Finite Group Databasebeta

← C2×#11722437 #13164630

Subgroups — C2×#11722437 #13164630

Order 1728.

1131 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 1131 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C2×#41⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#11722437C214yesmaximal, direct product factor
3C2×#88623C211yesmaximal, characteristic, complemented
4C2×#88622C211yesmaximal, characteristic, complemented
5C2×#41×C3⋊D3C311yesmaximal, characteristic, complemented
6C2×D3×#41⋊C3—31nomaximal
7C2×C4×(C3⋊D3)⋊C3—41nomaximal
8#41⋊H(3,𝔽3)C2212yescomplemented
9C2×Q8⋊H(3,𝔽3)C2211yescharacteristic, complemented
10#88623C2212yescomplemented
11#88622C2212yescomplemented
12C2×C32×#41C2×C311yescharacteristic, complemented
13#41×C3⋊D3—34no—
14#41×C3⋊D3C2×C314yescomplemented
15C2×#3832C2×C311yescharacteristic, complemented
16C2×Q8×C3⋊D3C2×C311yescharacteristic
17C2×#3830—31no—
18C2×D4×C3⋊D3—31no—
19C2×#3828—31no—
20C22×C4×C3⋊D3—31no—
21C2×C3×#41⋊C3—21no—
22C2×C3×#41⋊C3D311yescharacteristic, complemented
23D3×#41⋊C3—34no—
24C2×D3×SL(2,𝔽3)—31no—
25C2×C3⋊(#41⋊C3)—31no—
26C22×(C3⋊D3)⋊C3—41no—
27C2×C4×H(3,𝔽3)—41no—
28C2×(C3⋊Dic3)⋊C3—41no—
29C4×(C3⋊D3)⋊C3—44no—
30Q8⋊H(3,𝔽3)C2311yescharacteristic, complemented
31C2×D3×#41—31no—
32C2×D3×#41—91no—
33C22×#41⋊C3—91no—
34C23×C3⋊D3—31no—
35C32×#41—32no—
36C32×#41C22×C312yescomplemented
37C2×C32×Q8C22×C311yescharacteristic
38C2×C32×D4—31no—
39C22×C32×C4—31no—
40C2×#638—32no—
41C22×C3⋊Dic3—31no—
42#3832—32no—
43#3832C22×C312yescomplemented
44Q8×C3⋊D3—32no—
45Q8×C3⋊D3C22×C312yes—
46#3830—34no—
47#3830—34no—
48D4×C3⋊D3—34no—
49D4×C3⋊D3—34no—
50#3828—34no—
51#3828—34no—
52C2×C4⋊(C3⋊D3)—31no—
53C2×C4×C3⋊D3—31no—
54C2×C4×C3⋊D3—31no—
55C2×C4×C3⋊D3—31no—
56C2×C4×C3⋊D3—34no—
57C2×C4×C3⋊D3—34no—
58C2×C4×C3⋊D3A411yescharacteristic
59C2×#634—31no—
60C2×C3×C4×D3—121no—
61C3×#41⋊C3—22no—
62C3×#41⋊C3C2×D312yescomplemented
63C2×C3×SL(2,𝔽3)—21no—
64C2×C3×SL(2,𝔽3)C2×D311yescharacteristic, complemented
65D3×SL(2,𝔽3)—32no—
66C3⋊(#41⋊C3)—32no—
67C22×H(3,𝔽3)—41no—
68C2×(C3⋊D3)⋊C3—44no—
69C2×(C3⋊D3)⋊C3—42no—
70C4×H(3,𝔽3)—42no—
71(C3⋊Dic3)⋊C3—42no—
72C2×C3×#41—31no—
73C2×C3×#41C3×D311yescharacteristic
74D3×#41—34no—
75D3×#41—94no—
76D3×#41—912no—
77D3×#41—94no—
78C2×#239—31no—
79C2×#239—91no—
80C2×D3×Q8—31no—
81C2×D3×Q8—91no—
82C2×#237—93no—
83C2×#237—91no—
84C2×D3×D4—91no—
85C2×D3×D4—93no—
