Finite Group Databasebeta

← C2×#11724309 #13181078

Subgroups — C2×#11724309 #13181078

Order 1728.

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

normal subgroups only

This group has 928 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×#89040C211yesmaximal, characteristic, complemented
2C2×C3×#3798C211yesmaximal, characteristic
3C2×#88989C211yesmaximal, characteristic, complemented
4C2×#88938C212yesmaximal, complemented
5C2×#88937C212yesmaximal, complemented
6#11724309C218yesmaximal, direct product factor
7C2×#64226—31nomaximal
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C3×#634—21no—
10C2×C32×C4×D3—21no—
11C2×#6296C2212yes—
12C2×#6294—21no—
13#89040—22no—
14#89040C2214yescomplemented
15C2×C3×#627C2211yescharacteristic
16C2×C3×#626—21no—
17C2×C3×#625C2211yescharacteristic
18C3×#3798—22no—
19C3×#3798C2214yes—
20C2×C3⋊#622C2212yes—
21#88989C2214yescomplemented
22#88938C2218yescomplemented
23#88937C2218yescomplemented
24C2×C3×#235—61no—
25C2×#3798D311yescharacteristic
26C2×#3797—31no—
27C2×D3×Dic6—61no—
28C2×#3788—31no—
29C2×#3776—32no—
30C2×#3775—32no—
31#64226—38no—
32C22×C3×C3⋊D3D411yescharacteristic
33C22×C32×D3—21no—
34C2×C33×C4D411yescharacteristic
35C2×C3×C3⋊Dic3—22no—
36C2×C3×C3⋊Dic3C2311yescharacteristic
37C3×C4×C3⋊D3D412yes—
38C3×C4×C3⋊D3C2312yes—
39C3×#634—24no—
40C2×C32×Dic3—21no—
41C32×C4×D3—24no—
42#6296—22no—
43#6296C2314yes—
44#6294—24no—
45C3×#627—21no—
46C3×#627C2312yes—
47C3×#626—24no—
48C3×#625—21no—
49C3×#625C2312yes—
50C3⋊#622C2314yes—
51C2×#951—91nomaximal
52C22×C32×C4—61no—
53C2×C4×C3⋊D3C2×D311yescharacteristic
54C2×#634—21no—
55C2×C3×#67—61no—
56C2×C3×#67—61no—
57C3×#235—64no—
58C3×#235—64no—
59C2×C3×D12—61no—
60C2×C3×C4×D3—21no—
61C2×C3×C4×D3—21no—
62C2×C3×C4×D3—41no—
63C2×C3×C4×D3—61no—
64C2×C3×C4×D3—61no—
65C2×C3×Dic6—61no—
66C2×C3×Dic6—61no—
67C2×C3×Dic6—61no—
68C2×C3×Dic6—121no—
69C2×#627—32no—
70C2×#627—62no—
71C2×#627C2×D311yescharacteristic
72C2×#626—21no—
73C2×#625C2×D311yescharacteristic
74C2×#624—61no—
75C2×D3×Dic3—62no—
76#3798—22no—
77#3798C2×D314yes—
78#3797—34no—
79#3797—62no—
80D3×Dic6—64no—
81D3×Dic6—64no—
82C2×#622—32no—
83#3788—34no—
84#3776—38no—
85#3775—38no—
86C22×C33C2×D411yescharacteristic
87C2×C3×C3⋊D3—22no—
88C2×C3×C3⋊D3C2×D412yes—
89C2×C32×D3—22no—
90C2×C32×D3—42no—
91C33×C4C2×D412yes—
92C3×C3⋊Dic3—24no—
93C3×C3⋊Dic3C2×D411yescharacteristic
94C3×C3⋊Dic3C2411yescharacteristic
95C32×Dic3—22no—
96C2×C3×#41—91no—
97C2×D3×Q8—181no—
98C2×#235—61no—
99C22×Dic6—91no—
100#951—98no—
101C2×#216—92no—
102C2×#214—92no—
103C2×#208—91no—
104C23×C32—61no—
105C22×C3⋊D3#6711yescharacteristic
106C22×C3×D3—21no—
107C22×C3×D3—21no—
108C22×C3×D3—41no—
109C22×C3×D3—61no—
110C22×C3×D3—61no—
111C2×C32×C4—21no—
112C2×C32×C4—21no—
113C2×C32×C4—41no—
114C2×C32×C4—41no—
115C2×C32×C4—64no—
116C2×C32×C4—61no—
117C2×C32×C4#6711yescharacteristic
118C2×C3⋊Dic3—22no—
119C2×C3⋊Dic3C22×D311yescharacteristic
120C4×C3⋊D3#6712yes—
121C4×C3⋊D3C22×D312yes—
122#634—24no—
123C3×#67—62no—
124C3×#67—64no—
125C3×#67—62no—
