Finite Group Databasebeta

← C2×#11723705 #13178689

Subgroups — C2×#11723705 #13178689

Order 1728.

1386 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 1386 conjugacy classes of subgroups — too many to draw. Only its 199 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×C3×#3830C211yesmaximal, characteristic
2C2×#89040C211yesmaximal, characteristic, complemented
3C2×C3×#3754C212yesmaximal, complemented
4#11723705C218yesmaximal, direct product factor
5C2×#88812C212yesmaximal, complemented
6C2×#88809C211yesmaximal, characteristic, complemented
7C2×#63931—31nomaximal
8C2×#63929—32nomaximal
9C2×C33×D4C2211yescharacteristic
10C2×C3×#638—21no—
11C22×C3×C3⋊Dic3—21no—
12C3×#3830—22no—
13C3×#3830C2214yes—
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C3×#634C2212yes—
16C2×C3×#634C2211yescharacteristic
17C2×#6296—21no—
18C2×#6294—21no—
19#89040—22no—
20#89040C2214yescomplemented
21C3×#3754C2218yescomplemented
22C2×C3×C3⋊(C3⋊C8)C2212yes—
23#88812C2218yescomplemented
24#88809C2214yescomplemented
25C2×#3830D311yescharacteristic
26C2×C3×#237—32no—
27C2×C3×#237—61no—
28C2×#3797—31no—
29C2×D3×Dic6—32no—
30C2×#3754D312yescomplemented
31C2×C3×#214—32no—
32C2×C3×#214—32no—
33C2×C3×#214—62no—
34#63931—38no—
35#63929—316no—
36C2×#3719—32no—
37C2×#3719—32no—
38C2×#3718—32no—
39C2×#3712—31no—
40C2×#3711—32no—
41C23×C33—21no—
42C22×C3×C3⋊D3D411yescharacteristic
43C33×D4—21no—
44C33×D4C2312yes—
45C2×C33×C4C2311yescharacteristic
46C3×#638—24no—
47C2×C3×C3⋊Dic3—21no—
48C2×C3×C3⋊Dic3—22no—
49C2×C3×C3⋊Dic3—24no—
50C2×C3×C3⋊Dic3D411yescharacteristic
51C3×C4×C3⋊D3D412yes—
52C3×C4×C3⋊D3C2312yes—
53C3×#634—21no—
54C3×#634—22no—
55C3×#634C2314yes—
56C3×#634C2312yes—
57#6296—24no—
58#6294—24no—
59C3×C3⋊(C3⋊C8)C2314yes—
60C2×#951—91no—
61C2×#915—92no—
62C2×C32×D4—21no—
63C2×C32×D4—22no—
64C2×C32×D4—41no—
65C2×C32×D4C2×D311yescharacteristic
66C2×C32×D4C2×D312yes—
67C2×#638—21no—
68C22×C3⋊Dic3—21no—
69#3830—22no—
70#3830C2×D314yes—
71C2×C4×C3⋊D3C2×D311yescharacteristic
72C2×#634C2×D312yes—
73C2×#634C2×D311yescharacteristic
74C2×C3×#67—62no—
75C2×C3×#67—62no—
76C22×C3×Dic3—62no—
77C22×C3×Dic3—62no—
78C3×#237—38no—
79C3×#237—64no—
80C3×#237—64no—
81C3×#237—64no—
82C2×C3×C4×D3—32no—
83C2×C3×C4×D3—61no—
84C2×C3×Dic6—32no—
85C2×C3×Dic6—32no—
86C2×C3×Dic6—32no—
87C2×C3×Dic6—61no—
88C2×C3×Dic6—62no—
89C2×#627—62no—
90C2×#627—61no—
91C2×#624—61no—
92C2×D3×Dic3—62no—
93#3797—34no—
94#3797—62no—
95D3×Dic6—38no—
96D3×Dic6—64no—
97#3754C2×D318yescomplemented
98C2×C3⋊(C3⋊C8)C2×D312yes—
99C3×#214—38no—
100C3×#214—38no—
101C3×#214—68no—
102C2×C3×C3⋊C8—32no—
103C2×C3×C3⋊C8—32no—
104C2×C3×C3⋊C8—62no—
105#3719—38no—
106#3719—38no—
107#3718—38no—
108#3712—34no—
109#3711—38no—
110C22×C33—22no—