86C2×#235—93no—
87C2×#235—91no—
88C22×C4×D3—91no—
89C22×C4×D3—93no—
90C2×#41⋊C3—31no—
91C2×#41⋊C3—61no—
92C2×#41⋊C3—91no—
93C2×#41⋊C3—94no—
94C22×SL(2,𝔽3)—91no—
95C23×C32—31no—
96C22×C3⋊D3—31no—
97C22×C3⋊D3—34no—
98C22×C3⋊D3—64no—
99C22×C3⋊D3C2×A411yescharacteristic
100C22×C3×D3—121no—
101C32×Q8—31no—
102C32×Q8C23×C311yescharacteristic
103C32×D4—32no—
104C32×D4—32no—
105C2×C32×C4—31no—
106C2×C32×C4—32no—
107C2×C32×C4—32no—
108C2×C32×C4—41no—
109C2×C32×C4—81no—
110C2×C32×C4C2×A411yescharacteristic
111#638—38no—
112C2×C3⋊Dic3—31no—
113C2×C3⋊Dic3—34no—
114C2×C3⋊Dic3C2×A411yescharacteristic
115C4⋊(C3⋊D3)—32no—
116C4⋊(C3⋊D3)—32no—
117C4×C3⋊D3—32no—
118C4×C3⋊D3—32no—
119C4×C3⋊D3—34no—
120C4×C3⋊D3—32no—
121C4×C3⋊D3—32no—
122C4×C3⋊D3C2×A414yes—
123#634—32no—
124#634—32no—
125C2×C3×Dic3—121no—
126C3×C4×D3—124no—
127C3×SL(2,𝔽3)—21no—
128C3×SL(2,𝔽3)C22×D311yescharacteristic, complemented
129C22×#41—91noSylow
130C2×H(3,𝔽3)—42no—
131C2×H(3,𝔽3)—41no—
132(C3⋊D3)⋊C3—44no—
133C23×D3—91no—
134C23×D3—93no—
135C3×#41—36no—
136C3×#41—32no—
137C3×#41—32no—
138C3×#41C2×C3×D312yes—
139C2×C3×Q8—31no—
140C2×C3×Q8C2×C3×D311yescharacteristic
141C2×C3×D4—31no—
142C2×C3×D4—33no—
143C22×C3×C4—33no—
144C22×C3×C4—31no—
145C22×C3×C4—361no—
146C2×#67—96no—
147C2×#67—92no—
148C22×Dic3—91no—
149C22×Dic3—93no—
150#239—32no—
151#239—92no—
152#239—96no—
153#239—92no—
154D3×Q8—32no—
155D3×Q8—92no—
156D3×Q8—96no—
157D3×Q8—92no—
158#237—94no—
159#237—912no—
160#237—912no—
161#237—94no—
162D3×D4—94no—
163D3×D4—912no—
164D3×D4—94no—
165D3×D4—912no—
166#235—912no—
167#235—94no—
168#235—912no—
169#235—94no—
170C2×D12—91no—
171C2×D12—93no—
172C2×C4×D3—31no—
173C2×C4×D3—91no—
174C2×C4×D3—94no—
175C2×C4×D3—94no—
176C2×C4×D3—91no—
177C2×C4×D3—91no—
178C2×C4×D3—93no—
179C2×C4×D3—93no—
180C2×C4×D3—93no—
181C2×C4×D3—912no—
182C2×C4×D3—912no—
183C2×C4×D3—91no—
184C2×Dic6—93no—
185C2×Dic6—91no—
186#41⋊C3—32no—
187#41⋊C3—62no—
188#41⋊C3—92no—
189C2×SL(2,𝔽3)—31no—
190C2×SL(2,𝔽3)—61no—
191C2×SL(2,𝔽3)—92no—
192C22×C32—32no—
193C22×C32—41no—
194C22×C32—62no—
195C22×C32—81no—
196C22×C32C22×A411yescharacteristic
197C2×C3⋊D3—32no—
198C2×C3⋊D3—62no—
199C2×C3⋊D3—68no—
200C2×C3⋊D3#41⋊C314yescomplemented
201C2×C3⋊D3C22×A412yes—
202C2×C3×D3—122no—
203C2×C3×D3—124no—
204C32×C4—31no—
205C32×C4—31no—
206C32×C4—42no—
207C32×C4—82no—
208C32×C4C22×A412yes—
209C3⋊Dic3—32no—
210C3⋊Dic3C22×A412yes—
