Finite Group Databasebeta

← C2×#11724306 #13181068

Subgroups — C2×#11724306 #13181068

Order 1728.

1256 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 1256 conjugacy classes of subgroups — too many to draw. Only its 143 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×C4⋊#1113C211yesmaximal, characteristic, complemented
2C2×C3×#3802C211yesmaximal, characteristic, complemented
3C2×#88988C211yesmaximal, characteristic, complemented
4C2×#88935C214yesmaximal, complemented
5#11724306C218yesmaximal, direct product factor
6C2×#64225—31nomaximal
7C22×#1113—21no—
8C22×C3×D32—21no—
9C2×C3×C4⋊(C3⋊D3)—21no—
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C32×D12—21no—
12C2×#6295C2212yescomplemented
13C4⋊#1113—22no—
14C4⋊#1113C2214yescomplemented
15C2×C3×#625C2212yescomplemented
16C3×#3802—22no—
17C3×#3802C2214yescomplemented
18C2×C3⋊#622C2212yescomplemented
19#88988C2214yescomplemented
20#88935C22116yescomplemented
21C2×C3×D3×D4—61no—
22C2×#3802—31no—
23C2×#3802D311yescharacteristic, complemented
24C2×D3×D12—61no—
25C2×#3787—31no—
26C2×#3774—34no—
27#64225—38no—
28C22×C3×C3⋊D3—22no—
29C22×C3×C3⋊D3D411yescharacteristic
30C22×C32×D3—22no—
31C2×#1113—24no—
32C2×#1113—44no—
33C2×C3×D32—24no—
34C2×C3×D32—44no—
35C2×C33×C4D411yescharacteristic
36C2×C3×C3⋊Dic3C2311yescharacteristic
37C3×C4⋊(C3⋊D3)—24no—
38C3×C4×C3⋊D3D412yes—
39C3×C4×C3⋊D3C2312yes—
40C32×D12—24no—
41#6295—22no—
42#6295C2314yescomplemented
43C3×#625—22no—
44C3×#625C2314yescomplemented
45C3⋊#622C2314yescomplemented
46C22×#213—91nomaximal
47C23×C3×D3—62no—
48C22×D32—21no—
49C22×D32—62no—
50C22×D32—61no—
51C2×C32×D4—61no—
52C2×C4⋊(C3⋊D3)—21no—
53C2×C4×C3⋊D3C2×D311yescharacteristic
54C2×C3×#67—62no—
55C3×D3×D4—64no—
56C3×D3×D4—64no—
57C2×C3×D12—21no—
58C2×C3×D12—21no—
59C2×C3×D12—41no—
60C2×C3×D12—61no—
61C2×C3×D12—61no—
62C2×C3×D12—121no—
63C2×C3×C4×D3—61no—
64C2×C3×C4×D3—61no—
65C2×#626—62no—
66C2×#625—32no—
67C2×#625C2×D312yescomplemented
68#3802—22no—
69#3802—34no—
70#3802—62no—
71#3802C2×D314yescomplemented
72D3×D12—64no—
73D3×D12—64no—
74C2×#622—32no—
75#3787—34no—
76#3774—316no—
77C22×C33C2×D411yescharacteristic
78C2×C3×C3⋊D3—24no—
79C2×C3×C3⋊D3—44no—
80C2×C3×C3⋊D3D814yescomplemented
81C2×C3×C3⋊D3C2×D412yes—
82C2×C32×D3—24no—
83C2×C32×D3—44no—
84#1113—48no—
85C3×D32—48no—
86C33×C4C2×D412yes—
87C3×C3⋊Dic3C2×D411yescharacteristic
88C3×C3⋊Dic3C2411yescharacteristic
89C22×C3×D4—91no—
90C2×D3×D4—61no—
91C2×D3×D4—181no—
92C22×D12—91no—
93C2×#213—98no—
94C2×#213—94no—
95C22×C3⋊C8—91no—
96C23×C32—62no—
97C22×C3⋊D3—22no—
98C22×C3⋊D3#6711yescharacteristic
99C22×C3×D3—22no—
100C22×C3×D3—22no—
101C22×C3×D3—42no—
102C22×C3×D3—61no—
103C22×C3×D3—61no—
104C22×C3×D3—68no—
105C22×C3×D3—62no—
106C22×C3×D3—62no—
107C22×C3×D3—122no—
108C22×C3×D3—128no—
109C2×D32—24no—
110C2×D32—44no—
111C2×D32—68no—
112C2×D32—64no—
113C2×D32—128no—
114C2×D32—124no—
115C32×D4—64no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—41no—
119C2×C32×C4—41no—
120C2×C32×C4#6711yescharacteristic
121C2×C3⋊Dic3C22×D311yescharacteristic
122C4⋊(C3⋊D3)—24no—
123C4×C3⋊D3#6712yes—
124C4×C3⋊D3C22×D312yes—
125C3×#67—64no—