126C2×C3×Dic3—21no—
127C2×C3×Dic3—21no—
128C2×C3×Dic3—41no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—62no—
132C2×C3×Dic3—62no—
133C2×C3×Dic3—122no—
134C3×D12—64no—
135C3×C4×D3—24no—
136C3×C4×D3—24no—
137C3×C4×D3—44no—
138C3×C4×D3—62no—
139C3×C4×D3—62no—
140C3×C4×D3—62no—
141C3×C4×D3—62no—
142C3×Dic6—62no—
143C3×Dic6—62no—
144C3×Dic6—64no—
145C3×Dic6—64no—
146C3×Dic6—124no—
147#627—21no—
148#627—34no—
149#627—64no—
150#627—64no—
151#627—62no—
152#627C22×D312yes—
153#626—24no—
154#625—21no—
155#625C22×D312yes—
156#624—64no—
157D3×Dic3—68no—
158#622—34no—
159C2×#137—271noSylow
160C2×C33#13712yescomplemented
161C2×C33C22×D411yescharacteristic
162C3×C3⋊D3—22no—
163C32×D3—42no—
164C3×#41—94no—
165C3×#41—182no—
166C2×C3×Q8—91no—
167C2×C3×Q8—181no—
168C2×C3×D4—91no—
169C2×C3×D4—181no—
170C22×C3×C4—61no—
171C22×C3×C4—61no—
172C22×C3×C4—91no—
173C22×C3×C4—121no—
174C2×#67—61no—
175C2×#67—61no—
176C22×Dic3—181no—
177D3×Q8—184no—
178D3×Q8—184no—
179#235—64no—
180#235—64no—
181C2×D12—61no—
182C2×C4×D3—21no—
183C2×C4×D3—61no—
184C2×C4×D3—61no—
185C2×C4×D3—182no—
186C2×Dic6—61no—
187C2×Dic6—61no—
188C2×Dic6—61no—
189C2×Dic6—94no—
190C2×Dic6—92no—
191C2×Dic6—121no—
192C2×Dic6—182no—
193C2×Dic6—182no—
194#216—98no—
195#214—98no—
196#208—94no—
197C2×C3⋊C8—92no—
198C22×C32—21no—
199C22×C32—21no—
200C22×C32—41no—
201C22×C32—41no—
202C22×C32—62no—
203C22×C32—122no—
204C22×C32C2×#6711yescharacteristic
205C2×C3⋊D3—22no—
206C2×C3⋊D3C2×#6712yes—
207C2×C3×D3—22no—
208C2×C3×D3—22no—
209C2×C3×D3—42no—
210C2×C3×D3—42no—
211C2×C3×D3—44no—
212C2×C3×D3—42no—
213C2×C3×D3—62no—
214C2×C3×D3—62no—
215C2×C3×D3—64no—
216C2×C3×D3—122no—
217C32×C4—22no—
218C32×C4—22no—
219C32×C4—42no—
220C32×C4—42no—
221C32×C4—62no—
222C32×C4C2×#6712yes—
223C3⋊Dic3—24no—
224C3⋊Dic3C2×#6711yescharacteristic
225C3⋊Dic3C23×D311yescharacteristic
226C3×Dic3—22no—
227C3×Dic3—22no—
228C3×Dic3—42no—
229C3×Dic3—61no—
230C3×Dic3—61no—
231C3×Dic3—61no—
232C3×Dic3—61no—
233C3×Dic3—64no—
234C3×Dic3—64no—
235C3×Dic3—124no—
236C2×#41—91no—
237C22×Q8—271no—
238#137—278no—
239C2×Q16—272no—
240C2×SD16—272no—
241C2×M16—271no—
242C33C2×#13711yescharacteristic, complemented, Sylow
243C23×C3—61no—
244C23×C3—61no—
245C23×C3—91no—
246C23×C3—121no—
247C22×D3—21no—
248C22×D3—61no—
249C22×D3—61no—
250C3×Q8—92no—
251C3×Q8—184no—
252C3×Q8—181no—
253C3×D4—92no—
254C3×D4—184no—
255C3×D4—181no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—64no—
261C2×C3×C4—61no—
262C2×C3×C4—61no—
263C2×C3×C4—64no—
264C2×C3×C4—91no—
265C2×C3×C4—92no—
266C2×C3×C4—92no—
267C2×C3×C4—124no—
268C2×C3×C4—121no—
269C2×C3×C4—182no—
270C2×C3×C4GO+(4,𝔽2)11yescharacteristic
271#67—62no—
272#67—64no—
273#67—62no—
274C2×Dic3—21no—
275C2×Dic3—61no—
276C2×Dic3—61no—
277C2×Dic3—62no—
278C2×Dic3—62no—
279C2×Dic3—122no—
280C2×Dic3—184no—
281D12—64no—
282C4×D3—24no—
283C4×D3—62no—
284C4×D3—62no—
285C4×D3—62no—
286C4×D3—62no—
287C4×D3—188no—
288Dic6—62no—
289Dic6—62no—
290Dic6—64no—
291Dic6—64no—
292Dic6—94no—
293Dic6—124no—
294Dic6—184no—
295Dic6—182no—
296Dic6—184no—
