Finite Group Databasebeta

← C2×#11721294 #13152056

Subgroups — C2×#11721294 #13152056

Order 1728.

1391 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1391 conjugacy classes of subgroups — too many to draw. Only its 187 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
2C22×#6196C211yesmaximal, characteristic, complemented
3C2×#3757⋊C3C212yesmaximal, complemented
4C2×#88451C212yesmaximal, complemented
5C2×#88447C211yesmaximal, characteristic, complemented
6#11721294C218yesmaximal, direct product factor
7C2×#63959—32nomaximal
8C24×H(3,𝔽3)C2211yescharacteristic
9C23×#272C2211yescharacteristic
10C2×#6235—24no—
11C2×#6235C2214yes—
12C22×#1084C2211yescharacteristic
13C22×(C3⋊Dic3)⋊C3C2212yes—
14C22×(C3⋊Dic3)⋊C3C2212yes—
15C2×#6196—24no—
16C2×#6196C2214yescomplemented
17#3757⋊C3C2218yescomplemented
18#88451C2218yescomplemented
19#88447C2214yescomplemented
20C22×C3×#67—32no—
21C22×C3×#67—61no—
22C22×#627—32no—
23C2×#3757D312yescomplemented
24C2×C3×#217—32no—
25C2×(C2×D3)⋊Dic3—32no—
26C2×(C2×D3)⋊Dic3—32no—
27#63959—316no—
28C2×Dic32—32no—
29C23×H(3,𝔽3)—22no—
30C23×H(3,𝔽3)—24no—
31C23×H(3,𝔽3)C2312yes—
32C23×H(3,𝔽3)C2311yescharacteristic
33C22×#272—22no—
34C22×#272—24no—
35C22×#272C2312yes—
36#6235—28no—
37C2×#1084D414yes—
38C2×#1084C2312yes—
39C2×(C3⋊Dic3)⋊C3—24no—
40C2×(C3⋊Dic3)⋊C3—24no—
41C2×(C3⋊Dic3)⋊C3C2314yes—
42C2×(C3⋊Dic3)⋊C3C2314yes—
43#6196—28no—
44C2×#936—91no—
45C2×#885—92no—
46C24×C32—21no—
47C24×C32C2×D312yes—
48C23×C3×D3—32no—
49C23×C3×D3—61no—
50C22×C3⋊Dic3C2×D312yes—
51C22×C3⋊Dic3C2×D312yes—
52C2×C3×#67—38no—
53C2×C3×#67—68no—
54C2×C3×#67—64no—
55C2×C3×#67—68no—
56C22×C3×Dic3—32no—
57C22×C3×Dic3—32no—
58C22×C3×Dic3—32no—
59C22×C3×Dic3—61no—
60C2×#627—38no—
61C2×#627—68no—
62#3757C2×D318yescomplemented
63C3×#217—38no—
64(C2×D3)⋊Dic3—38no—
65(C2×D3)⋊Dic3—38no—
66Dic32—38no—
67C22×H(3,𝔽3)—22no—
68C22×H(3,𝔽3)—24no—
69C22×H(3,𝔽3)—44no—
70C22×H(3,𝔽3)C2×D412yes—
71C22×H(3,𝔽3)#4114yes—
72C22×H(3,𝔽3)C2411yescharacteristic
73C2×#272—22no—
74C2×#272—24no—
75C2×#272—44no—
76#1084C2×D414yes—
77(C3⋊Dic3)⋊C3—24no—
78(C3⋊Dic3)⋊C3—24no—
79C22×C3×D4—91no—
80C22×#67—32no—
81C22×#67—61no—
82C22×Dic6—91no—
83C22×Dic6—92no—
84C2×C3×C22⋊C4—92no—
85C2×#217—32no—
86C2×#217—92no—
87C2×#217—92no—
88#936—98no—
89C2×(C2×D3)⋊C4—92no—
90C2×(C2×D3)⋊C4—92no—
91C2×C4×Dic3—91no—
92C2×C4×Dic3—92no—
93#885—916no—
94C23×C32—24no—
95C23×C32—24no—
96C23×C32—24no—
97C23×C32—24no—
98C23×C32—21no—
99C23×C32—22no—
100C23×C32—44no—
101C23×C32C22×D314yes—
102C23×C32C22×D312yes—
103C22×C3×D3—34no—
104C22×C3×D3—62no—
105C22×C3×D3—64no—
106C22×C3×D3—64no—
107C22×C3×D3—64no—
108C22×C3×D3—64no—
