C2×C3⋊Dic3C22×D312yes—
134C3×#67—68no—
135C2×C3×Dic3—34no—
136C2×C3×Dic3—32no—
137C2×C3×Dic3—34no—
138C2×C3×Dic3—34no—
139C2×C3×Dic3—64no—
140C2×C3×Dic3—64no—
141C2×C3×Dic3—64no—
142C2×C3×Dic3—68no—
143C2×C3×Dic3—64no—
144C2×C3×Dic3—62no—
145C2×C3×Dic3—124no—
146C2×#123—271noSylow
147C2×H(3,𝔽3)—24no—
148C2×H(3,𝔽3)#12314yescomplemented
149C2×H(3,𝔽3)C22×D411yescharacteristic
150C2×H(3,𝔽3)C2×#4112yes—
151(C3⋊D3)⋊C3—42no—
152C24×C3—31no—
153C24×C3—31no—
154C24×C3—61no—
155C24×C3—91no—
156C24×C3D3211yescharacteristic
157C23×D3—31no—
158C23×D3—91no—
159C2×C3×D4—94no—
160C2×C3×D4—184no—
161C22×C3×C4—91no—
162C22×C3×C4—92no—
163C22×C3×C4—94no—
164C22×C3×C4—92no—
165C22×C3×C4—184no—
166C2×#67—34no—
167C2×#67—64no—
168C2×#67—94no—
169C2×#67—184no—
170C22×Dic3—32no—
171C22×Dic3—32no—
172C22×Dic3—34no—
173C22×Dic3—31no—
174C22×Dic3—32no—
175C22×Dic3—62no—
176C22×Dic3—64no—
177C22×Dic3—91no—
178C22×Dic3—92no—
179C22×Dic3—94no—
180C22×Dic3—184no—
181C3×C22⋊C4—94no—
182#217—34no—
183#217—34no—
184#217—64no—
185#217—98no—
186(C2×D3)⋊C4—98no—
187(C2×D3)⋊C4—98no—
188C4⋊Dic3—98no—
189Dic3⋊C4—98no—
190Dic3⋊C4—98no—
191C22×C32—28no—
192C22×C32—28no—
193C22×C32—24no—
194C22×C32—24no—
195C22×C32—21no—
196C22×C32—24no—
197C22×C32—22no—
198C22×C32—24no—
199C22×C32—44no—
200C22×C32—48no—
201C22×C32#23514yes—
202C22×C32#23714yes—
203C22×C32C2×#6714yes—
204C22×C32C2×#6714yes—
205C22×C32C2×#6712yes—
206C22×C32C2×#6712yes—
207C22×C32C23×D311yescharacteristic
208C22×C32C23×D311yescharacteristic
209C2×C3⋊D3—24no—
210C2×C3⋊D3—22no—
211C2×C3⋊D3—44no—
212C2×C3×D3—64no—
213C2×C3×D3—62no—
214C2×C3×D3—124no—
215C3⋊Dic3—24no—
216C3⋊Dic3—22no—
217C3×Dic3—64no—
218C3×Dic3—64no—
219C3×Dic3—64no—
220C3×Dic3—62no—
221C3×Dic3—124no—
222C22×D4—91no—
223C23×C4—91no—
224#123—278no—
225C2×C4⋊C4—272no—
226C2×C22⋊C4—91no—
227C2×C22⋊C4—272no—
228H(3,𝔽3)C2×#12311yescharacteristic, complemented, Sylow
229C23×C3—24no—
230C23×C3—34no—
231C23×C3—34no—
232C23×C3—32no—
233C23×C3—32no—
234C23×C3—31no—
235C23×C3—31no—
236C23×C3—64no—
237C23×C3—64no—
238C23×C3—64no—
239C23×C3—62no—
240C23×C3—61no—
241C23×C3—92no—
242C23×C3—124no—
243C23×C3—182no—
244C23×C3—184no—
245C23×C3#62514yes—
246C23×C3C2×D3212yes—
247C23×C3C2×D3211yescharacteristic
248C22×D3—32no—
249C22×D3—64no—
250C22×D3—62no—
251C22×D3—92no—
252C22×D3—182no—
253C22×D3—184no—
254C3×D4—188no—
255C2×C3×C4—92no—
256C2×C3×C4—94no—
257C2×C3×C4—94no—
258C2×C3×C4—184no—
259C2×C3×C4—188no—
260C2×C3×C4—182no—
261C2×C3×C4—184no—
262#67—68no—
263#67—188no—
264C2×Dic3—34no—
265C2×Dic3—34no—
266C2×Dic3—32no—
267C2×Dic3—34no—
268C2×Dic3—64no—
269C2×Dic3—64no—
270C2×Dic3—64no—
271C2×Dic3—68no—
272C2×Dic3—64no—
273C2×Dic3—62no—
274C2×Dic3—92no—
275C2×Dic3—94no—
276C2×Dic3—124no—
277C2×Dic3—184no—
278C2×Dic3—188no—
279C2×Dic3—182no—
280C2×C32—24no—
281C2×C32—24no—
282C2×C32—22no—
283C2×C32—21no—
284C2×C32—24no—
285C2×C32—44no—
286C2×C32#92914yescomplemented
287C2×C32#93514yescomplemented
288C2×C32C2×#23512yes—
289C2×C32C2×#23712yes—
290C2×C32C22×#6711yescharacteristic
