Finite Group Databasebeta

← C2×#11721301 #13152048

Subgroups — C2×#11721301 #13152048

Order 1728.

1433 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 1433 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×#6235C211yesmaximal, characteristic
2C23×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
3C2×#3757⋊C3C211yesmaximal, characteristic, complemented
4C2×#88451C212yesmaximal, complemented
5#11721301C218yesmaximal, direct product factor
6C2×#88448C212yesmaximal, complemented
7C2×#63962—31nomaximal
8C2×#63961—31nomaximal
9C24×H(3,𝔽3)C2211yescharacteristic
10C23×#272C2211yescharacteristic
11C2×#6235—24no—
12C2×#6235C2214yes—
13C22×#1084C2211yescharacteristic
14C22×(C3⋊Dic3)⋊C3—24no—
15C22×(C3⋊Dic3)⋊C3C2212yes—
16C22×(C3⋊Dic3)⋊C3C2212yes—
17C22×(C3⋊Dic3)⋊C3C2214yescomplemented
18#3757⋊C3C2214yescomplemented
19#88451C2218yescomplemented
20#88448C2218yescomplemented
21C23×C3⋊Dic3D311yescharacteristic, complemented
22C22×C3×#67—31no—
23C22×C3×#67—31no—
24C22×C3×#67—61no—
25C23×C3×Dic3—31no—
26C2×#3757D311yescharacteristic, complemented
27C2×C3×#217—31no—
28C2×#3724—32no—
29C2×Dic3⋊Dic3—32no—
30#63962—38no—
31#63961—38no—
32C2×(C2×D3)⋊Dic3—32no—
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×#272—22no—
39C22×#272—24no—
40C22×#272C2312yes—
41#6235—28no—
42C2×#1084—22no—
43C2×#1084C2312yes—
44C2×(C3⋊Dic3)⋊C3—24no—
45C2×(C3⋊Dic3)⋊C3—28no—
46C2×(C3⋊Dic3)⋊C3—24no—
47C2×(C3⋊Dic3)⋊C3C2314yes—
48C2×(C3⋊Dic3)⋊C3C2314yes—
49C2×#935—91no—
50C2×#929—91no—
51C2×#886—91no—
52C24×C32—21no—
53C24×C32C2×D311yescharacteristic
54C24×C32C2×D311yescharacteristic
55C23×C3×D3—31no—
56C23×C3×D3—31no—
57C23×C3×D3—61no—
58C22×C3⋊Dic3—24no—
59C22×C3⋊Dic3C2×D312yes—
60C22×C3⋊Dic3C2×D312yes—
61C22×C3⋊Dic3C2×D314yescomplemented
62C2×C3×#67—34no—
63C2×C3×#67—34no—
64C2×C3×#67—64no—
65C2×C3×#67—68no—
66C2×C3×#67—64no—
67C2×C3×#67—64no—
68C22×C3×Dic3—32no—
69C22×C3×Dic3—32no—
70C22×C3×Dic3—34no—
71C22×C3×Dic3—31no—
72C22×C3×Dic3—31no—
73C22×C3×Dic3—61no—
74C22×C3×Dic3—64no—
75#3757C2×D314yescomplemented
76C3×#217—34no—
77#3724—38no—
78Dic3⋊Dic3—38no—
79(C2×D3)⋊Dic3—38no—
80(C2×D3)⋊Dic3—38no—
81C22×H(3,𝔽3)—28no—
82C22×H(3,𝔽3)—24no—
83C22×H(3,𝔽3)C2×D414yes—
84C22×H(3,𝔽3)C2×D412yes—
85C22×H(3,𝔽3)#4114yes—
86C22×H(3,𝔽3)C2411yescharacteristic
87C2×#272—22no—
88C2×#272—24no—
89C2×#272—44no—
90#1084—22no—
91(C3⋊Dic3)⋊C3—24no—