126C3×#67—64no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C3×D12—24no—
130C3×D12—24no—
131C3×D12—44no—
132C3×D12—64no—
133C3×D12—64no—
134C3×D12—124no—
135C3×C4×D3—62no—
136C3×C4×D3—62no—
137C3×C4×D3—62no—
138C3×C4×D3—62no—
139#626—64no—
140#626—64no—
141#625—22no—
142#625—34no—
143#625—62no—
144#625C22×D314yescomplemented
145#622—34no—
146C22×D8—271noSylow
147C2×C33C2×D812yescomplemented
148C2×C33C22×D411yescharacteristic
149C3×C3⋊D3—44no—
150C3×C3⋊D3C2×D814yescomplemented
151C32×D3—44no—
152C24×C3—181no—
153C23×D3—62no—
154C23×D3—181no—
155C23×D3—182no—
156C2×C3×D4—61no—
157C2×C3×D4—61no—
158C2×C3×D4—94no—
159C2×C3×D4—92no—
160C2×C3×D4—121no—
161C2×C3×D4—181no—
162C2×C3×D4—182no—
163C22×C3×C4—91no—
164C2×#67—62no—
165C2×#67—182no—
166D3×D4—64no—
167D3×D4—64no—
168D3×D4—184no—
169D3×D4—184no—
170C2×D12—21no—
171C2×D12—61no—
172C2×D12—61no—
173C2×D12—94no—
174C2×D12—92no—
175C2×D12—121no—
176C2×D12—182no—
177C2×C4×D3—61no—
178C2×C4×D3—61no—
179#213—916no—
180C2×C3⋊C8—94no—
181C2×C3⋊C8—92no—
182C22×C32—21no—
183C22×C32—21no—
184C22×C32—41no—
185C22×C32—41no—
186C22×C32—64no—
187C22×C32—124no—
188C22×C32C2×#6711yescharacteristic
189C2×C3⋊D3—24no—
190C2×C3⋊D3—44no—
191C2×C3⋊D3#21314yescomplemented
192C2×C3⋊D3C2×#6712yes—
193C2×C3×D3—24no—
194C2×C3×D3—24no—
195C2×C3×D3—44no—
196C2×C3×D3—44no—
197C2×C3×D3—44no—
198C2×C3×D3—64no—
199C2×C3×D3—64no—
200C2×C3×D3—62no—
201C2×C3×D3—62no—
202C2×C3×D3—64no—
203C2×C3×D3—64no—
204C2×C3×D3—84no—
205C2×C3×D3—124no—
206C2×C3×D3—1216no—
207C2×C3×D3—124no—
208C2×C3×D3—124no—
209C2×C3×D3—244no—
210D32—48no—
211D32—1216no—
212D32—128no—
213C32×C4—22no—
214C32×C4—22no—
215C32×C4—42no—
216C32×C4—42no—
217C32×C4C2×#6712yes—
218C3⋊Dic3C2×#6711yescharacteristic
219C3⋊Dic3C23×D311yescharacteristic
220C3×Dic3—61no—
221C3×Dic3—61no—
222C3×Dic3—61no—
223C3×Dic3—61no—
224C22×D4—91no—
225C22×D4—271no—
226C2×D8—278no—
227C2×D8—274no—
228C22×C8—271no—
229C33C22×D811yescharacteristic, complemented, Sylow
230C23×C3—62no—
231C23×C3—62no—
232C23×C3—91no—
233C23×C3—122no—
234C23×C3—182no—
235C23×C3—184no—
236C23×C3—364no—
237C22×D3—22no—
238C22×D3—61no—
239C22×D3—61no—
240C22×D3—68no—
241C22×D3—62no—
242C22×D3—62no—
243C22×D3—122no—
244C22×D3—128no—
245C22×D3—184no—
246C22×D3—188no—
247C22×D3—368no—
248C22×D3—364no—
249C3×D4—64no—
250C3×D4—64no—
251C3×D4—94no—
252C3×D4—124no—
253C3×D4—184no—
254C3×D4—182no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—91no—
260C2×C3×C4—92no—
261C2×C3×C4—92no—
262C2×C3×C4GO+(4,𝔽2)11yescharacteristic
263#67—64no—
264#67—64no—
265#67—184no—
266#67—184no—
267C2×Dic3—61no—
268C2×Dic3—61no—
269D12—24no—
270D12—64no—
271D12—64no—
272D12—94no—
273D12—124no—
274D12—182no—
275C4×D3—62no—
276C4×D3—62no—
277C4×D3—62no—
278C4×D3—62no—
279C3⋊C8—94no—
280C2×C32—21no—
281C2×C32—21no—
282C2×C32—22no—
283C2×C32—22no—
284C2×C32—42no—
285C2×C32—42no—
286C2×C32—41no—
287C2×C32—41no—
288C2×C32—124no—
289C2×C32C2×#21312yescomplemented
290C2×C32C22×#6711yescharacteristic
291C3⋊D3—44no—
292C3⋊D3C2×#21314yescomplemented
293C3×D3—44no—
294C3×D3—44no—
295C3×D3—64no—
296C3×D3—64no—
297C3×D3—84no—
298C3×D3—124no—
299C3×D3—124no—
300C3×D3—244no—
