Finite Group Databasebeta

← C2×#11724124 #13180563

Subgroups — C2×#11724124 #13180563

Order 1728.

1576 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 1576 conjugacy classes of subgroups — too many to draw. Only its 156 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×#157⋊D3C211yesmaximal, characteristic, complemented
2C22×D32⋊D3C212yesmaximal, complemented
3C22×C3×GO+(4,𝔽2)C211yesmaximal, characteristic, complemented
4C2×#88953C212yesmaximal, complemented
5C2×Dic3×#157C211yesmaximal, characteristic, complemented
6#11724124C218yesmaximal, direct product factor
7C2×#64231—31nomaximal
8C22×#1113C2212yes—
9C22×C3×D32C2212yes—
10C22×C3⋊#157C2211yescharacteristic
11C22×C3×#157C2211yescharacteristic
12C2×#157⋊D3—24no—
13C2×#157⋊D3C2214yescomplemented
14C2×D32⋊D3—28no—
15C2×D32⋊D3C2218yescomplemented
16C2×C3×GO+(4,𝔽2)—24no—
17C2×C3×GO+(4,𝔽2)C2214yescomplemented
18C2×#6290—22no—
19C2×Dic3×C3⋊D3C2211yescharacteristic
20C2×C3×#626—22no—
21#88953—24no—
22#88953C2218yescomplemented
23Dic3×#157C2214yescomplemented
24C22×GO+(4,𝔽2)—32no—
25C22×GO+(4,𝔽2)—31no—
26C22×GO+(4,𝔽2)D311yescharacteristic, complemented
27C2×D3×#67—62no—
28C2×#3802—32no—
29C2×C4×#157—31no—
30#64231—38no—
31C22×C3×C3⋊D3—22no—
32C22×C3×C3⋊D3C2311yescharacteristic
33C22×C32×D3—22no—
34C2×#1113—24no—
35C2×#1113—24no—
36C2×#1113—22no—
37C2×#1113C2314yes—
38C2×C3×D32—24no—
39C2×C3×D32—24no—
40C2×C3×D32—22no—
41C2×C3×D32C2314yes—
42C2×C3⋊#157D414yescomplemented
43C2×C3⋊#157C2312yes—
44C2×C3×#157D414yescomplemented
45C2×C3×#157C2312yes—
46#157⋊D3—28no—
47D32⋊D3—216no—
48C3×GO+(4,𝔽2)—28no—
49C2×C3⋊(C3⋊Dic3)D411yescharacteristic, complemented
50C2×C32×Dic3D411yescharacteristic, complemented
51#6290—28no—
52Dic3×C3⋊D3D412yescomplemented
53Dic3×C3⋊D3C2312yes—
54C3×#626—28no—
55C2×#940—91nomaximal
56C23×C3×D3—62no—
57C22×D32—32no—
58C22×D32—62no—
59C22×D32C2×D312yes—
60C22×#157—31no—
61C22×#157C2×D311yescharacteristic
62C2×GO+(4,𝔽2)—24no—
63C2×GO+(4,𝔽2)—38no—
64C2×GO+(4,𝔽2)—34no—
65C2×GO+(4,𝔽2)—68no—
66C2×GO+(4,𝔽2)—64no—
67C2×GO+(4,𝔽2)C2×D314yescomplemented
68C2×#638—62no—
69C2×C4×C3⋊D3—31no—
70C2×C3×#67—62no—
71C2×C3×D12—62no—
72D3×#67—68no—
73D3×#67—68no—
74C2×#626—22no—
75C2×#625—62no—
76C2×#625—62no—
77C2×#625—122no—
78C2×D3×Dic3—62no—
79#3802—38no—
80#3802—64no—
81C4×#157—34no—
82C22×C33C2×D411yescharacteristic
83C2×C3×C3⋊D3—24no—
84C2×C3×C3⋊D3—44no—
85C2×C3×C3⋊D3C2×D411yescharacteristic
86C2×C3×C3⋊D3C2×D412yes—
87C2×C3×C3⋊D3C2×D412yes—
88C2×C3×C3⋊D3C2411yescharacteristic
89C2×C32×D3—24no—
90C2×C32×D3—44no—
91#1113—24no—
92#1113—24no—
93#1113—44no—
94C3×D32—24no—