211C3×Dic3—122no—
212C2×#41—94no—
213C2×#41—91no—
214C2×#41—271no—
215C2×#41—271no—
216C2×#41—274no—
217C2×#41(C3⋊D3)⋊C311yescharacteristic, complemented
218C22×Q8—91no—
219C22×D4—271no—
220C23×C4—271no—
221H(3,𝔽3)—41noSylow
222C23×C3—33no—
223C23×C3—31no—
224C23×C3—361no—
225C22×D3—31no—
226C22×D3—91no—
227C22×D3—94no—
228C22×D3—93no—
229C22×D3—912no—
230C22×D3—91no—
231C22×D3—184no—
232C22×D3—1812no—
233C3×Q8—31no—
234C3×Q8—33no—
235C3×Q8—31no—
236C3×Q8C22×C3×D311yescharacteristic
237C3×D4—32no—
238C3×D4—36no—
239C3×D4—36no—
240C3×D4—32no—
241C2×C3×C4—33no—
242C2×C3×C4—36no—
243C2×C3×C4—36no—
244C2×C3×C4—31no—
245C2×C3×C4—32no—
246C2×C3×C4—32no—
247C2×C3×C4—31no—
248C2×C3×C4—121no—
249C2×C3×C4—241no—
250C2×C3×C4—364no—
251C2×C3×C4—361no—
252C2×C3×C4D3×A411yescharacteristic
253#67—924no—
254#67—98no—
255C2×Dic3—31no—
256C2×Dic3—91no—
257C2×Dic3—94no—
258C2×Dic3—93no—
259C2×Dic3—912no—
260C2×Dic3—91no—
261D12—92no—
262D12—96no—
263D12—96no—
264D12—92no—
265C4×D3—34no—
266C4×D3—94no—
267C4×D3—92no—
268C4×D3—92no—
269C4×D3—92no—
270C4×D3—92no—
271C4×D3—912no—
272C4×D3—96no—
273C4×D3—96no—
274C4×D3—96no—
275C4×D3—96no—
276C4×D3—94no—
277Dic6—96no—
278Dic6—96no—
279Dic6—92no—
280Dic6—92no—
281SL(2,𝔽3)—31no—
282SL(2,𝔽3)—61no—
283C2×C32—42no—
284C2×C32—41no—
285C2×C32—62no—
286C2×C32—82no—
287C2×C32—81no—
288C2×C32C2×#41⋊C312yescomplemented
289C2×C32C23×A411yescharacteristic
290C3⋊D3—62no—
291C3⋊D3C2×#41⋊C314yescomplemented
292C3×D3—124no—
293C24—271no—
294#41—32no—
295#41—92no—
296#41—274no—
297#41—274no—
298#41—274no—
299#41—274no—
300#41—272no—
301#41C2×(C3⋊D3)⋊C312yescomplemented
302C2×Q8—92no—
303C2×Q8—271no—
304C2×Q8—272no—
305C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
306C2×D4—31no—
307C2×D4—271no—
308C2×D4—274no—
309C2×D4—274no—
310C2×D4—272no—
311C22×C4—31no—
312C22×C4—91no—
313C22×C4—271no—
314C22×C4—274no—
315C22×C4—274no—
316C22×C4—271no—
317C22×C4—271no—
318C22×C4—271no—
319C22×C3—36no—
320C22×C3—32no—
321C22×C3—31no—
322C22×C3—66no—
323C22×C3—62no—
324C22×C3—121no—
325C22×C3—241no—
326C22×C3—362no—
327C22×C3—364no—
328C22×C3C2×D3×A411yescharacteristic
329C2×D3—34no—
330C2×D3—32no—
331C2×D3—92no—
332C2×D3—94no—
333C2×D3—96no—
334C2×D3—92no—
335C2×D3—188no—
336C2×D3—1824no—
337C2×D3—186no—
338C2×D3—182no—
339C3×C4—33no—
340C3×C4—33no—
341C3×C4—31no—
342C3×C4—31no—
343C3×C4—32no—
344C3×C4—122no—
345C3×C4—242no—