109C22×C3×D3—124no—
110C2×C3⋊Dic3—24no—
111C2×C3⋊Dic3—24no—
112C2×C3⋊Dic3C22×D314yes—
113C2×C3⋊Dic3C22×D314yes—
114C3×#67—616no—
115C3×#67—616no—
116C2×C3×Dic3—38no—
117C2×C3×Dic3—34no—
118C2×C3×Dic3—34no—
119C2×C3×Dic3—34no—
120C2×C3×Dic3—64no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—64no—
123C2×C3×Dic3—64no—
124#627—616no—
125C2×#124—271noSylow
126C2×H(3,𝔽3)—42no—
127C2×H(3,𝔽3)#12414yescomplemented
128C2×H(3,𝔽3)C22×D411yescharacteristic
129C2×H(3,𝔽3)C2×#4112yes—
130#272—42no—
131C24×C3—32no—
132C24×C3—61no—
133C24×C3—91no—
134C24×C3D3211yescharacteristic
135C23×D3—32no—
136C23×D3—61no—
137C2×C3×D4—94no—
138C2×C3×D4—184no—
139C22×C3×C4—91no—
140C22×C3×C4—92no—
141C22×C3×C4—92no—
142C2×#67—38no—
143C2×#67—64no—
144C2×#67—68no—
145C2×#67—68no—
146C22×Dic3—32no—
147C22×Dic3—32no—
148C22×Dic3—32no—
149C22×Dic3—61no—
150C22×Dic3—92no—
151C22×Dic3—92no—
152C2×Dic6—94no—
153C2×Dic6—98no—
154C2×Dic6—188no—
155C2×Dic6—184no—
156C3×C22⋊C4—98no—
157#217—38no—
158#217—98no—
159#217—98no—
160(C2×D3)⋊C4—98no—
161(C2×D3)⋊C4—98no—
162C4×Dic3—94no—
163C4×Dic3—98no—
164C22×C32—24no—
165C22×C32—24no—
166C22×C32—24no—
167C22×C32—24no—
168C22×C32—21no—
169C22×C32—24no—
170C22×C32—22no—
171C22×C32—44no—
172C22×C32—48no—
173C22×C32—48no—
174C22×C32#23514yes—
175C22×C32D3×D414yes—
176C22×C32#23714yes—
177C22×C32C23×D312yes—
178C2×C3×D3—64no—
179C2×C3×D3—64no—
180C2×C3×D3—64no—
181C2×C3×D3—64no—
182C2×C3×D3—124no—
183C2×C3×D3—128no—
184C2×C3×D3—128no—
185C3⋊Dic3—24no—
186C3⋊Dic3—24no—
187C3×Dic3—38no—
188C3×Dic3—64no—
189C3×Dic3—64no—
190C3×Dic3—64no—
191C22×Q8—271no—
192C22×D4—91no—
193#124—278no—
194C2×C22⋊C4—92no—
195C2×C22⋊C4—272no—
196C2×C42—271no—
197H(3,𝔽3)C2×#12411yescharacteristic, complemented, Sylow
198C23×C3—24no—
199C23×C3—22no—
200C23×C3—34no—
201C23×C3—32no—
202C23×C3—64no—
203C23×C3—64no—
204C23×C3—64no—
205C23×C3—64no—
206C23×C3—62no—
207C23×C3—61no—
208C23×C3—92no—
209C23×C3—124no—
210C23×C3—184no—
211C23×C3—182no—
212C23×C3C2×D3212yes—
213C23×C3C2×D3211yescharacteristic
214C22×D3—34no—
215C22×D3—62no—
216C22×D3—64no—
217C22×D3—64no—
218C22×D3—64no—
219C22×D3—64no—
220C22×D3—124no—
221C3×D4—188no—
222C2×C3×C4—92no—
223C2×C3×C4—94no—
224C2×C3×C4—94no—
225C2×C3×C4—94no—
226C2×C3×C4—184no—
227C2×C3×C4—184no—
228#67—616no—
229#67—616no—
230C2×Dic3—34no—
231C2×Dic3—38no—
232C2×Dic3—34no—
233C2×Dic3—34no—
234C2×Dic3—62no—
235C2×Dic3—64no—
236C2×Dic3—64no—
237C2×Dic3—64no—
238C2×Dic3—94no—
239C2×Dic3—94no—
240C2×Dic3—184no—
241C2×Dic3—184no—
242Dic6—1816no—
243Dic6—188no—
244C2×C32—22no—
245C2×C32—21no—
246C2×C32—24no—
247C2×C32—44no—
248C2×C32—44no—
249C2×C32#88518yescomplemented
