Finite Group Databasebeta

← C2×#11720847 #13149825

Subgroups — C2×#11720847 #13149825

Order 1728.

1259 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1259 conjugacy classes of subgroups — too many to draw. Only its 171 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×#88654C212yesmaximal, complemented
2C2×#88529C211yesmaximal, characteristic, complemented
3C2×#3745⋊C3C211yesmaximal, characteristic, complemented
4C2×C8×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
5C2×#88436C212yesmaximal, complemented
6#11720847C218yesmaximal, direct product factor
7C2×#63797—31nomaximal
8C2×D3×D24—31nomaximal
9C22×((C3⋊D3)⋊C3)⋊C2—22no—
10C2×C4⋊#272C2212yescomplemented
11C2×C4⋊((C3⋊D3)⋊C3)C2212yescomplemented
12C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×#6200—22no—
14#88654—24no—
15#88654C2218yescomplemented
16C2×C8×H(3,𝔽3)C2211yescharacteristic, complemented
17#88529C2214yescomplemented
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic, complemented
19#3745⋊C3C2214yescomplemented
20C8×(C3⋊D3)⋊C3C2214yescomplemented
21#88436C2218yescomplemented
22C2×#3802—32no—
23C2×D3×D12—32no—
24C2×#3745D311yescharacteristic, complemented
25C2×C8×C3⋊D3D311yescharacteristic, complemented
26C2×C3×D24—31no—
27C2×C3×D24—31no—
28C2×C3×D24—31no—
29C2×C3×D24—61no—
30C2×C3×D3×C8—31no—
31C2×#3714—32no—
32C2×#3713—32no—
33#63797—38no—
34D3×D24—38no—
35C22×#272—22no—
36C22×(C3⋊D3)⋊C3—22no—
37C22×(C3⋊D3)⋊C3D411yescharacteristic
38C2×((C3⋊D3)⋊C3)⋊C2—28no—
39C2×((C3⋊D3)⋊C3)⋊C2—48no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C4⋊#272—22no—
42C4⋊#272C2314yescomplemented
43C2×(C3⋊Dic3)⋊C3D411yescharacteristic
44C4⋊((C3⋊D3)⋊C3)—22no—
45C4⋊((C3⋊D3)⋊C3)C2314yescomplemented
46C4×(C3⋊D3)⋊C3D412yes—
47C4×(C3⋊D3)⋊C3C2312yes—
48#6200—28no—
49C8×H(3,𝔽3)C2312yescomplemented
50(C3⋊(C3⋊C8))⋊C3C2312yescomplemented
51C2×D3×D8—91no—
52C2×D3×D8—91no—
53C22×D24—91no—
54C22×D32—62no—
55C22×D32—62no—
56C2×C4⋊(C3⋊D3)C2×D312yescomplemented
57C2×C4×C3⋊D3C2×D311yescharacteristic
58C2×C3×D12—32no—
59C2×C3×D12—32no—
60C2×C3×D12—32no—
61C2×C3×D12—62no—
62C2×C3×C4×D3—31no—
63C2×#626—62no—
64C2×#625—62no—
65#3802—38no—
66#3802—64no—
67D3×D12—38no—
68D3×D12—64no—
69C2×C32×C8—21no—
70C2×C32×C8C2×D311yescharacteristic, complemented
71C2×C32×C8C2×D311yescharacteristic, complemented
72C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
73#3745C2×D314yescomplemented
74C8×C3⋊D3C2×D314yescomplemented
75C2×C3×C3⋊C8—31no—
76C3×D24—34no—
77C3×D24—34no—
78C3×D24—34no—
79C3×D24—64no—
80C3×D3×C8—34no—
81#3714—38no—
82#3713—38no—
83C22×H(3,𝔽3)C2×D411yescharacteristic
84C2×#272—24no—
