Finite Group Databasebeta

← C2×#11722863 #13170634

Subgroups — C2×#11722863 #13170634

Order 1728.

1522 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 1522 conjugacy classes of subgroups — too many to draw. Only its 188 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×#89017C212yesmaximal, complemented
2C2×#88810C211yesmaximal, characteristic, complemented
3C2×#88785C211yesmaximal, characteristic, complemented
4C2×C3×#3718C211yesmaximal, characteristic, complemented
5C2×C3×#3717C212yesmaximal, complemented
6#11722863C218yesmaximal, direct product factor
7C2×#63950—32nomaximal
8C2×#63809—31nomaximal
9C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
10C2×C3×C4⋊(C3⋊D3)C2212yes—
11C2×C32×Dic6C2212yes—
12C2×#6292—22no—
13C2×Dic3×C3⋊D3—22no—
14#89017—24no—
15#89017C2218yescomplemented
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88810C2214yescomplemented
19#88785C2214yescomplemented
20C3×#3718C2214yescomplemented
21C3×#3717C2218yescomplemented
22C2×#3832—32no—
23C2×#3796—32no—
24C2×#3796—32no—
25C2×#3796—62no—
26C2×C8×C3⋊D3—31no—
27C2×C3×#216—32no—
28C2×C3×#215—32no—
29C2×C3×#204—32no—
30#63950—316no—
31C2×#3718D311yescharacteristic, complemented
32C2×#3717D312yescomplemented
33C2×#3714—32no—
34C2×#3713—31no—
35C2×#3710—32no—
36C2×#3710—61no—
37#63809—38no—
38C22×C3⋊(C3⋊D3)D411yescharacteristic
39C22×C3×C3⋊D3—22no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)D411yescharacteristic
42C4×C3⋊(C3⋊D3)D412yes—
43C4×C3⋊(C3⋊D3)C2312yes—
44C3×C4⋊(C3⋊D3)—22no—
45C3×C4⋊(C3⋊D3)C2314yes—
46C2×C32×Dic3—22no—
47C32×Dic6—22no—
48C32×Dic6C2314yes—
49#6292—28no—
50Dic3×C3⋊D3—28no—
51C3×C3⋊(C3⋊C8)C2312yes—
52C32×C3⋊C8C2312yes—
53C2×#919—91no—
54C2×#916—92no—
55C2×C32×Q8—32no—
56#3832—38no—
57#3832—64no—
58C2×C4⋊(C3⋊D3)—62no—
59C2×C4⋊(C3⋊D3)C2×D312yes—
60C2×C4×C3⋊D3—31no—
61C2×C4×C3⋊D3—32no—
62C2×C4×C3⋊D3—62no—
63C2×C4×C3⋊D3—62no—
64C2×C4×C3⋊D3—61no—
65C2×C4×C3⋊D3—121no—
66C2×C3×D12—32no—
67C2×C3×D12—32no—
68C2×C3×D12—62no—
69C2×C3×Dic6—22no—
70C2×C3×Dic6C2×D312yes—
71C2×C3×Dic6C2×D312yes—
72C2×#626—62no—
73C2×#626—62no—
74C2×#626—64no—
75C2×D3×Dic3—62no—
76C2×D3×Dic3—62no—
77C2×D3×Dic3—64no—
78#3796—38no—
79#3796—38no—
80#3796—64no—
81#3796—64no—
82#3796—68no—
83#3796—68no—
84C2×C32×C8—31no—
85C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
86C8×C3⋊D3—34no—
87C3×#216—38no—
88C3×#215—38no—
89C2×C3×C3⋊C8—21no—
90C2×C3×C3⋊C8—32no—
91C2×C3×C3⋊C8—61no—
92C2×C3×C3⋊C8C2×D312yes—
93C3×#204—38no—
94#3718C2×D314yescomplemented
95#3717C2×D318yescomplemented
96#3714—38no—
97#3713—34no—
98#3710—38no—
99#3710—64no—
100C22×C33C2×D411yescharacteristic
101C2×C3⋊(C3⋊D3)—22no—
102C2×C3⋊(C3⋊D3)C2×D412yes—
103C2×C3×C3⋊D3—24no—
104C2×C3×C3⋊D3—44no—
105C33×C4C2×D411yescharacteristic
106C33×C4C2411yescharacteristic
107C3⋊(C3⋊Dic3)C2×D412yes—