92(C3⋊Dic3)⋊C3—24no—
93C22×C3×D4—91no—
94C23×C3×C4—91no—
95C22×#67—31no—
96C22×#67—31no—
97C22×#67—61no—
98C23×Dic3—31no—
99C23×Dic3—91no—
100C2×C3×C22⋊C4—91no—
101C2×#217—31no—
102C2×#217—91no—
103C2×#217—92no—
104#935—98no—
105#929—98no—
106C2×(C2×D3)⋊C4—92no—
107C2×(C2×D3)⋊C4—92no—
108C2×C4⋊Dic3—92no—
109C2×Dic3⋊C4—92no—
110C2×Dic3⋊C4—92no—
111#886—98no—
112C23×C32—24no—
113C23×C32—24no—
114C23×C32—24no—
115C23×C32—21no—
116C23×C32—22no—
117C23×C32—44no—
118C23×C32D1214yes—
119C23×C32#6714yes—
120C23×C32C22×D312yes—
121C23×C32C22×D312yes—
122C23×C32C22×D311yescharacteristic
123C23×C32C22×D311yescharacteristic
124C22×C3×D3—32no—
125C22×C3×D3—32no—
126C22×C3×D3—62no—
127C22×C3×D3—64no—
128C22×C3×D3—62no—
129C22×C3×D3—64no—
130C22×C3×D3—64no—
131C22×C3×D3—62no—
132C22×C3×D3—124no—
133C2×C3⋊Dic3—24no—
134C2×C3⋊Dic3—24no—
135C2×C3⋊Dic3—28no—
136C2×C3⋊Dic3C22×D314yes—
137C2×C3⋊Dic3C22×D314yes—
138C3×#67—616no—
139C3×#67—68no—
140C3×#67—68no—
141C2×C3×Dic3—34no—
142C2×C3×Dic3—34no—
143C2×C3×Dic3—32no—
144C2×C3×Dic3—32no—
145C2×C3×Dic3—62no—
146C2×C3×Dic3—64no—
147C2×C3×Dic3—68no—
148C2×C3×Dic3—64no—
149C2×C3×Dic3—62no—
150C2×C3×Dic3—64no—
151C2×C3×Dic3—62no—
152C2×#123—271noSylow
153C2×H(3,𝔽3)—24no—
154C2×H(3,𝔽3)#12314yescomplemented
155C2×H(3,𝔽3)C22×D411yescharacteristic
156C2×H(3,𝔽3)C2×#4112yes—
157#272—42no—
158C24×C3—31no—
159C24×C3—31no—
160C24×C3—61no—
161C24×C3—91no—
162C24×C3D3211yescharacteristic
163C23×D3—31no—
164C23×D3—31no—
165C23×D3—61no—
166C2×C3×D4—94no—
167C2×C3×D4—184no—
168C22×C3×C4—92no—
169C22×C3×C4—92no—
170C22×C3×C4—94no—
171C22×C3×C4—91no—
172C22×C3×C4—184no—
173C2×#67—34no—
174C2×#67—34no—
175C2×#67—64no—
176C2×#67—64no—
177C2×#67—64no—
178C2×#67—68no—
179C22×Dic3—31no—
180C22×Dic3—31no—
181C22×Dic3—32no—
182C22×Dic3—32no—
183C22×Dic3—34no—
184C22×Dic3—61no—
185C22×Dic3—64no—
186C22×Dic3—94no—
187C22×Dic3—92no—
188C22×Dic3—92no—
189C22×Dic3—184no—
190C3×C22⋊C4—94no—
191#217—34no—
192#217—94no—
193#217—98no—
194(C2×D3)⋊C4—98no—
195(C2×D3)⋊C4—98no—
196C4⋊Dic3—98no—
197Dic3⋊C4—98no—
198Dic3⋊C4—98no—
199C22×C32—28no—
200C22×C32—28no—
201C22×C32—24no—
202C22×C32—24no—
203C22×C32—21no—
204C22×C32—24no—
205C22×C32—22no—
206C22×C32—24no—
207C22×C32—44no—
208C22×C32—48no—
209C22×C32C2×D1214yes—
210C22×C32C2×D1212yes—
211C22×C32#23514yes—
212C22×C32#23714yes—
213C22×C32C2×#6714yes—