95C3×D32—24no—
96C3×D32—44no—
97C3⋊#157C2×D414yescomplemented
98C3×#157C2×D414yescomplemented
99C3⋊(C3⋊Dic3)C2×D412yescomplemented
100C32×Dic3C2×D412yescomplemented
101C22×C3×D4—91no—
102C22×#67—92no—
103C22×#67—92no—
104C2×D3×D4—182no—
105C22×D12—91no—
106#940—98no—
107C2×C4×Dic3—91no—
108C23×C32—62no—
109C22×C3⋊D3—22no—
110C22×C3⋊D3C22×D311yescharacteristic
111C22×C3×D3—22no—
112C22×C3×D3—22no—
113C22×C3×D3—42no—
114C22×C3×D3—64no—
115C22×C3×D3—62no—
116C22×C3×D3—64no—
117C22×C3×D3—62no—
118C22×C3×D3—62no—
119C22×C3×D3—124no—
120C22×C3×D3—124no—
121C22×C3×D3—122no—
122C2×D32—22no—
123C2×D32—24no—
124C2×D32—24no—
125C2×D32—34no—
126C2×D32—64no—
127C2×D32—64no—
128C2×D32—62no—
129C2×D32—64no—
130C2×D32—64no—
131C2×D32—124no—
132C2×D32—124no—
133C2×D32C22×D314yes—
134C2×#157—34no—
135C2×#157—32no—
136C2×#157#6714yescomplemented
137C2×#157C22×D312yes—
138GO+(4,𝔽2)—28no—
139GO+(4,𝔽2)—616no—
140GO+(4,𝔽2)—68no—
141C2×C32×C4—31no—
142#638—68no—
143C2×C3⋊Dic3—31no—
144C2×C3⋊Dic3—62no—
145C2×C3⋊Dic3—122no—
146C4×C3⋊D3—32no—
147C4×C3⋊D3—32no—
148C3×#67—68no—
149C2×C3×Dic3—22no—
150C3×D12—68no—
151#626—28no—
152#625—68no—
153#625—68no—
154#625—128no—
155D3×Dic3—64no—
156D3×Dic3—64no—
157C2×#127—271noSylow
158C2×C33#12712yescomplemented
159C2×C33C22×D411yescharacteristic
160C3×C3⋊D3—44no—
161C3×C3⋊D3#12712yescomplemented
162C3×C3⋊D3C22×D411yescharacteristic
163C3×C3⋊D3C22×D411yescharacteristic
164C32×D3—44no—
165C24×C3—92no—
166C23×D3—62no—
167C23×D3—92no—
168C23×D3—182no—
169C2×C3×D4—94no—
170C2×C3×D4—182no—
171C2×C3×D4—184no—
172C22×C3×C4—91no—
173C2×#67—62no—
174C2×#67—98no—
175C2×#67—98no—
176C2×#67—184no—
177C2×#67—182no—
178C2×#67—188no—
179C2×#67—182no—
180C2×#67—182no—
181C2×#67—362no—
182C22×Dic3—91no—
183C22×Dic3—91no—
184D3×D4—188no—
185D3×D4—188no—
186C2×D12—62no—
187C2×D12—94no—
188C2×D12—184no—
189C2×C4×D3—182no—
190C4×Dic3—94no—
191C22×C32—22no—
192C22×C32—42no—
193C22×C32—64no—
194C22×C32—124no—
195C22×C32D3×D411yescharacteristic
196C2×C3⋊D3—24no—
197C2×C3⋊D3—44no—
198C2×C3⋊D3D3×D411yescharacteristic
199C2×C3⋊D3D3×D412yes—
200C2×C3⋊D3C2×#6712yes—
201C2×C3⋊D3C23×D311yescharacteristic
202C2×C3×D3—24no—
203C2×C3×D3—24no—
204C2×C3×D3—44no—
205C2×C3×D3—44no—
206C2×C3×D3—44no—
207C2×C3×D3—62no—
208C2×C3×D3—64no—
209C2×C3×D3—64no—
210C2×C3×D3—62no—
211C2×C3×D3—64no—
212C2×C3×D3—64no—
213C2×C3×D3—84no—
214C2×C3×D3—124no—
215C2×C3×D3—124no—
216C2×C3×D3—124no—
217C2×C3×D3—128no—
218C2×C3×D3—124no—
219C2×C3×D3—124no—
220C2×C3×D3—244no—
221D32—24no—
222D32—24no—
223D32—44no—
224D32—64no—
225D32—64no—
226D32—124no—
227D32—124no—
