Finite Group Databasebeta

← C2×#11724305 #13181067

Subgroups — C2×#11724305 #13181067

Order 1728.

1104 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 1104 conjugacy classes of subgroups — too many to draw. Only its 127 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×#3798C211yesmaximal, characteristic, complemented
3C2×#88989C211yesmaximal, characteristic, complemented
4C2×#88936C212yesmaximal, complemented
5C2×#88935C212yesmaximal, complemented
6#11724305C218yesmaximal, direct product factor
7C2×#64224—31nomaximal
8C22×#1113—21no—
9C2×C3×C4⋊(C3⋊D3)—21no—
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C32×C4×D3—21no—
12C2×#6295C2212yes—
13C4⋊#1113—22no—
14C4⋊#1113C2214yescomplemented
15C2×C3×#627C2211yescharacteristic, complemented
16C2×C3×#626—21no—
17C2×C3×#625C2211yescharacteristic, complemented
18C3×#3798—22no—
19C3×#3798C2214yescomplemented
20C2×C3⋊#622C2212yescomplemented
21#88989C2214yescomplemented
22#88936C2218yescomplemented
23#88935C2218yescomplemented
24C2×C3×#235—61no—
25C2×#3802—31no—
26C2×D3×D12—61no—
27C2×#3798D311yescharacteristic, complemented
28C2×#3788—31no—
29C2×#3775—32no—
30C2×#3774—32no—
31#64224—38no—
32C22×C3×C3⋊D3—22no—
33C22×C3×C3⋊D3D411yescharacteristic
34C22×C32×D3—21no—
35C2×#1113—24no—
36C2×#1113—44no—
37C2×C33×C4D411yescharacteristic
38C2×C3×C3⋊Dic3C2311yescharacteristic
39C3×C4⋊(C3⋊D3)—24no—
40C3×C4×C3⋊D3D412yes—
41C3×C4×C3⋊D3C2312yes—
42C2×C32×Dic3—21no—
43C32×C4×D3—24no—
44#6295—22no—
45#6295C2314yes—
46C3×#627—21no—
47C3×#627C2312yescomplemented
48C3×#626—24no—
49C3×#625—21no—
50C3×#625C2312yescomplemented
51C3⋊#622C2314yescomplemented
52C2×#949—91nomaximal
53C22×D32—62no—
54C22×D32—61no—
55C22×C32×C4—61no—
56C2×C4⋊(C3⋊D3)—21no—
57C2×C4×C3⋊D3C2×D311yescharacteristic
58C2×C3×#67—61no—
59C2×C3×#67—61no—
60C3×#235—64no—
61C3×#235—64no—
62C2×C3×D12—61no—
63C2×C3×D12—61no—
64C2×C3×D12—61no—
65C2×C3×D12—121no—
66C2×C3×C4×D3—21no—
67C2×C3×C4×D3—21no—
68C2×C3×C4×D3—41no—
69C2×C3×C4×D3—61no—
70C2×C3×C4×D3—61no—
71C2×C3×Dic6—61no—
72C2×#627C2×D311yescharacteristic, complemented
73C2×#626—21no—
74C2×#626—62no—
75C2×#625—32no—
76C2×#625C2×D311yescharacteristic, complemented
77#3802—34no—
78#3802—62no—
79D3×D12—64no—
80D3×D12—64no—
81#3798—22no—
82#3798C2×D314yescomplemented
83C2×#622—32no—
84#3788—34no—
85#3775—38no—
86#3774—38no—
87C22×C33C2×D411yescharacteristic
88C2×C3×C3⋊D3—22no—
89C2×C3×C3⋊D3—24no—
90C2×C3×C3⋊D3—44no—
91C2×C3×C3⋊D3C2×D412yes—
92C2×C32×D3—22no—
93C2×C32×D3—42no—
94#1113—48no—
95C33×C4C2×D412yes—