85C2×#272—44no—
86C2×(C3⋊D3)⋊C3—24no—
87C2×(C3⋊D3)⋊C3—44no—
88C2×(C3⋊D3)⋊C3D814yescomplemented
89C2×(C3⋊D3)⋊C3C2×D412yes—
90((C3⋊D3)⋊C3)⋊C2—416no—
91C4×H(3,𝔽3)C2×D411yescharacteristic
92C4×H(3,𝔽3)C2411yescharacteristic
93(C3⋊Dic3)⋊C3C2×D412yes—
94C2×D3×D4—92no—
95C2×D3×D4—92no—
96C22×D12—92no—
97C2×C3×D8—91no—
98C2×C3×D8—91no—
99C22×C3×C8—91no—
100C2×#213—92no—
101C2×#213—92no—
102D3×D8—98no—
103D3×D8—98no—
104C2×D24—31no—
105C2×D24—31no—
106C2×D24—31no—
107C2×D24—61no—
108C2×D24—91no—
109C2×D24—98no—
110C2×D24—92no—
111C2×D3×C8—31no—
112C2×D3×C8—91no—
113C22×C3⋊D3—22no—
114C22×C3⋊D3D1211yescharacteristic
115C22×C3×D3—31no—
116C22×C3×D3—62no—
117C22×C3×D3—62no—
118C22×C3×D3—62no—
119C22×C3×D3—122no—
120C2×D32—68no—
121C2×D32—68no—
122C2×D32—128no—
123C2×D32—128no—
124C2×C32×C4—21no—
125C2×C32×C4C22×D311yescharacteristic
126C2×C32×C4C22×D311yescharacteristic
127C2×C3⋊Dic3D1211yescharacteristic
128C4⋊(C3⋊D3)—22no—
129C4⋊(C3⋊D3)C22×D314yescomplemented
130C4×C3⋊D3D1212yes—
131C4×C3⋊D3C22×D312yes—
132C2×C3×Dic3—31no—
133C3×D12—34no—
134C3×D12—34no—
135C3×D12—34no—
136C3×D12—64no—
137C3×D12—62no—
138C3×D12—62no—
139C3×D12—62no—
140C3×D12—122no—
141C3×C4×D3—32no—
142C3×C4×D3—32no—
143#626—68no—
144#625—68no—
145C32×C8—22no—
146C32×C8C22×D312yescomplemented
147C32×C8C22×D312yescomplemented
148C3⋊(C3⋊C8)C22×D312yescomplemented
149C3×C3⋊C8—32no—
150C22×D8—271noSylow
151C2×H(3,𝔽3)C2×D812yescomplemented
152C2×H(3,𝔽3)C22×D411yescharacteristic
153#272—44no—
154(C3⋊D3)⋊C3—44no—
155(C3⋊D3)⋊C3C2×D814yescomplemented
156C23×D3—182no—
157C23×D3—182no—
158C23×D3—182no—
159C2×C3×D4—92no—
160C2×C3×D4—92no—
161C22×C3×C4—91no—
162C2×#67—182no—
163C2×#67—182no—
164D3×D4—98no—
165D3×D4—98no—
166D3×D4—184no—
167D3×D4—184no—
168C2×D12—32no—
169C2×D12—32no—
170C2×D12—32no—
171C2×D12—62no—
172C2×D12—92no—
173C2×D12—98no—
174C2×D12—184no—
175C2×D12—182no—
176C2×C4×D3—31no—
177C2×C4×D3—91no—
178C3×D8—94no—
179C3×D8—94no—
180C2×C3×C8—31no—
181C2×C3×C8—31no—
182C2×C3×C8—61no—
183C2×C3×C8—94no—
184C2×C3×C8—91no—
185C2×C3×C8D3211yescharacteristic
186#213—98no—
187#213—98no—
188C2×C3⋊C8—31no—
189C2×C3⋊C8—91no—
190D24—34no—
191D24—34no—
192D24—34no—
193D24—64no—
194D24—94no—
195D24—98no—
196D3×C8—34no—
197D3×C8—94no—
198C22×C32—21no—
199C22×C32C2×D1211yescharacteristic
200C22×C32D3×D411yescharacteristic
201C2×C3⋊D3—24no—
202C2×C3⋊D3—44no—
203C2×C3⋊D3D2414yescomplemented
204C2×C3⋊D3C2×D1212yes—
205C2×C3×D3—34no—
206C2×C3×D3—32no—
207C2×C3×D3—64no—
208C2×C3×D3—64no—
209C2×C3×D3—64no—
210C2×C3×D3—124no—
211C2×C3×D3—124no—
212C2×C3×D3—124no—
213C2×C3×D3—124no—
214C2×C3×D3—244no—