297C3⋊C8—94no—
298C2×C32—21no—
299C2×C32—21no—
300C2×C32—22no—
301C2×C32—22no—
302C2×C32—42no—
303C2×C32—42no—
304C2×C32—41no—
305C2×C32—41no—
306C2×C32—122no—
307C2×C32#95112yescomplemented
308C2×C32C22×#6711yescharacteristic
309C3⋊D3—22no—
310C3×D3—42no—
311C3×D3—44no—
312C3×D3—42no—
313C3×D3—64no—
314C3×D3—122no—
315#41—94no—
316#41—182no—
317C2×Q8—91no—
318C2×Q8—181no—
319C2×Q8—272no—
320C2×Q8—274no—
321C2×Q8—542no—
322C2×D4—91no—
323C2×D4—181no—
324C22×C4—61no—
325C22×C4—91no—
326C22×C4—541no—
327Q16—278no—
328SD16—278no—
329M16—274no—
330C2×C8—272no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—41no—
334C22×C3—41no—
335C22×C3—62no—
336C22×C3—62no—
337C22×C3—92no—
338C22×C3—122no—
339C22×C3—122no—
340C22×C3—124no—
341C22×C3—122no—
342C22×C3—182no—
343C22×C3C2×GO+(4,𝔽2)11yescharacteristic
344C2×D3—22no—
345C2×D3—42no—
346C2×D3—62no—
347C2×D3—62no—
348C2×D3—64no—
349C2×D3—122no—
350C3×C4—22no—
351C3×C4—22no—
352C3×C4—42no—
353C3×C4—42no—
354C3×C4—62no—
355C3×C4—62no—
356C3×C4—91no—
357C3×C4—91no—
358C3×C4—122no—
359C3×C4—184no—
360C3×C4C2×GO+(4,𝔽2)12yes—
361Dic3—22no—
362Dic3—61no—
363Dic3—61no—
364Dic3—61no—
365Dic3—61no—
366Dic3—64no—
367Dic3—64no—
368Dic3—124no—
369C32—21no—
370C32—21no—
371C32—41no—
372C32—41no—
373C32C2×#95111yescharacteristic, complemented
374C23—61no—
375C23—91no—
376Q8—92no—
377Q8—181no—
378Q8—184no—
379Q8—274no—
380Q8—542no—
381D4—92no—
382D4—184no—
383D4—181no—
384C2×C4—61no—
385C2×C4—64no—
386C2×C4—91no—
387C2×C4—92no—
388C2×C4—92no—
389C2×C4—182no—
390C2×C4—544no—
391C2×C4D32⋊D311yescharacteristic
392C8—274no—
393C2×C3—22no—
394C2×C3—22no—
395C2×C3—21no—
396C2×C3—21no—
397C2×C3—42no—
398C2×C3—42no—
399C2×C3—41no—
400C2×C3—41no—
401C2×C3—122no—
402C2×C3—124no—
403C2×C3—122no—
404C2×C3—182no—
405C2×C3#6422612yescomplemented
406C2×C3C22×GO+(4,𝔽2)11yescharacteristic
407D3—42no—
408D3—64no—
409D3—122no—
410C22—62no—
411C22—92no—
412C22—122no—
413C22—182no—
414C22C2×D32⋊D311yescharacteristic
415C4—62no—
416C4—91no—
417C4—91no—
418C4—184no—
419C4C2×D32⋊D312yes—
420C3—21no—
421C3—21no—
422C3—41no—
423C3—41no—
424C3C2×#6422611yescharacteristic, complemented
425C2—122no—
426C2—182no—
427C2#1172430912yesdirect product factor
428C2C22×D32⋊D311yescharacteristic
subgroupG/H
centerC22C2×D32⋊D3
commutator, cosocleC3×C3⋊Dic3C24
FrattiniC2C22×D32⋊D3
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×C3×#3798C2
hypercenterC2×C4D32⋊D3
hypocenterC33C2×#137
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×#89040C2semidirect
3. C2×#88989C2semidirect
4. C2×#88938C2semidirect
5. C2×#88937C2semidirect
6. #11724309C2direct (N × H)
14. #89040C22semidirect
21. #88989C22semidirect
22. #88938C22semidirect
23. #88937C22semidirect
160. C2×C33#137semidirect
242. C33C2×#137semidirect
307. C2×C32#951semidirect
373. C32C2×#951semidirect
405. C2×C3#64226semidirect
424. C3C2×#64226semidirect
427. C2#11724309direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890402
C2×C3×#37982
C2×#889892
C2×#889382
C2×#889372
#117243092
C2×#642263
C2×#9519