301C24—181no—
302C24—541no—
303C2×D4—61no—
304C2×D4—94no—
305C2×D4—92no—
306C2×D4—182no—
307C2×D4—181no—
308C2×D4—274no—
309C2×D4—272no—
310C2×D4—542no—
311C22×C4—91no—
312D8—2716no—
313C2×C8—274no—
314C2×C8—272no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3—64no—
320C22×C3—64no—
321C22×C3—92no—
322C22×C3—94no—
323C22×C3—124no—
324C22×C3—124no—
325C22×C3—124no—
326C22×C3—184no—
327C22×C3—244no—
328C22×C3—364no—
329C22×C3—368no—
330C22×C3C2×GO+(4,𝔽2)11yescharacteristic
331C2×D3—24no—
332C2×D3—44no—
333C2×D3—62no—
334C2×D3—62no—
335C2×D3—64no—
336C2×D3—64no—
337C2×D3—64no—
338C2×D3—64no—
339C2×D3—124no—
340C2×D3—1216no—
341C2×D3—124no—
342C2×D3—124no—
343C2×D3—244no—
344C2×D3—3616no—
345C2×D3—368no—
346C3×C4—22no—
347C3×C4—22no—
348C3×C4—42no—
349C3×C4—42no—
350C3×C4—91no—
351C3×C4—91no—
352C3×C4C2×GO+(4,𝔽2)12yes—
353Dic3—61no—
354Dic3—61no—
355Dic3—61no—
356Dic3—61no—
357C32—21no—
358C32—21no—
359C32—41no—
360C32—41no—
361C32C22×#21311yescharacteristic, complemented
362C23—62no—
363C23—91no—
364C23—184no—
365C23—182no—
366C23—364no—
367C23—544no—
368C23—1084no—
369D4—64no—
370D4—94no—
371D4—182no—
372D4—184no—
373D4—274no—
374D4—542no—
375C2×C4—91no—
376C2×C4—92no—
377C2×C4—92no—
378C2×C4D32⋊D311yescharacteristic
379C8—274no—
380C2×C3—22no—
381C2×C3—22no—
382C2×C3—21no—
383C2×C3—21no—
384C2×C3—42no—
385C2×C3—42no—
386C2×C3—41no—
387C2×C3—41no—
388C2×C3—94no—
389C2×C3—124no—
390C2×C3—124no—
391C2×C3—244no—
392C2×C3—364no—
393C2×C3#6422512yescomplemented
394C2×C3C22×GO+(4,𝔽2)11yescharacteristic
395D3—44no—
396D3—64no—
397D3—64no—
398D3—124no—
399D3—124no—
400D3—244no—
401C22—64no—
402C22—94no—
403C22—92no—
404C22—124no—
405C22—184no—
406C22—368no—
407C22—364no—
408C22—1088no—
409C22C2×D32⋊D311yescharacteristic
410C4—91no—
411C4—91no—
412C4C2×D32⋊D312yes—
413C3—21no—
414C3—21no—
415C3—41no—
416C3—41no—
417C3C2×#6422511yescharacteristic, complemented
418C2—94no—
419C2—124no—
420C2—364no—
421C2#1172430612yesdirect product factor
422C2C22×D32⋊D311yescharacteristic
subgroupG/H
centerC22C2×D32⋊D3
commutator, cosocleC3×C3⋊Dic3C24
FrattiniC2C22×D32⋊D3
FittingC2×C33×C4D4
socleC22×C33C2×D4
hypercenterC2×C4D32⋊D3
hypocenterC33C22×D8
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊#1113C2semidirect
2. C2×C3×#3802C2semidirect
3. C2×#88988C2semidirect
4. C2×#88935C2semidirect
5. #11724306C2direct (N × H)
12. C2×#6295C22semidirect
14. C4⋊#1113C22semidirect
15. C2×C3×#625C22semidirect
17. C3×#3802C22semidirect
18. C2×C3⋊#622C22semidirect
19. #88988C22semidirect
20. #88935C22semidirect
23. C2×#3802D3semidirect
42. #6295C23semidirect
44. C3×#625C23semidirect
45. C3⋊#622C23semidirect
67. C2×#625C2×D3semidirect
71. #3802C2×D3semidirect
80. C2×C3×C3⋊D3D8semidirect
144. #625C22×D3semidirect
147. C2×C33C2×D8semidirect
150. C3×C3⋊D3C2×D8semidirect
191. C2×C3⋊D3#213semidirect
229. C33C22×D8semidirect
289. C2×C32C2×#213semidirect
292. C3⋊D3C2×#213semidirect
361. C32C22×#213semidirect
393. C2×C3#64225semidirect
417. C3C2×#64225semidirect
421. C2#11724306direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4⋊#11132
C2×C3×#38022
C2×#889882
C2×#889352
#117243062
C2×#642253
C22×#2139