215D32—1216no—
216D32—1216no—
217C32×C4—21no—
218C32×C4—21no—
219C32×C4C2×D1211yescharacteristic
220C32×C4D3×D411yescharacteristic
221C32×C4C23×D311yescharacteristic
222C32×C4C23×D311yescharacteristic
223C3⋊Dic3C2×D1212yes—
224C3×Dic3—32no—
225C22×D4—272no—
226C2×D8—91no—
227C2×D8—91no—
228C2×D8—278no—
229C2×D8—272no—
230C22×C8—91no—
231H(3,𝔽3)C22×D811yescharacteristic, complemented, Sylow
232C23×C3—91no—
233C23×C3—182no—
234C23×C3—182no—
235C22×D3—31no—
236C22×D3—62no—
237C22×D3—62no—
238C22×D3—62no—
239C22×D3—91no—
240C22×D3—122no—
241C22×D3—182no—
242C22×D3—188no—
243C22×D3—188no—
244C22×D3—188no—
245C22×D3—368no—
246C22×D3—368no—
247C22×D3—368no—
248C3×D4—94no—
249C3×D4—94no—
250C3×D4—182no—
251C3×D4—182no—
252C2×C3×C4—31no—
253C2×C3×C4—31no—
254C2×C3×C4—61no—
255C2×C3×C4—92no—
256C2×C3×C4—92no—
257C2×C3×C4—91no—
258C2×C3×C4C2×D3211yescharacteristic
259#67—188no—
260#67—188no—
261C2×Dic3—31no—
262C2×Dic3—91no—
263D12—34no—
264D12—34no—
265D12—34no—
266D12—64no—
267D12—62no—
268D12—62no—
269D12—62no—
270D12—94no—
271D12—122no—
272D12—182no—
273D12—188no—
274C4×D3—32no—
275C4×D3—32no—
276C4×D3—92no—
277C4×D3—92no—
278C3×C8—32no—
279C3×C8—32no—
280C3×C8—62no—
281C3×C8—92no—
282C3×C8C2×D3212yes—
283C3⋊C8—32no—
284C3⋊C8—92no—
285C2×C32—21no—
286C2×C32—22no—
287C2×C32C2×D2412yescomplemented
288C2×C32D3×D812yescomplemented
289C2×C32C22×D1211yescharacteristic
290C2×C32C2×D3×D411yescharacteristic
291C3⋊D3—44no—
292C3⋊D3C2×D2414yescomplemented
293C3×D3—34no—
294C3×D3—124no—
295C3×D3—124no—
296C3×D3—124no—
297C3×D3—244no—
298C24—542no—
299C2×D4—92no—
300C2×D4—92no—
301C2×D4—278no—
302C2×D4—544no—
303C2×D4—542no—
304C22×C4—91no—
305D8—94no—
306D8—94no—
307D8—278no—
308C2×C8—91no—
309C2×C8—94no—
310C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
311C22×C3—31no—
312C22×C3—31no—
313C22×C3—61no—
314C22×C3—92no—
315C22×C3—94no—
316C22×C3—184no—
317C22×C3—184no—
318C22×C3—364no—
319C22×C3—364no—
320C22×C3D3×D1211yescharacteristic
321C2×D3—32no—
322C2×D3—34no—
323C2×D3—64no—
324C2×D3—64no—
325C2×D3—64no—
326C2×D3—92no—
327C2×D3—94no—
328C2×D3—124no—
329C2×D3—124no—
330C2×D3—124no—
331C2×D3—124no—
332C2×D3—184no—
333C2×D3—244no—
334C2×D3—364no—
335C2×D3—3616no—
336C2×D3—3616no—
337C2×D3—3616no—
338C3×C4—31no—
339C3×C4—31no—
340C3×C4—31no—
341C3×C4—31no—
342C3×C4—61no—
343C3×C4—61no—
344C3×C4—92no—
345C3×C4D3×D1211yescharacteristic
346C3×C4C22×D3211yescharacteristic
347Dic3—32no—
348Dic3—92no—
349C32—21no—
350C32C22×D2411yescharacteristic, complemented
351C32C2×D3×D811yescharacteristic, complemented
352C23—91no—
353C23—182no—
354C23—182no—