214C22×C32C2×#6712yes—
215C22×C32C23×D311yescharacteristic
216C22×C32C23×D311yescharacteristic
217C2×C3×D3—64no—
218C2×C3×D3—62no—
219C2×C3×D3—64no—
220C2×C3×D3—64no—
221C2×C3×D3—62no—
222C2×C3×D3—124no—
223C2×C3×D3—128no—
224C2×C3×D3—124no—
225C2×C3×D3—124no—
226C3⋊Dic3—24no—
227C3⋊Dic3—24no—
228C3×Dic3—64no—
229C3×Dic3—64no—
230C3×Dic3—62no—
231C3×Dic3—64no—
232C3×Dic3—62no—
233C22×D4—91no—
234C23×C4—91no—
235#123—278no—
236C2×C4⋊C4—272no—
237C2×C22⋊C4—91no—
238C2×C22⋊C4—272no—
239H(3,𝔽3)C2×#12311yescharacteristic, complemented, Sylow
240C23×C3—24no—
241C23×C3—34no—
242C23×C3—34no—
243C23×C3—32no—
244C23×C3—32no—
245C23×C3—31no—
246C23×C3—31no—
247C23×C3—64no—
248C23×C3—64no—
249C23×C3—64no—
250C23×C3—62no—
251C23×C3—61no—
252C23×C3—92no—
253C23×C3—124no—
254C23×C3—184no—
255C23×C3—182no—
256C23×C3#62614yes—
257C23×C3C2×D3212yes—
258C23×C3C2×D3211yescharacteristic
259C22×D3—32no—
260C22×D3—32no—
261C22×D3—62no—
262C22×D3—62no—
263C22×D3—62no—
264C22×D3—64no—
265C22×D3—64no—
266C22×D3—64no—
267C22×D3—124no—
268C3×D4—188no—
269C2×C3×C4—94no—
270C2×C3×C4—94no—
271C2×C3×C4—92no—
272C2×C3×C4—184no—
273C2×C3×C4—184no—
274C2×C3×C4—188no—
275C2×C3×C4—182no—
276#67—68no—
277#67—68no—
278#67—616no—
279C2×Dic3—32no—
280C2×Dic3—32no—
281C2×Dic3—34no—
282C2×Dic3—34no—
283C2×Dic3—62no—
284C2×Dic3—62no—
285C2×Dic3—62no—
286C2×Dic3—64no—
287C2×Dic3—64no—
288C2×Dic3—68no—
289C2×Dic3—64no—
290C2×Dic3—94no—
291C2×Dic3—94no—
292C2×Dic3—184no—
293C2×Dic3—184no—
294C2×Dic3—188no—
295C2×C32—24no—
296C2×C32—24no—
297C2×C32—22no—
298C2×C32—21no—
299C2×C32—24no—
300C2×C32—44no—
301C2×C32#88614yescomplemented
302C2×C32#92914yescomplemented
303C2×C32C22×D1211yescharacteristic
304C2×C32C2×#23512yes—
305C2×C32C2×#23712yes—
306C2×C32C22×#6711yescharacteristic
307C3×D3—124no—
308C3×D3—122no—
309C3×D3—122no—
310C24—91no—
311C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
312C2×D4—94no—
313C2×D4—184no—
314C22×C4—94no—
315C22×C4—92no—
316C22×C4—92no—
317C22×C4—91no—
318C22×C4—184no—
319C4⋊C4—278no—
320C22⋊C4—94no—
321C22⋊C4—278no—
322C22×C3—24no—
323C22×C3—28no—
324C22×C3—31no—
325C22×C3—31no—
326C22×C3—34no—
327C22×C3—34no—
328C22×C3—32no—
329C22×C3—32no—
330C22×C3—34no—
331C22×C3—34no—
332C22×C3—64no—
333C22×C3—64no—
334C22×C3—68no—
335C22×C3—68no—
336C22×C3—61no—
337C22×C3—64no—
338C22×C3—62no—
339C22×C3—64no—
340C22×C3—124no—
341C22×C3—128no—
