Finite Group Databasebeta

← C2×#11721307 #13152051

Subgroups — C2×#11721307 #13152051

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
1C23×#1084C211yesmaximal, characteristic, complemented
2C22×#638⋊C3C211yesmaximal, characteristic
3C2×#3757⋊C3C211yesmaximal, characteristic, complemented
4#11721307C218yesmaximal, direct product factor
5C2×#88450C212yesmaximal, complemented
6C2×#88448C212yesmaximal, complemented
7C2×#63966—31nomaximal
8C2×#63961—31nomaximal
9C24×H(3,𝔽3)C2211yescharacteristic
10C23×(C3⋊D3)⋊C3C2211yescharacteristic
11C22×#1084—24no—
12C22×#1084C2212yes—
13C22×#1084C2214yescomplemented
14C2×#638⋊C3—24no—
15C2×#638⋊C3C2214yes—
16C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C22×(C3⋊Dic3)⋊C3C2212yes—
18#3757⋊C3C2214yescomplemented
19#88450C2218yescomplemented
20#88448C2218yescomplemented
21C22×#638D311yescharacteristic
22C22×C3×#67—31no—
23C23×C3×Dic3—31no—
24C23×C3×Dic3—31no—
25C23×C3×Dic3—61no—
26C2×#3757D311yescharacteristic, complemented
27C2×C3×#217—31no—
28C2×#3724—32no—
29#63966—38no—
30C2×Dic3⋊Dic3—32no—
31C2×#3722—32no—
32#63961—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×#1084—28no—
43C2×#1084C2314yes—
44#638⋊C3—28no—
45C2×(C3⋊Dic3)⋊C3—22no—
46C2×(C3⋊Dic3)⋊C3—24no—
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⋊Dic3C2×D311yescharacteristic
60C22×C3⋊Dic3C2×D312yes—
61C2×C3×#67—34no—
62C2×C3×#67—64no—
63C22×C3×Dic3—32no—
64C22×C3×Dic3—32no—
65C22×C3×Dic3—32no—
66C22×C3×Dic3—34no—
67C22×C3×Dic3—34no—
68C22×C3×Dic3—31no—
69C22×C3×Dic3—62no—
70C22×C3×Dic3—64no—
71C22×C3×Dic3—68no—
72C22×C3×Dic3—64no—
73C22×C3×Dic3—64no—
74#3757C2×D314yescomplemented
75C3×#217—34no—
76#3724—38no—
77Dic3⋊Dic3—38no—
78#3722—38no—
79(C2×D3)⋊Dic3—38no—
80C22×H(3,𝔽3)—28no—
81C22×H(3,𝔽3)—24no—
82C22×H(3,𝔽3)C2×D414yes—
83C22×H(3,𝔽3)C2×D412yes—
84C22×H(3,𝔽3)#4114yes—
85C22×H(3,𝔽3)C2411yescharacteristic
86C2×(C3⋊D3)⋊C3—22no—
87C2×(C3⋊D3)⋊C3—24no—
88C2×(C3⋊D3)⋊C3—44no—
89#1084—24no—
90(C3⋊Dic3)⋊C3—22no—
91(C3⋊Dic3)⋊C3—24no—
92C22×C3×D4—91no—
93C23×C3×C4—91no—
94C22×#67—31no—
95C22×#67—91no—
96C23×Dic3—31no—
97C23×Dic3—31no—
98C23×Dic3—61no—
99C2×C3×C22⋊C4—91no—
100C2×#217—31no—
101C2×#217—91no—
102C2×#217—92no—
103#935—98no—
104#929—98no—
105C2×(C2×D3)⋊C4—92no—
106C2×(C2×D3)⋊C4—92no—
107C2×C4⋊Dic3—92no—
108C2×Dic3⋊C4—92no—
109C2×Dic3⋊C4—92no—
110#886—98no—
111C23×C32—28no—
112C23×C32—24no—
113C23×C32—24no—
114C23×C32—24no—
115C23×C32—22no—
116C23×C32—21no—
117C23×C32D1214yes—
118C23×C32#6714yes—
119C23×C32C22×D312yes—
120C23×C32C22×D312yes—
121C23×C32C22×D311yescharacteristic
122C23×C32C22×D311yescharacteristic
123C22×C3⋊D3—24no—
124C22×C3⋊D3—22no—
125C22×C3⋊D3C22×D312yes—
126C22×C3×D3—32no—
127C22×C3×D3—62no—
128C22×C3×D3—64no—
129#638—28no—
130C2×C3⋊Dic3—24no—
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—68no—
141C2×C3×Dic3—616no—
142C2×C3×Dic3—62no—
143C2×C3×Dic3—68no—