111C22×C33—42no—
112C22×C33C2×D411yescharacteristic
113C2×C3×C3⋊D3—22no—
114C2×C3×C3⋊D3C2×D412yes—
115C33×C4C2×D411yescharacteristic
116C33×C4C2411yescharacteristic
117C3×C3⋊Dic3—22no—
118C3×C3⋊Dic3—24no—
119C3×C3⋊Dic3C2×D412yes—
120C2×C3×#41—91no—
121C2×D3×Q8—92no—
122C2×#237—32no—
123C2×#237—61no—
124C22×Dic6—91no—
125C2×C3×SD16—92no—
126#951—98no—
127C2×#216—92no—
128C2×#216—92no—
129C2×#214—32no—
130C2×#214—32no—
131C2×#214—62no—
132C2×#208—91no—
133#915—916no—
134C2×Dic12—92no—
135C2×ΓL(1,𝔽25)—92no—
136C23×C32—21no—
137C23×C32—22no—
138C23×C32—22no—
139C23×C32—42no—
140C23×C32—42no—
141C22×C3⋊D3#6711yescharacteristic
142C22×C3×D3—32no—
143C22×C3×D3—61no—
144C32×D4—21no—
145C32×D4—22no—
146C32×D4—22no—
147C32×D4—22no—
148C32×D4—24no—
149C32×D4—42no—
150C32×D4—42no—
151C32×D4—42no—
152C32×D4C22×D312yes—
153C32×D4C22×D314yes—
154C2×C32×C4—22no—
155C2×C32×C4—21no—
156C2×C32×C4—41no—
157C2×C32×C4C22×D312yes—
158C2×C32×C4C22×D311yescharacteristic
159#638—24no—
160C2×C3⋊Dic3—21no—
161C2×C3⋊Dic3—24no—
162C2×C3⋊Dic3—22no—
163C2×C3⋊Dic3#6711yescharacteristic
164C4×C3⋊D3#6712yes—
165C4×C3⋊D3C22×D312yes—
166#634—21no—
167#634—22no—
168#634C22×D312yes—
169#634C22×D314yes—
170C3×#67—68no—
171C3×#67—68no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—68no—
175C2×C3×Dic3—62no—
176C2×C3×Dic3—68no—
177C2×C3×Dic3—62no—
178C2×C3×Dic3—62no—
179C2×C3×Dic3—62no—
180C2×C3×Dic3—122no—
181C3×C4×D3—34no—
182C3×C4×D3—34no—
183C3×C4×D3—62no—
184C3×C4×D3—62no—
185C3×Dic6—34no—
186C3×Dic6—34no—
187C3×Dic6—34no—
188C3×Dic6—62no—
189C3×Dic6—64no—
190C3×Dic6—62no—
191C3×Dic6—62no—
192C3×Dic6—62no—
193C3×Dic6—62no—
194C3×Dic6—122no—
195#627—68no—
196#627—64no—
197#624—64no—
198D3×Dic3—68no—
199C3⋊(C3⋊C8)C22×D314yes—
200C3×C3⋊C8—34no—
201C3×C3⋊C8—34no—
202C3×C3⋊C8—64no—
203C2×#137—271noSylow
204C2×C33—42no—
205C2×C33#13712yescomplemented
206C2×C33C22×D411yescharacteristic
207C3×C3⋊D3—22no—
208C3×#41—94no—
209C3×#41—182no—
210C2×C3×Q8—91no—
211C2×C3×Q8—92no—
212C2×C3×D4—21no—
213C2×C3×D4—22no—
214C2×C3×D4—41no—
215C2×C3×D4—181no—
216C2×C3×D4D3211yescharacteristic
217C2×C3×D4D3212yes—
218C22×C3×C4—91no—
219C22×C3×C4—181no—
220C2×#67—62no—
221C2×#67—62no—
222C22×Dic3—62no—
223C22×Dic3—62no—
224C22×Dic3—181no—
225D3×Q8—98no—
226D3×Q8—184no—
227#237—38no—
228#237—64no—
229#237—64no—
230#237—64no—
231C2×C4×D3—32no—
232C2×C4×D3—61no—
233C2×C4×D3—182no—
234C2×Dic6—32no—
235C2×Dic6—32no—
236C2×Dic6—32no—
237C2×Dic6—61no—
238C2×Dic6—62no—
239C2×Dic6—94no—
240C2×Dic6—182no—
241C2×Dic6—182no—
242C2×Dic6—181no—
243C3×SD16—98no—
244C2×C3×C8—92no—
245#216—98no—
246#216—98no—
247#214—38no—
248#214—38no—
249#214—68no—
250#208—94no—
251C2×C3⋊C8—32no—
252C2×C3⋊C8—32no—
253C2×C3⋊C8—62no—
254Dic12—98no—
255ΓL(1,𝔽25)—98no—