96C3×C3⋊Dic3C2×D411yescharacteristic
97C3×C3⋊Dic3C2411yescharacteristic
98C32×Dic3—22no—
99C2×C3×#41—91no—
100C2×D3×D4—181no—
101C2×#235—61no—
102C22×D12—91no—
103#949—98no—
104C2×#215—92no—
105C2×#213—92no—
106C2×#208—91no—
107C23×C32—61no—
108C22×C3⋊D3—22no—
109C22×C3⋊D3#6711yescharacteristic
110C22×C3×D3—21no—
111C22×C3×D3—21no—
112C22×C3×D3—41no—
113C22×C3×D3—61no—
114C22×C3×D3—61no—
115C22×C3×D3—62no—
116C22×C3×D3—62no—
117C22×C3×D3—122no—
118C2×D32—68no—
119C2×D32—64no—
120C2×D32—128no—
121C2×D32—124no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—41no—
125C2×C32×C4—41no—
126C2×C32×C4—64no—
127C2×C32×C4—61no—
128C2×C32×C4#6711yescharacteristic
129C2×C3⋊Dic3C22×D311yescharacteristic
130C4⋊(C3⋊D3)—24no—
131C4×C3⋊D3#6712yes—
132C4×C3⋊D3C22×D312yes—
133C3×#67—62no—
134C3×#67—64no—
135C3×#67—62no—
136C2×C3×Dic3—21no—
137C2×C3×Dic3—21no—
138C2×C3×Dic3—41no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C3×D12—64no—
142C3×D12—64no—
143C3×D12—64no—
144C3×D12—124no—
145C3×C4×D3—24no—
146C3×C4×D3—24no—
147C3×C4×D3—44no—
148C3×C4×D3—62no—
149C3×C4×D3—62no—
150C3×C4×D3—62no—
151C3×C4×D3—62no—
152C3×Dic6—62no—
153C3×Dic6—62no—
154#627—21no—
155#627C22×D312yescomplemented
156#626—24no—
157#626—64no—
158#626—64no—
159#625—21no—
160#625—34no—
161#625—62no—
162#625C22×D312yescomplemented
163#622—34no—
164C2×AGL(1,ℤ/8)—271noSylow
165C2×C33AGL(1,ℤ/8)12yescomplemented
166C2×C33C22×D411yescharacteristic
167C3×C3⋊D3—22no—
168C3×C3⋊D3—44no—
169C32×D3—42no—
170C23×D3—181no—
171C23×D3—182no—
172C3×#41—94no—
173C3×#41—182no—
174C2×C3×Q8—91no—
175C2×C3×D4—91no—
176C2×C3×D4—181no—
177C2×C3×D4—181no—
178C22×C3×C4—61no—
179C22×C3×C4—61no—
180C22×C3×C4—91no—
181C22×C3×C4—121no—
182C2×#67—61no—
183C2×#67—61no—
184C2×#67—182no—
185D3×D4—184no—
186D3×D4—184no—
187#235—64no—
188#235—64no—
189C2×D12—61no—
190C2×D12—61no—
191C2×D12—61no—
192C2×D12—94no—
193C2×D12—92no—
194C2×D12—121no—
195C2×D12—182no—
196C2×C4×D3—21no—
197C2×C4×D3—61no—
198C2×C4×D3—61no—
199C2×Dic6—61no—
200#215—98no—
201#213—98no—
202#208—94no—
203C2×C3⋊C8—92no—
204C22×C32—21no—
205C22×C32—21no—
206C22×C32—41no—
207C22×C32—41no—
208C22×C32—62no—
209C22×C32—122no—
210C22×C32C2×#6711yescharacteristic
211C2×C3⋊D3—22no—
212C2×C3⋊D3—24no—
213C2×C3⋊D3—44no—
214C2×C3⋊D3C2×#6712yes—
215C2×C3×D3—22no—
216C2×C3×D3—22no—
217C2×C3×D3—42no—
218C2×C3×D3—42no—
219C2×C3×D3—44no—
220C2×C3×D3—42no—
221C2×C3×D3—62no—
222C2×C3×D3—62no—
223C2×C3×D3—64no—
224C2×C3×D3—64no—
225C2×C3×D3—64no—
226C2×C3×D3—122no—
227C2×C3×D3—124no—
228C2×C3×D3—124no—