250C2×C32C2×#23512yes—
251C2×C32C2×D3×D412yes—
252C2×C32C2×#23712yes—
253C3×D3—124no—
254C3×D3—124no—
255C24—91no—
256C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
257C2×Q8—274no—
258C2×Q8—544no—
259C2×D4—94no—
260C2×D4—184no—
261C22×C4—91no—
262C22×C4—92no—
263C22×C4—92no—
264C22⋊C4—98no—
265C22⋊C4—278no—
266C42—274no—
267C22×C3—24no—
268C22×C3—22no—
269C22×C3—32no—
270C22×C3—34no—
271C22×C3—34no—
272C22×C3—34no—
273C22×C3—44no—
274C22×C3—64no—
275C22×C3—64no—
276C22×C3—64no—
277C22×C3—64no—
278C22×C3—61no—
279C22×C3—64no—
280C22×C3—62no—
281C22×C3—124no—
282C22×C3—128no—
283C22×C3—128no—
284C22×C3—184no—
285C22×C3—182no—
286C22×C3—364no—
287C22×C3#380414yes—
288C22×C3#381112yes—
289C22×C3C22×D3211yescharacteristic
290C2×D3—64no—
291C2×D3—64no—
292C2×D3—64no—
293C2×D3—64no—
294C2×D3—124no—
295C2×D3—128no—
296C2×D3—128no—
297C3×C4—94no—
298C3×C4—184no—
299C3×C4—184no—
300Dic3—38no—
301Dic3—64no—
302Dic3—64no—
303Dic3—64no—
304Dic3—184no—
305Dic3—184no—
306C32—21no—
307C32C2×#88512yescomplemented
308C23—24no—
309C23—22no—
310C23—92no—
311C23—184no—
312C23—182no—
313C23C2×((C3⋊D3)⋊C3)⋊C212yes—
314C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
315Q8—548no—
316D4—188no—
317C2×C4—92no—
318C2×C4—94no—
319C2×C4—94no—
320C2×C4—94no—
321C2×C4—184no—
322C2×C4—184no—
323C2×C3—32no—
324C2×C3—32no—
325C2×C3—32no—
326C2×C3—38no—
327C2×C3—42no—
328C2×C3—62no—
329C2×C3—61no—
330C2×C3—64no—
331C2×C3—124no—
332C2×C3—124no—
333C2×C3—362no—
334C2×C3#6395314yes—
335C2×C3C2×#380412yes—
336C2×C3C2×#381111yescharacteristic
337D3—124no—
338D3—124no—
339C22—24no—
340C22—22no—
341C22—44no—
342C22—184no—
343C22—182no—
344C22—364no—
345C22#8867114yes—
346C22#8867712yes—
347C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
348C4—94no—
349C4—184no—
350C4—184no—
351C3—32no—
352C3—61no—
353C3C2×#6395311yescharacteristic
354C2—42no—
355C2—362no—
356C2#1172129414yesdirect product factor
357C2C2×#8867112yes—
358C2C2×#8867711yescharacteristic
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×#124
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. C22×#6196C2semidirect
3. C2×#3757⋊C3C2semidirect
4. C2×#88451C2semidirect
5. C2×#88447C2semidirect
6. #11721294C2direct (N × H)
16. C2×#6196C22semidirect
17. #3757⋊C3C22semidirect
18. #88451C22semidirect
19. #88447C22semidirect
23. C2×#3757D3semidirect
62. #3757C2×D3semidirect
127. C2×H(3,𝔽3)#124semidirect
197. H(3,𝔽3)C2×#124semidirect
249. C2×C32#885semidirect
307. C32C2×#885semidirect
356. C2#11721294direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62352
C22×#61962
C2×#3757⋊C32
C2×#884512
C2×#884472
#117212942
C2×#639593