Finite Group Databasebeta

← C2×#11722432 #13164622

Subgroups — C2×#11722432 #13164622

Order 1728.

1051 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 1051 conjugacy classes of subgroups — too many to draw. Only its 82 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, complemented
2#11722432C214yesmaximal, direct product factor
3C2×#88623C211yesmaximal, characteristic, complemented
4C2×#88622C211yesmaximal, characteristic, complemented
5C2×#64364C311yesmaximal, characteristic, complemented
6C2×#1007⋊C3—31nomaximal
7C2×#638⋊C3—41nomaximal
8C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
9C2×Q8⋊H(3,𝔽3)C2212yescomplemented
10#88623C2212yescomplemented
11#88622C2212yescomplemented
12C22×C32×Q8C2×C311yescharacteristic, complemented
13#64364—34no—
14#64364C2×C314yescomplemented
15C2×#3832—31no—
16C2×#3832C2×C311yescharacteristic, complemented
17C2×Q8×C3⋊D3—31no—
18C2×Q8×C3⋊D3C2×C311yescharacteristic, complemented
19C2×#3828—31no—
20C2×#3828—31no—
21C22×C3×SL(2,𝔽3)—21no—
22C22×C3×SL(2,𝔽3)D311yescharacteristic, complemented
23#1007⋊C3—34no—
24C2×D3×SL(2,𝔽3)—31no—
25C2×C3⋊(#41⋊C3)—31no—
26C23×H(3,𝔽3)—41no—
27C22×(C3⋊D3)⋊C3—41no—
28#638⋊C3—44no—
29C2×(C3⋊Dic3)⋊C3—41no—
30Q8⋊H(3,𝔽3)C2311yescharacteristic
31C2×#1007—31no—
32C2×#1007—91no—
33C2×2−1+4⋊C3—91no—
34C2×C32×Q8—31no—
35C2×C32×Q8—32no—
36C2×C32×Q8C22×C311yescharacteristic
37C2×C32×Q8C22×C312yescomplemented
38C22×C32×C4—31no—
39C2×#638—31no—
40C2×#638A411yescharacteristic
41#3832—34no—
42#3832—34no—
43#3832—32no—
44#3832C22×C312yescomplemented
45Q8×C3⋊D3—34no—
46Q8×C3⋊D3—32no—
47Q8×C3⋊D3—34no—
48Q8×C3⋊D3C22×C312yescomplemented
49#3828—34no—
50#3828—34no—
51#3828—34no—
52#3828—34no—
53C2×C4⋊(C3⋊D3)—31no—
54C2×C4⋊(C3⋊D3)—31no—
55C2×C4×C3⋊D3—32no—
56C2×C4×C3⋊D3—31no—
57C2×C4×C3⋊D3—31no—
58C2×#634—31no—
59C2×#634—31no—
60C2×C3×#67—121no—
61C2×C3×SL(2,𝔽3)—21no—
62C2×C3×SL(2,𝔽3)—22no—
63C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
64C2×C3×SL(2,𝔽3)C2×D312yescomplemented
65D3×SL(2,𝔽3)—32no—
66C3⋊(#41⋊C3)—32no—
67C22×H(3,𝔽3)—41no—
68C22×H(3,𝔽3)—42no—
69C22×H(3,𝔽3)—82no—
70C2×(C3⋊D3)⋊C3—42no—
71C2×(C3⋊D3)⋊C3—82no—
72(C3⋊Dic3)⋊C3—42no—
73C22×C3×Q8—31no—
74C22×C3×Q8C3×D311yescharacteristic
75#1007—34no—
76#1007—94no—
77#1007—912no—
78#1007—94no—
79C2×#239—31no—
80C2×#239—91no—
81C2×#239—93no—
82C2×#239—91no—