229C2×C3×D3—124no—
230C2×C3×D3—244no—
231D32—1216no—
232D32—128no—
233C32×C4—22no—
234C32×C4—22no—
235C32×C4—42no—
236C32×C4—42no—
237C32×C4—62no—
238C32×C4C2×#6712yes—
239C3⋊Dic3C2×#6711yescharacteristic
240C3⋊Dic3C23×D311yescharacteristic
241C3×Dic3—22no—
242C3×Dic3—22no—
243C3×Dic3—42no—
244C3×Dic3—61no—
245C3×Dic3—61no—
246C3×Dic3—61no—
247C3×Dic3—61no—
248C2×#41—91no—
249C22×D4—271no—
250AGL(1,ℤ/8)—278no—
251C2×SD16—272no—
252C2×D8—272no—
253C2×M16—271no—
254C33C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
255C23×C3—61no—
256C23×C3—61no—
257C23×C3—91no—
258C23×C3—121no—
259C23×C3—182no—
260C22×D3—21no—
261C22×D3—61no—
262C22×D3—61no—
263C22×D3—62no—
264C22×D3—62no—
265C22×D3—122no—
266C22×D3—184no—
267C22×D3—188no—
268C22×D3—368no—
269C22×D3—364no—
270C3×Q8—92no—
271C3×Q8—181no—
272C3×D4—92no—
273C3×D4—184no—
274C3×D4—184no—
275C3×D4—181no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—64no—
281C2×C3×C4—61no—
282C2×C3×C4—61no—
283C2×C3×C4—64no—
284C2×C3×C4—91no—
285C2×C3×C4—92no—
286C2×C3×C4—92no—
287C2×C3×C4—124no—
288C2×C3×C4—121no—
289C2×C3×C4GO+(4,𝔽2)11yescharacteristic
290#67—62no—
291#67—64no—
292#67—62no—
293#67—184no—
294#67—184no—
295C2×Dic3—21no—
296C2×Dic3—61no—
297C2×Dic3—61no—
298D12—64no—
299D12—64no—
300D12—64no—
301D12—94no—
302D12—124no—
303D12—182no—
304C4×D3—24no—
305C4×D3—62no—
306C4×D3—62no—
307C4×D3—62no—
308C4×D3—62no—
309Dic6—62no—
310Dic6—62no—
311C3⋊C8—94no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32—22no—
315C2×C32—22no—
316C2×C32—42no—
317C2×C32—42no—
318C2×C32—41no—
319C2×C32—41no—
320C2×C32—122no—
321C2×C32#94912yescomplemented
322C2×C32C22×#6711yescharacteristic
323C3⋊D3—22no—
324C3⋊D3—44no—
325C3×D3—42no—
326C3×D3—44no—
327C3×D3—42no—
328C3×D3—64no—
329C3×D3—122no—
330C3×D3—124no—
331C3×D3—124no—
332C3×D3—244no—
333C24—541no—
334#41—94no—
335#41—182no—
336C2×Q8—91no—
337C2×D4—91no—
338C2×D4—181no—
339C2×D4—181no—
340C2×D4—274no—
341C2×D4—272no—
342C2×D4—542no—
343C22×C4—61no—
344C22×C4—91no—
345SD16—278no—
346D8—278no—
347M16—274no—
348C2×C8—272no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—41no—
352C22×C3—41no—
353C22×C3—62no—
354C22×C3—62no—
355C22×C3—92no—
356C22×C3—122no—
357C22×C3—122no—
358C22×C3—124no—
359C22×C3—122no—
360C22×C3—184no—
361C22×C3—182no—
362C22×C3—364no—
363C22×C3C2×GO+(4,𝔽2)11yescharacteristic
364C2×D3—22no—
365C2×D3—42no—
366C2×D3—62no—
367C2×D3—62no—
368C2×D3—64no—
369C2×D3—64no—
370C2×D3—64no—
371C2×D3—122no—
372C2×D3—124no—
373C2×D3—124no—
374C2×D3—124no—
375C2×D3—244no—
