Finite Group Databasebeta

← C2×#11724338 #13181138

Subgroups — C2×#11724338 #13181138

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×#89040C211yesmaximal, characteristic, complemented
3C2×C3×#3788C211yesmaximal, characteristic, complemented
4C2×#88943C212yesmaximal, complemented
5C2×#88942C212yesmaximal, complemented
6#11724338C218yesmaximal, direct product factor
7C2×#64226—31nomaximal
8C2×C3×C4×C3⋊D3—21no—
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C3×#634—21no—
11C2×#6296C2212yes—
12C2×#6296C2211yescharacteristic
13C2×#6295—21no—
14C2×#6295C2211yescharacteristic
15C2×#6294—21no—
16#89042—22no—
17#89042C2214yes—
18#89040—22no—
19#89040C2214yescomplemented
20C2×C3×#622C2212yes—
21C3×#3788C2214yescomplemented
22#88943C2218yescomplemented
23#88942C2218yescomplemented
24C2×#3798—31no—
25C2×#3798—61no—
26C2×#3797—31no—
27C2×D3×Dic6—61no—
28C2×#3788D311yescharacteristic, complemented
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—22no—
36C2×C3×C3⋊Dic3—21no—
37C2×C3×C3⋊Dic3C2311yescharacteristic
38C3×C4×C3⋊D3—24no—
39C3×C4×C3⋊D3D412yes—
40C3×C4×C3⋊D3C2312yes—
41C3×#634—24no—
42#6296—22no—
43#6296—21no—
44#6296C2314yes—
45#6296C2312yes—
46#6295—24no—
47#6295—21no—
48#6295C2312yes—
49#6294—24no—
50C3×#622C2314yes—
51C2×#909—91nomaximal
52C2×C4×C3⋊D3—21no—
53C2×C4×C3⋊D3C2×D311yescharacteristic
54C2×#634—21no—
55C2×C3×C4×D3—61no—
56C2×C3×C4×D3—61no—
57C2×C3×C4×D3—61no—
58C2×C3×C4×D3—61no—
59C2×C3×C4×D3—121no—
60C2×C3×Dic6—61no—
61C2×C3×Dic6—61no—
62C2×C3×Dic6—121no—
63C2×#627—32no—
64C2×#627—31no—
65C2×#627—62no—
66C2×#627—61no—
67C2×#626—61no—
68C2×#626—61no—
69C2×#626—61no—
70C2×#625—31no—
71C2×#625—61no—
72C2×#624—61no—
73C2×D3×Dic3—62no—
74#3798—34no—
75#3798—64no—
76#3798—64no—
77#3798—62no—
78#3797—34no—
79#3797—62no—
80D3×Dic6—64no—
81D3×Dic6—64no—
82C2×#622C2×D312yes—
83#3788C2×D314yescomplemented
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—24no—
93C3×C3⋊Dic3—22no—
94C3×C3⋊Dic3C2×D411yescharacteristic
95C3×C3⋊Dic3C2411yescharacteristic
96C2×D3×Q8—181no—
97C2×#235—91no—
98C2×#235—181no—
99C22×Dic6—91no—
100C2×C3×M16—91no—
101#909—98no—
102C2×Dic12—92no—
103C2×#204—92no—
104C22×C3⋊D3—21no—
105C22×C3⋊D3D1211yescharacteristic
106C22×C3×D3—61no—
107C22×C3×D3—61no—
108C22×C3×D3—61no—
109C22×C3×D3—61no—
110C22×C3×D3—121no—
111C2×C32×C4—21no—
112C2×C32×C4—21no—
113C2×C32×C4—41no—
114C2×C32×C4—41no—
115C2×C32×C4D1211yescharacteristic
116C2×C3⋊Dic3—22no—
117C2×C3⋊Dic3—21no—
118C2×C3⋊Dic3C22×D311yescharacteristic
119C4×C3⋊D3—24no—
120C4×C3⋊D3D1212yes—
121C4×C3⋊D3C22×D312yes—
122#634—24no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—62no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—62no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—122no—
130C2×C3×Dic3—121no—
131C3×C4×D3—62no—
132C3×C4×D3—62no—
133C3×C4×D3—62no—
134C3×C4×D3—62no—
135C3×C4×D3—64no—
136C3×C4×D3—64no—
137C3×C4×D3—124no—
138C3×Dic6—64no—
139C3×Dic6—64no—
140C3×Dic6—124no—
141#627—34no—
142#627—32no—
143#627—64no—
144#627—62no—
145#627—64no—
146#627—62no—
147#627—62no—
148#627—61no—
149#626—62no—
150#626—62no—
151#626—64no—
152#626—64no—
153#625—32no—
154#625—64no—
155#625—61no—
156#624—64no—
157D3×Dic3—68no—
158#622C22×D314yes—
159C2×#137—271noSylow
160C2×C33#13712yescomplemented
161C2×C33C22×D411yescharacteristic
162C3×C3⋊D3—22no—
163C3×C3⋊D3—42no—
164C2×C3×Q8—181no—
165C22×C3×C4—91no—
166C22×C3×C4—181no—
167C2×#67—181no—
168C2×#67—181no—
169C2×#67—181no—
170C22×Dic3—181no—
171D3×Q8—184no—
172D3×Q8—184no—
173#235—94no—
174#235—184no—
175#235—182no—
176#235—184no—
177C2×D12—91no—
178C2×D12—181no—
179C2×C4×D3—61no—
180C2×C4×D3—61no—
181C2×C4×D3—61no—
182C2×C4×D3—61no—
183C2×C4×D3—121no—
184C2×C4×D3—182no—
185C2×Dic6—61no—
186C2×Dic6—61no—
187C2×Dic6—94no—
188C2×Dic6—92no—