83C2×D3×Q8—31no—
84C2×D3×Q8—91no—
85C2×D3×Q8—93no—
86C2×D3×Q8—91no—
87C2×#235—91no—
88C2×#235—93no—
89C2×#235—93no—
90C2×#235—91no—
912−1+4⋊C3—94no—
92C2×#41⋊C3—91no—
93C22×SL(2,𝔽3)—31no—
94C22×SL(2,𝔽3)—61no—
95C22×SL(2,𝔽3)—91no—
96C23×C32—41no—
97C23×C32—81no—
98C23×C32C2×A411yescharacteristic
99C22×C3⋊D3—31no—
100C22×C3⋊D3C2×A411yescharacteristic
101C22×C3×D3—121no—
102C32×Q8—31no—
103C32×Q8—32no—
104C32×Q8—32no—
105C32×Q8C23×C311yescharacteristic
106C2×C32×C4—31no—
107C2×C32×C4—31no—
108C2×C32×C4—32no—
109C2×C32×C4—32no—
110#638—34no—
111#638C2×A414yes—
112C2×C3⋊Dic3—31no—
113C2×C3⋊Dic3C2×A411yescharacteristic
114C4⋊(C3⋊D3)—32no—
115C4⋊(C3⋊D3)—32no—
116C4⋊(C3⋊D3)—34no—
117C4×C3⋊D3—38no—
118C4×C3⋊D3—32no—
119C4×C3⋊D3—32no—
120C4×C3⋊D3—32no—
121C4×C3⋊D3—32no—
122#634—32no—
123#634—34no—
124#634—32no—
125C3×#67—124no—
126C2×C3×Dic3—121no—
127C3×SL(2,𝔽3)—21no—
128C3×SL(2,𝔽3)C22×D311yescharacteristic
129C2×2−1+4—91noSylow
130C2×H(3,𝔽3)—42no—
131C2×H(3,𝔽3)—41no—
132C2×H(3,𝔽3)—82no—
133(C3⋊D3)⋊C3—82no—
134C2×C3×Q8—33no—
135C2×C3×Q8—36no—
136C2×C3×Q8—31no—
137C2×C3×Q8—32no—
138C2×C3×Q8—31no—
139C2×C3×Q8—32no—
140C2×C3×Q8C2×C3×D311yescharacteristic
141C2×C3×Q8C2×C3×D312yes—
142C2×C3×D4—361no—
143C22×C3×C4—33no—
144C22×C3×C4—31no—
145C2×#67—31no—
146C2×#67—91no—
147C2×#67—93no—
148C2×#67—91no—
149#239—32no—
150#239—92no—
151#239—94no—
152#239—912no—
153#239—912no—
154#239—96no—
155#239—94no—
156#239—92no—
157D3×Q8—32no—
158D3×Q8—92no—
159D3×Q8—912no—
160D3×Q8—94no—
161D3×Q8—912no—
162D3×Q8—96no—
163D3×Q8—92no—
164D3×Q8—94no—
165#235—94no—
166#235—94no—
167#235—912no—
168#235—912no—
169#235—912no—
170#235—94no—
171#235—912no—
172#235—94no—
173C2×D12—91no—
174C2×D12—91no—
175C2×D12—93no—
176C2×D12—93no—
177C2×C4×D3—91no—
178C2×C4×D3—91no—
179C2×C4×D3—93no—
180C2×C4×D3—93no—
181C2×C4×D3—96no—
182C2×C4×D3—92no—
183C2×Dic6—91no—
184C2×Dic6—93no—
185C2×Dic6—93no—
186C2×Dic6—91no—
187#41⋊C3—92no—
188C2×SL(2,𝔽3)—31no—
189C2×SL(2,𝔽3)—32no—
190C2×SL(2,𝔽3)—61no—
191C2×SL(2,𝔽3)—62no—
192C2×SL(2,𝔽3)—92no—
193C22×C32—22no—
194C22×C32—41no—
195C22×C32—42no—
196C22×C32—82no—
197C22×C32—84no—
198C22×C32—82no—
199C22×C32—81no—
200C22×C32C22×A412yes—
201C22×C32C22×A411yescharacteristic
202C2×C3⋊D3—22no—
203C2×C3⋊D3—32no—
204C2×C3⋊D3—62no—
205C2×C3⋊D3C22×A412yes—
206C2×C3×D3—122no—
207C2×C3×D3—242no—
208C32×C4—31no—