376C2×D3—3616no—
377C2×D3—368no—
378C3×C4—22no—
379C3×C4—22no—
380C3×C4—42no—
381C3×C4—42no—
382C3×C4—62no—
383C3×C4—62no—
384C3×C4—91no—
385C3×C4—91no—
386C3×C4—122no—
387C3×C4C2×GO+(4,𝔽2)12yes—
388Dic3—22no—
389Dic3—61no—
390Dic3—61no—
391Dic3—61no—
392Dic3—61no—
393C32—21no—
394C32—21no—
395C32—41no—
396C32—41no—
397C32C2×#94911yescharacteristic, complemented
398C23—61no—
399C23—91no—
400C23—182no—
401C23—544no—
402C23—1084no—
403Q8—92no—
404Q8—181no—
405D4—92no—
406D4—184no—
407D4—181no—
408D4—184no—
409D4—274no—
410D4—542no—
411C2×C4—61no—
412C2×C4—64no—
413C2×C4—91no—
414C2×C4—92no—
415C2×C4—92no—
416C2×C4D32⋊D311yescharacteristic
417C8—274no—
418C2×C3—22no—
419C2×C3—22no—
420C2×C3—21no—
421C2×C3—21no—
422C2×C3—42no—
423C2×C3—42no—
424C2×C3—41no—
425C2×C3—41no—
426C2×C3—122no—
427C2×C3—124no—
428C2×C3—122no—
429C2×C3—182no—
430C2×C3—364no—
431C2×C3#6422412yescomplemented
432C2×C3C22×GO+(4,𝔽2)11yescharacteristic
433D3—42no—
434D3—64no—
435D3—122no—
436D3—124no—
437D3—124no—
438D3—244no—
439C22—62no—
440C22—92no—
441C22—122no—
442C22—182no—
443C22—184no—
444C22—364no—
445C22—1088no—
446C22C2×D32⋊D311yescharacteristic
447C4—62no—
448C4—91no—
449C4—91no—
450C4C2×D32⋊D312yes—
451C3—21no—
452C3—21no—
453C3—41no—
454C3—41no—
455C3C2×#6422411yescharacteristic, complemented
456C2—122no—
457C2—182no—
458C2—364no—
459C2#1172430512yesdirect product factor
460C2C22×D32⋊D311yescharacteristic
subgroupG/H
centerC22C2×D32⋊D3
commutator, cosocleC3×C3⋊Dic3C24
FrattiniC2C22×D32⋊D3
FittingC2×C33×C4D4
socleC22×C33C2×D4
hypercenterC2×C4D32⋊D3
hypocenterC33C2×AGL(1,ℤ/8)
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×#3798C2semidirect
3. C2×#88989C2semidirect
4. C2×#88936C2semidirect
5. C2×#88935C2semidirect
6. #11724305C2direct (N × H)
14. C4⋊#1113C22semidirect
15. C2×C3×#627C22semidirect
17. C2×C3×#625C22semidirect
19. C3×#3798C22semidirect
20. C2×C3⋊#622C22semidirect
21. #88989C22semidirect
22. #88936C22semidirect
23. #88935C22semidirect
27. C2×#3798D3semidirect
47. C3×#627C23semidirect
50. C3×#625C23semidirect
51. C3⋊#622C23semidirect
72. C2×#627C2×D3semidirect
76. C2×#625C2×D3semidirect
82. #3798C2×D3semidirect
155. #627C22×D3semidirect
162. #625C22×D3semidirect
165. C2×C33AGL(1,ℤ/8)semidirect
254. C33C2×AGL(1,ℤ/8)semidirect
321. C2×C32#949semidirect
397. C32C2×#949semidirect
431. C2×C3#64224semidirect
455. C3C2×#64224semidirect
459. C2#11724305direct (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×#37982
C2×#889892
C2×#889362
C2×#889352
#117243052
C2×#642243
C2×#9499