346C3×C4—362no—
347C3×C4C2×D3×A412yes—
348Dic3—32no—
349Dic3—92no—
350Dic3—96no—
351Dic3—92no—
352C32—41no—
353C32—81no—
354C32C22×#41⋊C311yescharacteristic, complemented
355C23—31no—
356C23—91no—
357C23—274no—
358C23—271no—
359C23—544no—
360Q8—31no—
361Q8—272no—
362Q8—272no—
363Q8C22×(C3⋊D3)⋊C311yescharacteristic
364D4—32no—
365D4—32no—
366D4—272no—
367D4—272no—
368D4—278no—
369C2×C4—31no—
370C2×C4—32no—
371C2×C4—32no—
372C2×C4—91no—
373C2×C4—94no—
374C2×C4—272no—
375C2×C4—272no—
376C2×C4—272no—
377C2×C4—272no—
378C2×C4—274no—
379C2×C4—271no—
380C2×C4—274no—
381C2×C4(C3⋊D3)⋊A411yescharacteristic
382C2×C3—31no—
383C2×C3—32no—
384C2×C3—66no—
385C2×C3—62no—
386C2×C3—121no—
387C2×C3—122no—
388C2×C3—241no—
389C2×C3—242no—
390C2×C3—364no—
391C2×C3D3×#41⋊C312yes—
392C2×C3C22×D3×A411yescharacteristic
393D3—34no—
394D3—94no—
395D3—186no—
396D3—182no—
397C22—32no—
398C22—62no—
399C22—92no—
400C22—94no—
401C22—272no—
402C22—548no—
403C22—542no—
404C22C2×(C3⋊D3)⋊A411yescharacteristic
405C4—31no—
406C4—31no—
407C4—92no—
408C4—272no—
409C4C2×(C3⋊D3)⋊A412yes—
410C3—31no—
411C3—121no—
412C3—241no—
413C3C2×D3×#41⋊C311yescharacteristic
414C2—62no—
415C2—94no—
416C2—542no—
417C2#1172243712yesdirect product factor
418C2C22×(C3⋊D3)⋊A411yescharacteristic
subgroupG/H
center, hypercenterC2×C4(C3⋊D3)⋊A4
commutator, cosocle, hypocenterC32×Q8C23×C3
FrattiniC2×C3C22×D3×A4
FittingC2×C32×#41C2×C3
socleC22×C3C2×D3×A4
LCSG ▷ C32×Q8
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#41⋊H(3,𝔽3)C2semidirect
2. #11722437C2direct (N × H)
3. C2×#88623C2semidirect
4. C2×#88622C2semidirect
5. C2×#41×C3⋊D3C3semidirect
8. #41⋊H(3,𝔽3)C22semidirect
9. C2×Q8⋊H(3,𝔽3)C22semidirect
10. #88623C22semidirect
11. #88622C22semidirect
12. C2×C32×#41C2×C3semidirect
14. #41×C3⋊D3C2×C3semidirect
15. C2×#3832C2×C3semidirect
22. C2×C3×#41⋊C3D3semidirect
30. Q8⋊H(3,𝔽3)C23semidirect
36. C32×#41C22×C3semidirect
43. #3832C22×C3semidirect
62. C3×#41⋊C3C2×D3semidirect
64. C2×C3×SL(2,𝔽3)C2×D3semidirect
128. C3×SL(2,𝔽3)C22×D3semidirect
200. C2×C3⋊D3#41⋊C3semidirect
217. C2×#41(C3⋊D3)⋊C3semidirect
288. C2×C32C2×#41⋊C3semidirect
291. C3⋊D3C2×#41⋊C3semidirect
301. #41C2×(C3⋊D3)⋊C3semidirect
354. C32C22×#41⋊C3semidirect
417. C2#11722437direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#41⋊H(3,𝔽3)2
#117224372
C2×#886232
C2×#886222
C2×#41×C3⋊D33
C2×D3×#41⋊C33
C2×C4×(C3⋊D3)⋊C34