228D32—128no—
229D32—124no—
230#157—34no—
231#157C2×#6714yescomplemented
232C32×C4—32no—
233C3⋊Dic3—32no—
234C3⋊Dic3—64no—
235C3⋊Dic3—124no—
236C3×Dic3—24no—
237C22×D4—91no—
238C22×D4—271no—
239C22×D4—272no—
240C22×D4—272no—
241#127—278no—
242C2×C42—271no—
243C33C2×#12711yescharacteristic, complemented, Sylow
244C23×C3—62no—
245C23×C3—62no—
246C23×C3—94no—
247C23×C3—91no—
248C23×C3—122no—
249C23×C3—182no—
250C23×C3—182no—
251C23×C3—184no—
252C23×C3—184no—
253C22×D3—22no—
254C22×D3—64no—
255C22×D3—62no—
256C22×D3—64no—
257C22×D3—62no—
258C22×D3—62no—
259C22×D3—94no—
260C22×D3—124no—
261C22×D3—124no—
262C22×D3—122no—
263C22×D3—184no—
264C22×D3—182no—
265C22×D3—184no—
266C22×D3—184no—
267C22×D3—184no—
268C22×D3—364no—
269C22×D3—364no—
270C3×D4—188no—
271C3×D4—188no—
272C2×C3×C4—62no—
273C2×C3×C4—94no—
274C2×C3×C4—92no—
275#67—68no—
276#67—188no—
277#67—1816no—
278#67—188no—
279#67—188no—
280#67—368no—
281C2×Dic3—94no—
282C2×Dic3—92no—
283C2×Dic3—91no—
284C2×Dic3—92no—
285C2×Dic3—92no—
286C2×Dic3—182no—
287C2×Dic3—362no—
288C2×Dic3GO+(4,𝔽2)11yescharacteristic, complemented
289D12—68no—
290D12—188no—
291C4×D3—184no—
292C4×D3—184no—
293C2×C32—22no—
294C2×C32—24no—
295C2×C32—44no—
296C2×C32—42no—
297C2×C32—124no—
298C2×C32#94012yescomplemented
299C2×C32C2×D3×D411yescharacteristic
300C3⋊D3—44no—
301C3⋊D3#94012yescomplemented
302C3⋊D3C2×D3×D411yescharacteristic
303C3⋊D3C22×#6711yescharacteristic
304C3×D3—44no—
305C3×D3—44no—
306C3×D3—64no—
307C3×D3—62no—
308C3×D3—62no—
309C3×D3—84no—
310C3×D3—124no—
311C3×D3—124no—
312C3×D3—244no—
313C24—92no—
314C24—272no—
315C2×D4—94no—
316C2×D4—182no—
317C2×D4—184no—
318C2×D4—274no—
319C2×D4—278no—
320C2×D4—278no—
321C2×D4—544no—
322C2×D4—544no—
323C2×D4—542no—
324C2×D4—548no—
325C22×C4—91no—
326C22×C4—271no—
327C22×C4—271no—
328C42—274no—
329C22×C3—22no—
330C22×C3—42no—
331C22×C3—64no—
332C22×C3—64no—
333C22×C3—91no—
334C22×C3—91no—
335C22×C3—92no—
336C22×C3—92no—
337C22×C3—124no—
338C22×C3—124no—
339C22×C3—124no—
340C22×C3—184no—
341C22×C3—184no—
342C22×C3—184no—
343C22×C3—244no—
344C22×C3—364no—
345C22×C3—364no—
346C22×C3C2×GO+(4,𝔽2)11yescharacteristic
347C2×D3—24no—
348C2×D3—44no—
349C2×D3—62no—
350C2×D3—62no—
351C2×D3—64no—
352C2×D3—64no—
353C2×D3—64no—
354C2×D3—64no—
355C2×D3—124no—
356C2×D3—124no—
357C2×D3—124no—
358C2×D3—128no—
359C2×D3—124no—
360C2×D3—124no—
361C2×D3—184no—
362C2×D3—184no—
363C2×D3—244no—
364C2×D3—364no—
365C2×D3—364no—
366C2×D3—368no—
367C2×D3—364no—
368C3×C4—64no—
369C3×C4—94no—
370Dic3—94no—
371Dic3—92no—
372Dic3—184no—
373Dic3—364no—
374Dic3C2×GO+(4,𝔽2)12yescomplemented