144C2×C3×Dic3—64no—
145C2×C3×Dic3—64no—
146C2×C3×Dic3—68no—
147C2×C3×Dic3—64no—
148C2×#123—271noSylow
149C2×H(3,𝔽3)—24no—
150C2×H(3,𝔽3)#12314yescomplemented
151C2×H(3,𝔽3)C22×D411yescharacteristic
152C2×H(3,𝔽3)C2×#4112yes—
153(C3⋊D3)⋊C3—42no—
154C24×C3—31no—
155C24×C3—31no—
156C24×C3—61no—
157C24×C3—91no—
158C24×C3D3211yescharacteristic
159C23×D3—31no—
160C23×D3—91no—
161C2×C3×D4—94no—
162C2×C3×D4—184no—
163C22×C3×C4—91no—
164C22×C3×C4—92no—
165C22×C3×C4—92no—
166C22×C3×C4—94no—
167C22×C3×C4—184no—
168C2×#67—34no—
169C2×#67—64no—
170C2×#67—94no—
171C2×#67—184no—
172C22×Dic3—34no—
173C22×Dic3—32no—
174C22×Dic3—32no—
175C22×Dic3—34no—
176C22×Dic3—31no—
177C22×Dic3—32no—
178C22×Dic3—64no—
179C22×Dic3—62no—
180C22×Dic3—64no—
181C22×Dic3—64no—
182C22×Dic3—68no—
183C22×Dic3—92no—
184C22×Dic3—91no—
185C3×C22⋊C4—94no—
186#217—34no—
187#217—94no—
188#217—98no—
189(C2×D3)⋊C4—98no—
190(C2×D3)⋊C4—98no—
191C4⋊Dic3—98no—
192Dic3⋊C4—98no—
193Dic3⋊C4—98no—
194C22×C32—28no—
195C22×C32—216no—
196C22×C32—28no—
197C22×C32—28no—
198C22×C32—24no—
199C22×C32—24no—
200C22×C32—21no—
201C22×C32—24no—
202C22×C32—24no—
203C22×C32—22no—
204C22×C32C2×D1214yes—
205C22×C32C2×D1212yes—
206C22×C32#23714yes—
207C22×C32#23714yes—
208C22×C32C2×#6714yes—
209C22×C32C2×#6712yes—
210C22×C32C23×D311yescharacteristic
211C22×C32C23×D311yescharacteristic
212C2×C3⋊D3—24no—
213C2×C3⋊D3—22no—
214C2×C3⋊D3—44no—
215C2×C3×D3—62no—
216C2×C3×D3—64no—
217C2×C3×D3—124no—
218C3⋊Dic3—24no—
219C3⋊Dic3—22no—
220C3×Dic3—68no—
221C3×Dic3—62no—
222C3×Dic3—64no—
223C3×Dic3—64no—
224C3×Dic3—64no—
225C22×D4—91no—
226C23×C4—91no—
227#123—278no—
228C2×C4⋊C4—272no—
229C2×C22⋊C4—91no—
230C2×C22⋊C4—272no—
231H(3,𝔽3)C2×#12311yescharacteristic, complemented, Sylow
232C23×C3—24no—
233C23×C3—34no—
234C23×C3—34no—
235C23×C3—32no—
236C23×C3—32no—
237C23×C3—31no—
238C23×C3—31no—
239C23×C3—64no—
240C23×C3—64no—
241C23×C3—68no—
242C23×C3—64no—
243C23×C3—62no—
244C23×C3—61no—
245C23×C3—92no—
246C23×C3—182no—
247C23×C3—184no—
248C23×C3#62614yes—
249C23×C3C2×D3212yes—
250C23×C3C2×D3211yescharacteristic
251C22×D3—32no—
252C22×D3—64no—
253C22×D3—62no—
254C22×D3—92no—
255C22×D3—182no—
256C22×D3—184no—
257C3×D4—188no—
258C2×C3×C4—92no—
259C2×C3×C4—94no—
260C2×C3×C4—94no—
261C2×C3×C4—184no—
262C2×C3×C4—182no—
263C2×C3×C4—188no—
264C2×C3×C4—184no—
265#67—68no—
266#67—188no—
267C2×Dic3—34no—
268C2×Dic3—34no—
269C2×Dic3—32no—
270C2×Dic3—34no—
271C2×Dic3—64no—
272C2×Dic3—64no—
273C2×Dic3—64no—
274C2×Dic3—68no—
275C2×Dic3—68no—
276C2×Dic3—616no—
277C2×Dic3—68no—
278C2×Dic3—64no—
279C2×Dic3—62no—
280C2×Dic3—94no—
281C2×Dic3—92no—
282C2×Dic3—184no—
283C2×Dic3—182no—
284C2×C32—28no—
285C2×C32—24no—
286C2×C32—24no—
287C2×C32—22no—
288C2×C32—21no—
289C2×C32—24no—
290C2×C32#88614yescomplemented
291C2×C32#93514yescomplemented
292C2×C32C22×D1211yescharacteristic
293C2×C32C2×#23712yes—
