Finite Group Databasebeta

← C2×#11724307 #13181075

Subgroups — C2×#11724307 #13181075

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×#89042C211yesmaximal, characteristic
2C2×C3×#3797C211yesmaximal, characteristic, complemented
3C2×#88989C211yesmaximal, characteristic, complemented
4C2×#88938C212yesmaximal, complemented
5C2×#88936C212yesmaximal, complemented
6#11724307C218yesmaximal, direct product factor
7C2×#64226—31nomaximal
8C2×C3×C4×C3⋊D3—21no—
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C32×Dic6—21no—
11C2×#6296C2211yescharacteristic
12C2×#6295—21no—
13C2×#6295C2211yescharacteristic
14#89042—22no—
15#89042C2214yes—
16C2×C3×#627C2212yes—
17C2×C3×#624—21no—
18C3×#3797—22no—
19C3×#3797C2214yescomplemented
20C2×C3⋊#622C2212yes—
21#88989C2214yescomplemented
22#88938C2218yescomplemented
23#88936C2218yescomplemented
24C2×C3×D3×Q8—61no—
25C2×#3798—31no—
26C2×#3798—61no—
27C2×#3797D311yescharacteristic, complemented
28C2×#3788—31no—
29C2×#3776—32no—
30C2×#3775—32no—
31#64226—38no—
32C22×C3×C3⋊D3—21no—
33C22×C3×C3⋊D3D411yescharacteristic
34C2×C33×C4D411yescharacteristic
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3C2311yescharacteristic
37C3×C4×C3⋊D3—24no—
38C3×C4×C3⋊D3D412yes—
39C3×C4×C3⋊D3C2312yes—
40C2×C32×Dic3—22no—
41C32×Dic6—24no—
42#6296—21no—
43#6296C2312yes—
44#6295—24no—
45#6295—21no—
46#6295C2312yes—
47C3×#627—22no—
48C3×#627C2314yes—
49C3×#624—24no—
50C3⋊#622C2314yes—
51C2×#942—91nomaximal
52C2×C32×Q8—61no—
53C2×C4×C3⋊D3—21no—
54C2×C4×C3⋊D3C2×D311yescharacteristic
55C3×D3×Q8—64no—
56C3×D3×Q8—64no—
57C2×C3×C4×D3—62no—
58C2×C3×C4×D3—61no—
59C2×C3×C4×D3—61no—
60C2×C3×C4×D3—61no—
61C2×C3×C4×D3—61no—
62C2×C3×C4×D3—121no—
63C2×C3×Dic6—21no—
64C2×C3×Dic6—21no—
65C2×C3×Dic6—41no—
66C2×C3×Dic6—62no—
67C2×#627—31no—
68C2×#627—61no—
69C2×#627C2×D312yes—
70C2×#626—61no—
71C2×#626—61no—
72C2×#626—61no—
73C2×#625—31no—
74C2×#625—61no—
75C2×#624—21no—
76#3798—34no—
77#3798—64no—
78#3798—64no—
79#3798—62no—
80#3797—22no—
81#3797C2×D314yescomplemented
82C2×#622—32no—
83#3788—34no—
84#3776—38no—
85#3775—38no—
86C22×C33C2×D411yescharacteristic
87C2×C3×C3⋊D3—22no—
88C2×C3×C3⋊D3—22no—
89C2×C3×C3⋊D3—42no—
90C2×C3×C3⋊D3C2×D412yes—
91C33×C4C2×D412yes—
92C3×C3⋊Dic3—22no—
93C3×C3⋊Dic3C2×D411yescharacteristic
94C3×C3⋊Dic3C2411yescharacteristic
95C32×Dic3—24no—
96C22×C3×Q8—91no—
97C2×D3×Q8—61no—
98C2×#235—91no—
99C2×#235—181no—
100C2×#216—92no—
101#942—98no—
102C2×#215—92no—
103C2×#208—91no—
104C22×C3⋊D3—21no—
105C22×C3⋊D3#6711yescharacteristic
106C22×C3×D3—61no—
107C22×C3×D3—61no—
108C22×C3×D3—61no—
109C22×C3×D3—61no—
110C22×C3×D3—121no—
111C32×Q8—64no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—41no—
115C2×C32×C4—41no—
116C2×C32×C4—62no—
117C2×C32×C4#6711yescharacteristic
118C2×C3⋊Dic3—21no—
119C2×C3⋊Dic3C22×D311yescharacteristic
120C4×C3⋊D3—24no—
121C4×C3⋊D3#6712yes—