256C22×C32—22no—
257C22×C32—24no—
258C22×C32—24no—
259C22×C32—22no—
260C22×C32—21no—
261C22×C32—41no—
262C22×C32—44no—
263C22×C32—44no—
264C22×C32—42no—
265C22×C32—44no—
266C22×C32—44no—
267C22×C32—84no—
268C22×C32—84no—
269C22×C32D3×D412yes—
270C22×C32C2×#6711yescharacteristic
271C2×C3⋊D3—22no—
272C2×C3⋊D3C2×#6712yes—
273C2×C3×D3—34no—
274C2×C3×D3—62no—
275C2×C3×D3—64no—
276C2×C3×D3—122no—
277C32×C4—22no—
278C32×C4—21no—
279C32×C4—22no—
280C32×C4—21no—
281C32×C4—41no—
282C32×C4—41no—
283C32×C4D3×D412yes—
284C32×C4C2×#6711yescharacteristic
285C32×C4C23×D312yes—
286C32×C4C23×D311yescharacteristic
287C3⋊Dic3—22no—
288C3⋊Dic3—24no—
289C3⋊Dic3C2×#6712yes—
290C3×Dic3—34no—
291C3×Dic3—62no—
292C3×Dic3—64no—
293C3×Dic3—64no—
294C3×Dic3—64no—
295C3×Dic3—64no—
296C3×Dic3—124no—
297C2×#41—91no—
298C22×Q8—271no—
299#137—278no—
300C2×Q16—272no—
301C2×SD16—92no—
302C2×M16—271no—
303C33C2×#13711yescharacteristic, complemented, Sylow
304C23×C3—22no—
305C23×C3—22no—
306C23×C3—21no—
307C23×C3—42no—
308C23×C3—42no—
309C23×C3—91no—
310C22×D3—32no—
311C22×D3—61no—
312C3×Q8—92no—
313C3×Q8—94no—
314C3×Q8—182no—
315C3×Q8—181no—
316C3×D4—22no—
317C3×D4—22no—
318C3×D4—21no—
319C3×D4—22no—
320C3×D4—24no—
321C3×D4—42no—
322C3×D4—42no—
323C3×D4—42no—
324C3×D4—184no—
325C3×D4C2×D3212yes—
326C3×D4C2×D3214yes—
327C2×C3×C4—22no—
328C2×C3×C4—21no—
329C2×C3×C4—41no—
330C2×C3×C4—91no—
331C2×C3×C4—92no—
332C2×C3×C4—92no—
333C2×C3×C4—184no—
334C2×C3×C4—182no—
335C2×C3×C4—181no—
336C2×C3×C4C2×D3212yes—
337C2×C3×C4C2×D3211yescharacteristic
338#67—68no—
339#67—68no—
340C2×Dic3—32no—
341C2×Dic3—61no—
342C2×Dic3—68no—
343C2×Dic3—62no—
344C2×Dic3—68no—
345C2×Dic3—62no—
346C2×Dic3—62no—
347C2×Dic3—62no—
348C2×Dic3—122no—
349C2×Dic3—184no—
350C4×D3—34no—
351C4×D3—34no—
352C4×D3—62no—
353C4×D3—62no—
354C4×D3—188no—
355Dic6—34no—
356Dic6—34no—
357Dic6—34no—
358Dic6—62no—
359Dic6—64no—
360Dic6—62no—
361Dic6—62no—
362Dic6—62no—
363Dic6—62no—
364Dic6—122no—
365Dic6—188no—
366Dic6—184no—
367C3×C8—94no—
368C3⋊C8—34no—
369C3⋊C8—34no—
370C3⋊C8—64no—
371C2×C32—22no—
372C2×C32—21no—
373C2×C32—24no—
374C2×C32—22no—
375C2×C32—42no—
376C2×C32—41no—
377C2×C32—42no—
378C2×C32—44no—
379C2×C32—44no—
380C2×C32—84no—
381C2×C32—84no—
382C2×C32#91514yescomplemented
383C2×C32#95112yescomplemented
384C2×C32C2×D3×D412yes—
385C2×C32C22×#6711yescharacteristic
386C3⋊D3—22no—
387C3×D3—64no—
388C3×D3—122no—
389#41—94no—
390#41—182no—
391C2×Q8—92no—
392C2×Q8—91no—
393C2×Q8—274no—
394C2×Q8—541no—
395C2×Q8—542no—
396C2×D4—181no—
397C2×D4#111311yescharacteristic
398C22×C4—91no—
399C22×C4—181no—
400C22×C4—541no—
401Q16—278no—
402SD16—98no—
403M16—274no—
404C2×C8—92no—
405C22×C3—24no—
406C22×C3—24no—
407C22×C3—22no—
408C22×C3—21no—