375C32—22no—
376C32—42no—
377C32C2×#94011yescharacteristic, complemented
378C23—62no—
379C23—94no—
380C23—91no—
381C23—182no—
382C23—184no—
383C23—184no—
384C23—182no—
385C23—274no—
386C23—542no—
387C23—544no—
388C23—544no—
389D4—188no—
390D4—188no—
391D4—548no—
392D4—548no—
393D4—5416no—
394C2×C4—31no—
395C2×C4—92no—
396C2×C4—94no—
397C2×C4—272no—
398C2×C4—272no—
399C2×C4—272no—
400C2×C4—274no—
401C2×C4—271no—
402C2×C3—24no—
403C2×C3—22no—
404C2×C3—44no—
405C2×C3—42no—
406C2×C3—91no—
407C2×C3—91no—
408C2×C3—92no—
409C2×C3—124no—
410C2×C3—124no—
411C2×C3—244no—
412C2×C3—364no—
413C2×C3#6423112yescomplemented
414C2×C3C22×GO+(4,𝔽2)11yescharacteristic
415D3—44no—
416D3—64no—
417D3—62no—
418D3—62no—
419D3—124no—
420D3—124no—
421D3—244no—
422C22—64no—
423C22—92no—
424C22—92no—
425C22—91no—
426C22—91no—
427C22—124no—
428C22—184no—
429C22—184no—
430C22—184no—
431C22—364no—
432C22—364no—
433C22—544no—
434C22—544no—
435C22—1084no—
436C22D3×GO+(4,𝔽2)11yescharacteristic
437C4—32no—
438C4—94no—
439C4—274no—
440C4—272no—
441C3—22no—
442C3—42no—
443C3C2×#6423111yescharacteristic, complemented
444C2—91no—
445C2—91no—
446C2—92no—
447C2—124no—
448C2—364no—
449C2#1172412412yesdirect product factor
450C2C2×D3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
center, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊D3C24
FrattiniC2C2×D3×GO+(4,𝔽2)
Fitting, socleC22×C33C2×D4
hypocenterC33C2×#127
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×#157⋊D3C2semidirect
2. C22×D32⋊D3C2semidirect
3. C22×C3×GO+(4,𝔽2)C2semidirect
4. C2×#88953C2semidirect
5. C2×Dic3×#157C2semidirect
6. #11724124C2direct (N × H)
13. C2×#157⋊D3C22semidirect
15. C2×D32⋊D3C22semidirect
17. C2×C3×GO+(4,𝔽2)C22semidirect
22. #88953C22semidirect
23. Dic3×#157C22semidirect
26. C22×GO+(4,𝔽2)D3semidirect
42. C2×C3⋊#157D4semidirect
44. C2×C3×#157D4semidirect
49. C2×C3⋊(C3⋊Dic3)D4semidirect
50. C2×C32×Dic3D4semidirect
52. Dic3×C3⋊D3D4semidirect
67. C2×GO+(4,𝔽2)C2×D3semidirect
97. C3⋊#157C2×D4semidirect
98. C3×#157C2×D4semidirect
99. C3⋊(C3⋊Dic3)C2×D4semidirect
100. C32×Dic3C2×D4semidirect
136. C2×#157#67semidirect
158. C2×C33#127semidirect
161. C3×C3⋊D3#127semidirect
231. #157C2×#67semidirect
243. C33C2×#127semidirect
288. C2×Dic3GO+(4,𝔽2)semidirect
298. C2×C32#940semidirect
301. C3⋊D3#940semidirect
374. Dic3C2×GO+(4,𝔽2)semidirect
377. C32C2×#940semidirect
413. C2×C3#64231semidirect
443. C3C2×#64231semidirect
449. C2#11724124direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#157⋊D32
C22×D32⋊D32
C22×C3×GO+(4,𝔽2)2
C2×#889532
C2×Dic3×#1572
#117241242
C2×#642313
C2×#9409