122C4×C3⋊D3C22×D312yes—
123C2×C3×Dic3—22no—
124C2×C3×Dic3—22no—
125C2×C3×Dic3—42no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—121no—
131C3×C4×D3—62no—
132C3×C4×D3—62no—
133C3×C4×D3—62no—
134C3×C4×D3—62no—
135C3×C4×D3—68no—
136C3×C4×D3—64no—
137C3×C4×D3—64no—
138C3×C4×D3—124no—
139C3×Dic6—24no—
140C3×Dic6—24no—
141C3×Dic6—44no—
142C3×Dic6—64no—
143C3×Dic6—64no—
144#627—22no—
145#627—32no—
146#627—62no—
147#627—62no—
148#627—61no—
149#627C22×D314yes—
150#626—62no—
151#626—62no—
152#626—64no—
153#626—64no—
154#625—32no—
155#625—64no—
156#625—61no—
157#624—24no—
158#622—34no—
159C2×#137—271noSylow
160C2×C33#13712yescomplemented
161C2×C33C22×D411yescharacteristic
162C3×C3⋊D3—22no—
163C3×C3⋊D3—42no—
164C2×C3×Q8—61no—
165C2×C3×Q8—61no—
166C2×C3×Q8—92no—
167C2×C3×Q8—94no—
168C2×C3×Q8—121no—
169C2×C3×Q8—182no—
170C22×C3×C4—91no—
171C22×C3×C4—181no—
172C22×C3×C4—181no—
173C2×#67—181no—
174C2×#67—181no—
175C2×#67—181no—
176D3×Q8—64no—
177D3×Q8—64no—
178#235—94no—
179#235—184no—
180#235—182no—
181#235—184no—
182C2×D12—91no—
183C2×D12—181no—
184C2×C4×D3—61no—
185C2×C4×D3—61no—
186C2×C4×D3—62no—
187C2×C4×D3—61no—
188C2×C4×D3—61no—
189C2×C4×D3—121no—
190C2×Dic6—21no—
191C2×Dic6—62no—
192C2×Dic6—91no—
193C2×Dic6—181no—
194#216—98no—
195#215—98no—
196#208—94no—
197C2×C3⋊C8—92no—
198C22×C32—21no—
199C22×C32—21no—
200C22×C32—41no—
201C22×C32—41no—
202C22×C32C2×#6711yescharacteristic
203C2×C3⋊D3—22no—
204C2×C3⋊D3—22no—
205C2×C3⋊D3—42no—
206C2×C3⋊D3C2×#6712yes—
207C2×C3×D3—62no—
208C2×C3×D3—62no—
209C2×C3×D3—64no—
210C2×C3×D3—62no—
211C2×C3×D3—62no—
212C2×C3×D3—122no—
213C2×C3×D3—122no—
214C2×C3×D3—122no—
215C2×C3×D3—124no—
216C2×C3×D3—122no—
217C32×C4—22no—
218C32×C4—22no—
219C32×C4—42no—
220C32×C4—42no—
221C32×C4—64no—
222C32×C4C2×#6712yes—
223C3⋊Dic3—22no—
224C3⋊Dic3C2×#6711yescharacteristic
225C3⋊Dic3C23×D311yescharacteristic
226C3×Dic3—24no—
227C3×Dic3—24no—
228C3×Dic3—44no—
229C3×Dic3—61no—
230C3×Dic3—61no—
231C3×Dic3—61no—
232C3×Dic3—61no—
233C3×Dic3—62no—
234C3×Dic3—62no—
235C3×Dic3—122no—
236C2×#41—271no—
237C22×Q8—91no—
238#137—278no—
239C2×Q16—272no—
240C2×SD16—272no—
241C2×M16—271no—
242C33C2×#13711yescharacteristic, complemented, Sylow
243C23×C3—91no—
244C23×C3—181no—
245C22×D3—61no—
246C22×D3—61no—
247C22×D3—61no—
248C22×D3—61no—
249C22×D3—121no—
250C3×Q8—64no—
251C3×Q8—64no—
252C3×Q8—94no—
253C3×Q8—124no—
254C3×Q8—182no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—62no—
260C2×C3×C4—62no—
261C2×C3×C4—91no—
262C2×C3×C4—92no—
263C2×C3×C4—92no—
264C2×C3×C4—122no—
265C2×C3×C4—184no—
266C2×C3×C4—184no—
267C2×C3×C4—181no—
268C2×C3×C4GO+(4,𝔽2)11yescharacteristic
269#67—182no—
270#67—184no—
271#67—182no—
272#67—184no—
273C2×Dic3—22no—
274C2×Dic3—61no—
275C2×Dic3—61no—
276C2×Dic3—61no—
277C2×Dic3—61no—
278C2×Dic3—121no—
279D12—92no—
280D12—181no—
281D12—184no—
282C4×D3—62no—
283C4×D3—62no—
284C4×D3—62no—