294C2×C32C2×#23712yes—
295C2×C32C22×#6711yescharacteristic
296C3⋊D3—42no—
297C3×D3—122no—
298C24—91no—
299C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
300C2×D4—94no—
301C2×D4—184no—
302C22×C4—94no—
303C22×C4—92no—
304C22×C4—92no—
305C22×C4—91no—
306C22×C4—184no—
307C4⋊C4—278no—
308C22⋊C4—94no—
309C22⋊C4—278no—
310C22×C3—24no—
311C22×C3—28no—
312C22×C3—31no—
313C22×C3—31no—
314C22×C3—34no—
315C22×C3—34no—
316C22×C3—32no—
317C22×C3—32no—
318C22×C3—34no—
319C22×C3—34no—
320C22×C3—64no—
321C22×C3—64no—
322C22×C3—68no—
323C22×C3—68no—
324C22×C3—68no—
325C22×C3—616no—
326C22×C3—61no—
327C22×C3—64no—
328C22×C3—64no—
329C22×C3—62no—
330C22×C3—182no—
331C22×C3—184no—
332C22×C3—364no—
333C22×C3#380514yes—
334C22×C3C2×#62614yes—
335C22×C3C2×#62612yes—
336C22×C3C22×D3211yescharacteristic
337C2×D3—64no—
338C2×D3—62no—
339C2×D3—124no—
340C2×D3—182no—
341C2×D3—184no—
342C2×D3—364no—
343C3×C4—182no—
344C3×C4—184no—
345C3×C4—184no—
346Dic3—64no—
347Dic3—64no—
348Dic3—68no—
349Dic3—64no—
350Dic3—62no—
351Dic3—182no—
352Dic3—184no—
353C32—21no—
354C32C2×#88611yescharacteristic, complemented
355C32C2×#93511yescharacteristic, complemented
356C23—24no—
357C23—92no—
358C23—184no—
359C23—182no—
360C23#620014yes—
361C23C2×((C3⋊D3)⋊C3)⋊C212yes—
362C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
363D4—188no—
364C2×C4—94no—
365C2×C4—94no—
366C2×C4—92no—
367C2×C4—184no—
368C2×C4—188no—
369C2×C4—184no—
370C2×C4—182no—
371C2×C3—24no—
372C2×C3—32no—
373C2×C3—32no—
374C2×C3—31no—
375C2×C3—31no—
376C2×C3—34no—
377C2×C3—34no—
378C2×C3—64no—
379C2×C3—64no—
380C2×C3—68no—
381C2×C3—62no—
382C2×C3—61no—
383C2×C3—64no—
384C2×C3—362no—
385C2×C3#6396214yes—
386C2×C3C2×#380512yes—
387C2×C3C22×#62611yescharacteristic
388D3—122no—
389D3—362no—
390C22—24no—
391C22—28no—
392C22—184no—
393C22—182no—
394C22—364no—
395C22#8867214yes—
396C22C2×#620014yes—
397C22C2×#620012yes—
398C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
399C4—184no—
400C4—184no—
401C4—182no—
402C3—31no—
403C3—31no—
404C3—61no—
405C3C2×#6396211yescharacteristic
406C2—24no—
407C2—362no—
408C2#1172130714yesdirect product factor
409C2C2×#8867212yes—
410C2C22×#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×#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
1. C23×#1084C2semidirect
3. C2×#3757⋊C3C2semidirect
4. #11721307C2direct (N × H)
5. C2×#88450C2semidirect
6. C2×#88448C2semidirect
13. C22×#1084C22semidirect
18. #3757⋊C3C22semidirect
19. #88450C22semidirect
20. #88448C22semidirect
26. C2×#3757D3semidirect
74. #3757C2×D3semidirect
150. C2×H(3,𝔽3)#123semidirect
231. H(3,𝔽3)C2×#123semidirect
290. C2×C32#886semidirect
291. C2×C32#935semidirect
354. C32C2×#886semidirect
355. C32C2×#935semidirect
408. C2#11721307direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#10842
C22×#638⋊C32
C2×#3757⋊C32
#117213072
C2×#884502
C2×#884482
C2×#639663
C2×#639613