291C2×C32C22×#6711yescharacteristic
292C3⋊D3—42no—
293C3×D3—122no—
294C24—91no—
295C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
296C2×D4—94no—
297C2×D4—184no—
298C22×C4—94no—
299C22×C4—92no—
300C22×C4—92no—
301C22×C4—91no—
302C22×C4—184no—
303C4⋊C4—278no—
304C22⋊C4—94no—
305C22⋊C4—278no—
306C22×C3—24no—
307C22×C3—28no—
308C22×C3—31no—
309C22×C3—31no—
310C22×C3—34no—
311C22×C3—34no—
312C22×C3—32no—
313C22×C3—32no—
314C22×C3—34no—
315C22×C3—34no—
316C22×C3—64no—
317C22×C3—64no—
318C22×C3—68no—
319C22×C3—68no—
320C22×C3—61no—
321C22×C3—64no—
322C22×C3—62no—
323C22×C3—64no—
324C22×C3—124no—
325C22×C3—128no—
326C22×C3—182no—
327C22×C3—184no—
328C22×C3—364no—
329C22×C3#380414yes—
330C22×C3C2×#62514yes—
331C22×C3C2×#62512yes—
332C22×C3C22×D3211yescharacteristic
333C2×D3—64no—
334C2×D3—62no—
335C2×D3—124no—
336C2×D3—182no—
337C2×D3—184no—
338C2×D3—364no—
339C3×C4—184no—
340C3×C4—182no—
341C3×C4—184no—
342Dic3—64no—
343Dic3—64no—
344Dic3—64no—
345Dic3—62no—
346Dic3—124no—
347Dic3—184no—
348Dic3—182no—
349C32—21no—
350C32C2×#92911yescharacteristic, complemented
351C32C2×#93511yescharacteristic, complemented
352C23—24no—
353C23—92no—
354C23—184no—
355C23—182no—
356C23#619814yes—
357C23C2×((C3⋊D3)⋊C3)⋊C212yes—
358C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
359D4—188no—
360C2×C4—94no—
361C2×C4—94no—
362C2×C4—92no—
363C2×C4—184no—
364C2×C4—188no—
365C2×C4—184no—
366C2×C4—182no—
367C2×C3—24no—
368C2×C3—32no—
369C2×C3—32no—
370C2×C3—31no—
371C2×C3—31no—
372C2×C3—34no—
373C2×C3—34no—
374C2×C3—64no—
375C2×C3—64no—
376C2×C3—62no—
377C2×C3—61no—
378C2×C3—64no—
379C2×C3—124no—
380C2×C3—362no—
381C2×C3#6396114yes—
382C2×C3C2×#380412yes—
383C2×C3C22×#62511yescharacteristic
384D3—122no—
385D3—362no—
386C22—24no—
387C22—28no—
388C22—184no—
389C22—182no—
390C22—364no—
391C22#8867114yes—
392C22C2×#619814yes—
393C22C2×#619812yes—
394C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
395C4—184no—
396C4—184no—
397C4—182no—
398C3—31no—
399C3—31no—
400C3—61no—
401C3C2×#6396111yescharacteristic
402C2—24no—
403C2—362no—
404C2#1172130614yesdirect product factor
405C2C2×#8867112yes—
406C2C22×#619811yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
FrattiniC22×C3C22×D32
FittingC24×H(3,𝔽3)C22
socleC23×C3C2×D32
prime radicalC22×#638⋊C3C2
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#123
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C23×(C3⋊Dic3)⋊C3C2semidirect
3. C2×#88552C2semidirect
4. #11721306C2direct (N × H)
5. C2×#88450C2semidirect
6. C2×#88449C2semidirect
17. C22×(C3⋊Dic3)⋊C3C22semidirect
18. #88552C22semidirect
19. #88450C22semidirect
20. #88449C22semidirect
22. C23×C3⋊Dic3D3semidirect
62. C22×C3⋊Dic3C2×D3semidirect
148. C2×H(3,𝔽3)#123semidirect
228. H(3,𝔽3)C2×#123semidirect
286. C2×C32#929semidirect
287. C2×C32#935semidirect
350. C32C2×#929semidirect
351. C32C2×#935semidirect
404. C2#11721306direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#638⋊C32
C23×(C3⋊Dic3)⋊C32
C2×#885522
#117213062
C2×#884502
C2×#884492
C2×#639663
C2×#639623