285C4×D3—62no—
286C4×D3—68no—
287C4×D3—64no—
288C4×D3—64no—
289C4×D3—124no—
290Dic6—24no—
291Dic6—64no—
292Dic6—64no—
293Dic6—92no—
294Dic6—182no—
295Dic6—181no—
296Dic6—182no—
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#94212yescomplemented
307C2×C32C22×#6711yescharacteristic
308C3⋊D3—22no—
309C3⋊D3—42no—
310C3×D3—64no—
311C3×D3—122no—
312C3×D3—122no—
313C3×D3—124no—
314C3×D3—122no—
315#41—274no—
316#41—542no—
317C2×Q8—61no—
318C2×Q8—92no—
319C2×Q8—94no—
320C2×Q8—182no—
321C2×Q8—271no—
322C2×D4—271no—
323C2×D4—541no—
324C22×C4—91no—
325C22×C4—181no—
326C22×C4—181no—
327Q16—278no—
328SD16—278no—
329M16—274no—
330C2×C8—272no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—41no—
334C22×C3—41no—
335C22×C3—92no—
336C22×C3—182no—
337C22×C3—182no—
338C22×C3—362no—
339C22×C3C2×GO+(4,𝔽2)11yescharacteristic
340C2×D3—62no—
341C2×D3—62no—
342C2×D3—64no—
343C2×D3—62no—
344C2×D3—62no—
345C2×D3—122no—
346C2×D3—122no—
347C2×D3—122no—
348C2×D3—124no—
349C2×D3—122no—
350C3×C4—22no—
351C3×C4—22no—
352C3×C4—42no—
353C3×C4—42no—
354C3×C4—64no—
355C3×C4—64no—
356C3×C4—91no—
357C3×C4—91no—
358C3×C4—124no—
359C3×C4—182no—
360C3×C4C2×GO+(4,𝔽2)12yes—
361Dic3—24no—
362Dic3—61no—
363Dic3—61no—
364Dic3—61no—
365Dic3—61no—
366Dic3—62no—
367Dic3—62no—
368Dic3—122no—
369C32—21no—
370C32—21no—
371C32—41no—
372C32—41no—
373C32C2×#94211yescharacteristic, complemented
374C23—91no—
375C23—181no—
376Q8—64no—
377Q8—94no—
378Q8—182no—
379Q8—272no—
380Q8—541no—
381D4—272no—
382D4—541no—
383D4—544no—
384C2×C4—62no—
385C2×C4—91no—
386C2×C4—92no—
387C2×C4—92no—
388C2×C4—184no—
389C2×C4—181no—
390C2×C4—184no—
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—182no—
402C2×C3—362no—
403C2×C3#6422612yescomplemented
404C2×C3C22×GO+(4,𝔽2)11yescharacteristic
405D3—64no—
406D3—122no—
407D3—122no—
408D3—124no—
409D3—122no—
410C22—92no—
411C22—182no—
412C22—182no—
413C22—362no—
414C22C2×D32⋊D311yescharacteristic
415C4—64no—
416C4—91no—
417C4—91no—
418C4—182no—
419C4C2×D32⋊D312yes—
420C3—21no—
421C3—21no—
422C3—41no—
423C3—41no—
424C3C2×#6422611yescharacteristic, complemented
425C2—182no—
426C2—362no—
427C2#1172430712yesdirect 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×#89042C2
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
2. C2×C3×#3797C2semidirect
3. C2×#88989C2semidirect
4. C2×#88938C2semidirect
5. C2×#88936C2semidirect
6. #11724307C2direct (N × H)
19. C3×#3797C22semidirect
21. #88989C22semidirect
22. #88938C22semidirect
23. #88936C22semidirect
27. C2×#3797D3semidirect
81. #3797C2×D3semidirect
160. C2×C33#137semidirect
242. C33C2×#137semidirect
306. C2×C32#942semidirect
373. C32C2×#942semidirect
403. C2×C3#64226semidirect
424. C3C2×#64226semidirect
427. C2#11724307direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890422
C2×C3×#37972
C2×#889892
C2×#889382
C2×#889362
#117243072
C2×#642263
C2×#9429