409C22×C3—22no—
410C22×C3—44no—
411C22×C3—44no—
412C22×C3—44no—
413C22×C3—44no—
414C22×C3—42no—
415C22×C3—41no—
416C22×C3—84no—
417C22×C3—84no—
418C22×C3—92no—
419C22×C3—182no—
420C22×C3D3×#6712yes—
421C22×C3#381111yescharacteristic
422C2×D3—34no—
423C2×D3—62no—
424C2×D3—64no—
425C2×D3—122no—
426C3×C4—22no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—22no—
430C3×C4—41no—
431C3×C4—41no—
432C3×C4—92no—
433C3×C4—184no—
434C3×C4—182no—
435C3×C4D3×#6712yes—
436C3×C4#381111yescharacteristic
437C3×C4C22×D3212yes—
438C3×C4C22×D3211yescharacteristic
439Dic3—34no—
440Dic3—62no—
441Dic3—64no—
442Dic3—64no—
443Dic3—64no—
444Dic3—64no—
445Dic3—124no—
446C32—22no—
447C32—21no—
448C32—41no—
449C32C2×#91512yescomplemented
450C32C2×#95111yescharacteristic, complemented
451C23—21no—
452C23—91no—
453Q8—94no—
454Q8—92no—
455Q8—182no—
456Q8—181no—
457Q8—544no—
458D4—21no—
459D4—184no—
460D4C2×#111312yes—
461C2×C4—91no—
462C2×C4—92no—
463C2×C4—92no—
464C2×C4—184no—
465C2×C4—181no—
466C2×C4—182no—
467C2×C4—544no—
468C2×C4C2×#111311yescharacteristic
469C8—94no—
470C2×C3—21no—
471C2×C3—22no—
472C2×C3—22no—
473C2×C3—24no—
474C2×C3—44no—
475C2×C3—44no—
476C2×C3—42no—
477C2×C3—41no—
478C2×C3—42no—
479C2×C3—84no—
480C2×C3—84no—
481C2×C3—182no—
482C2×C3#6392914yescomplemented
483C2×C3#6393112yescomplemented
484C2×C3C2×D3×#6712yes—
485C2×C3C2×#381111yescharacteristic
486D3—64no—
487D3—122no—
488C22—22no—
489C22—42no—
490C22—92no—
491C22—182no—
492C22#8904911yescharacteristic
493C4—92no—
494C4—182no—
495C4—184no—
496C4#8904911yescharacteristic
497C4C22×#111311yescharacteristic
498C3—21no—
499C3—22no—
500C3—41no—
501C3C2×#6392912yescomplemented
502C3C2×#6393111yescharacteristic, complemented
503C2—42no—
504C2—182no—
505C2#1172370512yesdirect product factor
506C2C2×#8904911yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC33×C4C24
FrattiniC4C22×#1113
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC2×C3×#3830C2
hypercenterC2×D4#1113
hypocenterC33C2×#137
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×#89040C2semidirect
3. C2×C3×#3754C2semidirect
4. #11723705C2direct (N × H)
5. C2×#88812C2semidirect
6. C2×#88809C2semidirect
20. #89040C22semidirect
21. C3×#3754C22semidirect
23. #88812C22semidirect
24. #88809C22semidirect
30. C2×#3754D3semidirect
97. #3754C2×D3semidirect
205. C2×C33#137semidirect
303. C33C2×#137semidirect
382. C2×C32#915semidirect
383. C2×C32#951semidirect
449. C32C2×#915semidirect
450. C32C2×#951semidirect
482. C2×C3#63929semidirect
483. C2×C3#63931semidirect
501. C3C2×#63929semidirect
502. C3C2×#63931semidirect
505. C2#11723705direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C2×#890402
C2×C3×#37542
#117237052
C2×#888122
C2×#888092
C2×#639313
C2×#639293