189C2×Dic6—91no—
190C2×Dic6—121no—
191C2×Dic6—181no—
192C2×Dic6—182no—
193C2×Dic6—182no—
194C3×M16—94no—
195C2×C3×C8—92no—
196Dic12—98no—
197#204—98no—
198C22×C32—21no—
199C22×C32—21no—
200C22×C32—41no—
201C22×C32—41no—
202C22×C32C2×D1211yescharacteristic
203C2×C3⋊D3—22no—
204C2×C3⋊D3—22no—
205C2×C3⋊D3—42no—
206C2×C3⋊D3C2×D1212yes—
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×C4C2×D1212yes—
222C3⋊Dic3—24no—
223C3⋊Dic3—22no—
224C3⋊Dic3C2×D1211yescharacteristic
225C3⋊Dic3C23×D311yescharacteristic
226C3×Dic3—61no—
227C3×Dic3—61no—
228C3×Dic3—61no—
229C3×Dic3—61no—
230C3×Dic3—64no—
231C3×Dic3—62no—
232C3×Dic3—64no—
233C3×Dic3—62no—
234C3×Dic3—124no—
235C3×Dic3—122no—
236C2×#41—271no—
237C22×Q8—271no—
238#137—278no—
239C2×Q16—272no—
240C2×SD16—272no—
241C2×M16—91no—
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—184no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—91no—
256C2×C3×C4—92no—
257C2×C3×C4—92no—
258C2×C3×C4—184no—
259C2×C3×C4—182no—
260C2×C3×C4—181no—
261C2×C3×C4GO+(4,𝔽2)11yescharacteristic
262#67—182no—
263#67—184no—
264#67—182no—
265#67—184no—
266C2×Dic3—61no—
267C2×Dic3—61no—
268C2×Dic3—62no—
269C2×Dic3—61no—
270C2×Dic3—62no—
271C2×Dic3—61no—
272C2×Dic3—122no—
273C2×Dic3—121no—
274C2×Dic3—184no—
275D12—92no—
276D12—181no—
277D12—184no—
278C4×D3—62no—
279C4×D3—62no—
280C4×D3—62no—
281C4×D3—62no—
282C4×D3—64no—
283C4×D3—64no—
284C4×D3—124no—
285C4×D3—188no—
286Dic6—64no—
287Dic6—64no—
288Dic6—94no—
289Dic6—92no—
290Dic6—124no—
291Dic6—182no—
292Dic6—184no—
293Dic6—182no—
294Dic6—181no—
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#90912yescomplemented
307C2×C32C22×D1211yescharacteristic
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—181no—
318C2×Q8—272no—
319C2×Q8—271no—
320C2×Q8—274no—
321C2×Q8—542no—
322C2×D4—271no—
323C2×D4—541no—
324C22×C4—91no—
325C22×C4—181no—
326C22×C4—541no—
327Q16—278no—
328SD16—278no—
329M16—94no—
330C2×C8—92no—
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—91no—
355C3×C4—91no—
356C3×C4—184no—
357C3×C4—182no—
358C3×C4C2×GO+(4,𝔽2)12yes—
359Dic3—61no—
360Dic3—61no—
361Dic3—61no—
362Dic3—61no—
363Dic3—64no—
364Dic3—62no—
365Dic3—64no—
366Dic3—62no—
367Dic3—124no—
368Dic3—122no—
369C32—21no—
370C32—21no—
371C32—41no—
372C32—41no—
373C32C2×#90911yescharacteristic, complemented
374C23—91no—
375C23—181no—
376Q8—184no—
377Q8—274no—
378Q8—272no—
379Q8—542no—
380Q8—541no—
381D4—272no—
382D4—541no—
383D4—544no—
384C2×C4—91no—
385C2×C4—92no—
386C2×C4—92no—
387C2×C4—181no—
388C2×C4—182no—
389C2×C4—184no—
390C2×C4—544no—
391C2×C4#157⋊D311yescharacteristic
392C8—94no—
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×#157⋊D311yescharacteristic
415C4—91no—
416C4—91no—
417C4—182no—
418C4—184no—
419C4C2×#157⋊D312yes—
420C3—21no—
421C3—21no—
422C3—41no—
423C3—41no—
424C3C2×#6422611yescharacteristic, complemented
425C2—182no—
426C2—362no—
427C2#1172433812yesdirect product factor
428C2C22×#157⋊D311yescharacteristic
subgroupG/H
centerC22C2×#157⋊D3
commutator, cosocleC3×C3⋊Dic3C24
FrattiniC2C22×#157⋊D3
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×#89042C2
hypercenterC2×C4#157⋊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×#89040C2semidirect
3. C2×C3×#3788C2semidirect
4. C2×#88943C2semidirect
5. C2×#88942C2semidirect
6. #11724338C2direct (N × H)
19. #89040C22semidirect
21. C3×#3788C22semidirect
22. #88943C22semidirect
23. #88942C22semidirect
28. C2×#3788D3semidirect
83. #3788C2×D3semidirect
160. C2×C33#137semidirect
242. C33C2×#137semidirect
306. C2×C32#909semidirect
373. C32C2×#909semidirect
403. C2×C3#64226semidirect
424. C3C2×#64226semidirect
427. C2#11724338direct (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×#890402
C2×C3×#37882
C2×#889432
C2×#889422
#117243382
C2×#642263
C2×#9099