209C32×C4—32no—
210C32×C4—31no—
211C3⋊Dic3—32no—
212C3⋊Dic3C22×A412yes—
213C3×Dic3—122no—
2142−1+4—94no—
2152−1+4—274no—
216C2×#41—91no—
217C2×#41—271no—
218C2×#41—271no—
219C2×#41—271no—
220C22×Q8—91no—
221C22×Q8—271no—
222C22×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
223H(3,𝔽3)—41noSylow
224C23×C3—31no—
225C23×C3—121no—
226C23×C3—241no—
227C23×C3—361no—
228C23×C3D3×A411yescharacteristic
229C22×D3—31no—
230C22×D3—91no—
231C22×D3—93no—
232C22×D3—91no—
233C3×Q8—31no—
234C3×Q8—32no—
235C3×Q8—33no—
236C3×Q8—36no—
237C3×Q8—36no—
238C3×Q8—32no—
239C3×Q8—31no—
240C3×Q8C22×C3×D311yescharacteristic
241C3×D4—364no—
242C2×C3×C4—33no—
243C2×C3×C4—33no—
244C2×C3×C4—36no—
245C2×C3×C4—36no—
246C2×C3×C4—31no—
247C2×C3×C4—31no—
248C2×C3×C4—32no—
249C2×C3×C4—32no—
250C2×C3×C4—361no—
251#67—34no—
252#67—94no—
253#67—912no—
254#67—94no—
255C2×Dic3—31no—
256C2×Dic3—91no—
257C2×Dic3—93no—
258C2×Dic3—91no—
259D12—94no—
260D12—92no—
261D12—912no—
262D12—96no—
263D12—96no—
264D12—92no—
265C4×D3—92no—
266C4×D3—92no—
267C4×D3—92no—
268C4×D3—92no—
269C4×D3—96no—
270C4×D3—96no—
271C4×D3—96no—
272C4×D3—96no—
273C4×D3—924no—
274C4×D3—98no—
275Dic6—94no—
276Dic6—912no—
277Dic6—96no—
278Dic6—96no—
279Dic6—92no—
280Dic6—92no—
281SL(2,𝔽3)—31no—
282SL(2,𝔽3)—61no—
283C2×C32—22no—
284C2×C32—42no—
285C2×C32—41no—
286C2×C32—82no—
287C2×C32—84no—
288C2×C32—82no—
289C2×C32—81no—
290C2×C322−1+4⋊C312yescomplemented
291C2×C32C23×A411yescharacteristic
292C3⋊D3—22no—
293C3⋊D3—62no—
294C3×D3—242no—
295#41—92no—
296#41—274no—
297#41—274no—
298#41—274no—
299#41—274no—
300#41—274no—
301#41—274no—
302#41—272no—
303C2×Q8—31no—
304C2×Q8—32no—
305C2×Q8—92no—
306C2×Q8—271no—
307C2×Q8—271no—
308C2×Q8—274no—
309C2×Q8—274no—
310C2×Q8—272no—
311C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
312C2×Q8C2×(C3⋊D3)⋊C312yescomplemented
313C2×D4—91no—
314C2×D4—271no—
315C2×D4—271no—
316C2×D4—271no—
317C22×C4—31no—
318C22×C4—271no—
319C22×C4—271no—
320C22×C4—272no—
321C22×C3—22no—
322C22×C3—32no—
323C22×C3—31no—
324C22×C3—62no—
325C22×C3—122no—
326C22×C3—121no—
327C22×C3—242no—
328C22×C3—244no—
329C22×C3—242no—
330C22×C3—241no—
331C22×C3—362no—
332C22×C3—722no—
333C22×C3C2×D3×A412yes—
334C22×C3C2×D3×A411yescharacteristic
335C2×D3—32no—
336C2×D3—62no—
337C2×D3—92no—
338C2×D3—96no—
339C2×D3—92no—
340C2×D3—182no—
341C2×D3—186no—