342C22×C3—184no—
343C22×C3—182no—
344C22×C3—364no—
345C22×C3#380414yes—
346C22×C3C2×#62614yes—
347C22×C3C2×#62612yes—
348C22×C3C22×D3211yescharacteristic
349C2×D3—62no—
350C2×D3—62no—
351C2×D3—64no—
352C2×D3—64no—
353C2×D3—64no—
354C2×D3—124no—
355C2×D3—124no—
356C2×D3—124no—
357C2×D3—128no—
358C3×C4—184no—
359C3×C4—184no—
360C3×C4—182no—
361Dic3—62no—
362Dic3—62no—
363Dic3—64no—
364Dic3—64no—
365Dic3—64no—
366Dic3—184no—
367Dic3—184no—
368C32—21no—
369C32C2×#88611yescharacteristic, complemented
370C32C2×#92911yescharacteristic, complemented
371C23—24no—
372C23—92no—
373C23—184no—
374C23—182no—
375C23#620014yes—
376C23C2×((C3⋊D3)⋊C3)⋊C212yes—
377C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
378D4—188no—
379C2×C4—94no—
380C2×C4—94no—
381C2×C4—92no—
382C2×C4—184no—
383C2×C4—188no—
384C2×C4—184no—
385C2×C4—182no—
386C2×C3—24no—
387C2×C3—32no—
388C2×C3—32no—
389C2×C3—31no—
390C2×C3—31no—
391C2×C3—34no—
392C2×C3—34no—
393C2×C3—64no—
394C2×C3—64no—
395C2×C3—62no—
396C2×C3—61no—
397C2×C3—64no—
398C2×C3—124no—
399C2×C3—362no—
400C2×C3#6396614yes—
401C2×C3C2×#380412yes—
402C2×C3C22×#62611yescharacteristic
403D3—122no—
404D3—122no—
405D3—124no—
406C22—24no—
407C22—28no—
408C22—184no—
409C22—182no—
410C22—364no—
411C22#8867114yes—
412C22C2×#620014yes—
413C22C2×#620012yes—
414C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
415C4—184no—
416C4—184no—
417C4—182no—
418C3—31no—
419C3—31no—
420C3—61no—
421C3C2×#6396611yescharacteristic
422C2—24no—
423C2—362no—
424C2#1172130114yesdirect product factor
425C2C2×#8867112yes—
426C2C22×#620011yescharacteristic
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×#6235C2
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×#3757⋊C3C2semidirect
4. C2×#88451C2semidirect
5. #11721301C2direct (N × H)
6. C2×#88448C2semidirect
17. C22×(C3⋊Dic3)⋊C3C22semidirect
18. #3757⋊C3C22semidirect
19. #88451C22semidirect
20. #88448C22semidirect
21. C23×C3⋊Dic3D3semidirect
26. C2×#3757D3semidirect
61. C22×C3⋊Dic3C2×D3semidirect
75. #3757C2×D3semidirect
154. C2×H(3,𝔽3)#123semidirect
239. H(3,𝔽3)C2×#123semidirect
301. C2×C32#886semidirect
302. C2×C32#929semidirect
369. C32C2×#886semidirect
370. C32C2×#929semidirect
424. C2#11721301direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62352
C23×(C3⋊Dic3)⋊C32
C2×#3757⋊C32
C2×#884512
#117213012
C2×#884482
C2×#639623
C2×#639613