355C23—548no—
356C23—1088no—
357D4—94no—
358D4—94no—
359D4—182no—
360D4—182no—
361D4—548no—
362C2×C4—91no—
363C2×C4—92no—
364C2×C4—92no—
365C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
366C8—92no—
367C8C2×((C3⋊D3)⋊C3)⋊C212yescomplemented
368C2×C3—32no—
369C2×C3—32no—
370C2×C3—31no—
371C2×C3—31no—
372C2×C3—62no—
373C2×C3—61no—
374C2×C3—94no—
375C2×C3—364no—
376C2×C3—364no—
377C2×C3D3×D2412yes—
378C2×C3C2×D3×D1211yescharacteristic
379D3—34no—
380D3—94no—
381D3—124no—
382D3—124no—
383D3—124no—
384D3—244no—
385D3—364no—
386C22—94no—
387C22—92no—
388C22—184no—
389C22—184no—
390C22—364no—
391C22—364no—
392C22—10816no—
393C22#8865411yescharacteristic
394C4—92no—
395C4#8865411yescharacteristic
396C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
397C3—31no—
398C3—31no—
399C3—61no—
400C3C2×D3×D2411yescharacteristic
401C2—94no—
402C2—364no—
403C2—364no—
404C2#1172084712yesdirect product factor
405C2C2×#8865411yescharacteristic
subgroupG/H
centerC22#88654
commutator, cosocleC4×H(3,𝔽3)C24
FrattiniC3×C4C22×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C22×D8
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×#88654C2semidirect
2. C2×#88529C2semidirect
3. C2×#3745⋊C3C2semidirect
4. C2×C8×(C3⋊D3)⋊C3C2semidirect
5. C2×#88436C2semidirect
6. #11720847C2direct (N × H)
10. C2×C4⋊#272C22semidirect
11. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
15. #88654C22semidirect
16. C2×C8×H(3,𝔽3)C22semidirect
17. #88529C22semidirect
18. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
19. #3745⋊C3C22semidirect
20. C8×(C3⋊D3)⋊C3C22semidirect
21. #88436C22semidirect
24. C2×#3745D3semidirect
25. C2×C8×C3⋊D3D3semidirect
42. C4⋊#272C23semidirect
45. C4⋊((C3⋊D3)⋊C3)C23semidirect
49. C8×H(3,𝔽3)C23semidirect
50. (C3⋊(C3⋊C8))⋊C3C23semidirect
56. C2×C4⋊(C3⋊D3)C2×D3semidirect
70. C2×C32×C8C2×D3semidirect
71. C2×C32×C8C2×D3semidirect
72. C2×C3⋊(C3⋊C8)C2×D3semidirect
73. #3745C2×D3semidirect
74. C8×C3⋊D3C2×D3semidirect
88. C2×(C3⋊D3)⋊C3D8semidirect
129. C4⋊(C3⋊D3)C22×D3semidirect
146. C32×C8C22×D3semidirect
147. C32×C8C22×D3semidirect
148. C3⋊(C3⋊C8)C22×D3semidirect
151. C2×H(3,𝔽3)C2×D8semidirect
155. (C3⋊D3)⋊C3C2×D8semidirect
203. C2×C3⋊D3D24semidirect
231. H(3,𝔽3)C22×D8semidirect
287. C2×C32C2×D24semidirect
288. C2×C32D3×D8semidirect
292. C3⋊D3C2×D24semidirect
310. C2×C8((C3⋊D3)⋊C3)⋊C2semidirect
350. C32C22×D24semidirect
351. C32C2×D3×D8semidirect
367. C8C2×((C3⋊D3)⋊C3)⋊C2semidirect
404. C2#11720847direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#886542
C2×#885292
C2×#3745⋊C32
C2×C8×(C3⋊D3)⋊C32
C2×#884362
#117208472
C2×#637973
C2×D3×D243