342C2×D3—182no—
343C3×C4—33no—
344C3×C4—36no—
345C3×C4—33no—
346C3×C4—31no—
347C3×C4—32no—
348C3×C4—31no—
349C3×C4—362no—
350Dic3—32no—
351Dic3—92no—
352Dic3—96no—
353Dic3—92no—
354C32—41no—
355C32—81no—
356C32C2×2−1+4⋊C311yescharacteristic, complemented
357C23—91no—
358C23—271no—
359C23(C3⋊D3)⋊A411yescharacteristic
360Q8—31no—
361Q8—32no—
362Q8—32no—
363Q8—274no—
364Q8—272no—
365Q8—272no—
366Q8C22×(C3⋊D3)⋊C311yescharacteristic
367D4—94no—
368D4—274no—
369D4—272no—
370D4—272no—
371D4—274no—
372C2×C4—32no—
373C2×C4—32no—
374C2×C4—31no—
375C2×C4—31no—
376C2×C4—91no—
377C2×C4—272no—
378C2×C4—272no—
379C2×C4—272no—
380C2×C4—272no—
381C2×C4—278no—
382C2×C4—271no—
383C2×C3—22no—
384C2×C3—31no—
385C2×C3—32no—
386C2×C3—62no—
387C2×C3—121no—
388C2×C3—122no—
389C2×C3—242no—
390C2×C3—244no—
391C2×C3—241no—
392C2×C3—242no—
393C2×C3—722no—
394C2×C3#1007⋊C312yes—
395C2×C3C22×D3×A411yescharacteristic
396D3—62no—
397D3—182no—
398D3—186no—
399D3—182no—
400C22—22no—
401C22—92no—
402C22—182no—
403C22—272no—
404C22—542no—
405C22C2×(C3⋊D3)⋊A412yes—
406C22C2×(C3⋊D3)⋊A411yescharacteristic
407C4—31no—
408C4—32no—
409C4—31no—
410C4—92no—
411C4—272no—
412C3—31no—
413C3—121no—
414C3—241no—
415C3C2×#1007⋊C311yescharacteristic
416C2—22no—
417C2—182no—
418C2—542no—
419C2#1172243212yesdirect product factor
420C2C22×(C3⋊D3)⋊A411yescharacteristic
subgroupG/H
centerC22C2×(C3⋊D3)⋊A4
commutator, cosocle, hypocenterC32×Q8C23×C3
FrattiniC2×C3C22×D3×A4
FittingC22×C32×Q8C2×C3
socleC22×C3C2×D3×A4
hypercenterC23(C3⋊D3)⋊A4
LCSG ▷ C32×Q8
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×Q8⋊H(3,𝔽3)C2semidirect
2. #11722432C2direct (N × H)
3. C2×#88623C2semidirect
4. C2×#88622C2semidirect
5. C2×#64364C3semidirect
9. C2×Q8⋊H(3,𝔽3)C22semidirect
10. #88623C22semidirect
11. #88622C22semidirect
12. C22×C32×Q8C2×C3semidirect
14. #64364C2×C3semidirect
16. C2×#3832C2×C3semidirect
18. C2×Q8×C3⋊D3C2×C3semidirect
22. C22×C3×SL(2,𝔽3)D3semidirect
37. C2×C32×Q8C22×C3semidirect
44. #3832C22×C3semidirect
48. Q8×C3⋊D3C22×C3semidirect
64. C2×C3×SL(2,𝔽3)C2×D3semidirect
222. C22×Q8(C3⋊D3)⋊C3semidirect
290. C2×C322−1+4⋊C3semidirect
312. C2×Q8C2×(C3⋊D3)⋊C3semidirect
356. C32C2×2−1+4⋊C3semidirect
419. C2#11722432direct (N × H)

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
#117224322
C2×#886232
C2×#886222
C2×#643643
C2×#1007⋊C33
C2×#638⋊C34