108C32×Dic3—24no—
109C2×#239—92no—
110C2×#239—92no—
111C2×#239—182no—
112C2×#237—92no—
113C2×C3×Q16—91no—
114C2×C3×SD16—92no—
115C2×#216—32no—
116C2×#215—32no—
117C2×#213—92no—
118#919—98no—
119#916—916no—
120C2×D24—91no—
121C2×#204—32no—
122C2×D3×C8—92no—
123C2×D3×C8—91no—
124C2×D3×C8—181no—
125C22×C3⋊D3—22no—
126C22×C3⋊D3—31no—
127C22×C3⋊D3—32no—
128C22×C3⋊D3—61no—
129C22×C3⋊D3—62no—
130C22×C3⋊D3—121no—
131C22×C3×D3—64no—
132C22×C3×D3—62no—
133C22×C3×D3—62no—
134C32×Q8—34no—
135C32×Q8—62no—
136C2×C32×C4—21no—
137C2×C32×C4—22no—
138C2×C32×C4—41no—
139C2×C32×C4—62no—
140C2×C32×C4C22×D312yes—
141C2×C32×C4C22×D311yescharacteristic
142C2×C3⋊Dic3—31no—
143C2×C3⋊Dic3—32no—
144C2×C3⋊Dic3—62no—
145C2×C3⋊Dic3—61no—
146C2×C3⋊Dic3—121no—
147C4⋊(C3⋊D3)—22no—
148C4⋊(C3⋊D3)—68no—
149C4⋊(C3⋊D3)C22×D314yes—
150C4×C3⋊D3—32no—
151C4×C3⋊D3—34no—
152C4×C3⋊D3—34no—
153C4×C3⋊D3—32no—
154C4×C3⋊D3—62no—
155C4×C3⋊D3—64no—
156C4×C3⋊D3—62no—
157C4×C3⋊D3—64no—
158C4×C3⋊D3—68no—
159C4×C3⋊D3—122no—
160C4×C3⋊D3—122no—
161C2×C3×Dic3—24no—
162C2×C3×Dic3—22no—
163C2×C3×Dic3—22no—
164C3×D12—34no—
165C3×D12—34no—
166C3×D12—64no—
167C3×D12—64no—
168C3×D12—62no—
169C3×D12—62no—
170C3×Dic6—24no—
171C3×Dic6—22no—
172C3×Dic6—22no—
173C3×Dic6—24no—
174C3×Dic6C22×D314yes—
175C3×Dic6C22×D314yes—
176#626—68no—
177#626—68no—
178#626—616no—
179D3×Dic3—68no—
180D3×Dic3—68no—
181D3×Dic3—616no—
182C32×C8—32no—
183C3⋊(C3⋊C8)C22×D312yes—
184C3×C3⋊C8—22no—
185C3×C3⋊C8—34no—
186C3×C3⋊C8—62no—
187C3×C3⋊C8C22×D314yes—
188C2×#135—271noSylow
189C2×C33#13512yescomplemented
190C2×C33C22×D411yescharacteristic
191C3⋊(C3⋊D3)—22no—
192C3×C3⋊D3—44no—
193C2×C3×Q8—32no—
194C2×C3×Q8—32no—
195C2×C3×Q8—62no—
196C2×C3×D4—92no—
197C2×#67—182no—
198C22×Dic3—182no—
199#239—98no—
200#239—98no—
201#239—188no—
202#239—184no—
203#239—184no—
204#239—188no—
205#237—98no—
206#237—184no—
207C2×D12—32no—
208C2×D12—32no—
209C2×D12—62no—
210C2×D12—182no—
211C2×D12—182no—
212C2×D12—184no—
213C2×C4×D3—92no—
214C2×C4×D3—91no—
215C2×C4×D3—181no—
216C2×C4×D3—182no—
217C2×C4×D3—182no—
218C2×C4×D3—182no—
219C2×C4×D3—184no—
220C2×C4×D3—361no—
221C2×Dic6D3212yes—
222C3×Q16—94no—
223C3×SD16—98no—
224C2×C3×C8—32no—
225C2×C3×C8—61no—
226C2×C3×C8—91no—
227#216—38no—
228#215—38no—
229#213—98no—
230C2×C3⋊C8—32no—
231C2×C3⋊C8—61no—
232C2×C3⋊C8D3211yescharacteristic
233D24—94no—
234#204—38no—
235D3×C8—98no—
236D3×C8—94no—
237D3×C8—184no—
238C22×C32—21no—
239C22×C32—22no—
240C22×C32—41no—
241C22×C32D3×D412yes—
242C22×C32D3×D411yescharacteristic
243C2×C3⋊D3—24no—
244C2×C3⋊D3—32no—
245C2×C3⋊D3—34no—
246C2×C3⋊D3—44no—
247C2×C3⋊D3—62no—
248C2×C3⋊D3—64no—
249C2×C3⋊D3—62no—
250C2×C3⋊D3—64no—
251C2×C3⋊D3—64no—
252C2×C3⋊D3—122no—
253C2×C3⋊D3—124no—
254C2×C3⋊D3—124no—
255C2×C3×D3—68no—
256C2×C3×D3—64no—
257C2×C3×D3—64no—
258C2×C3×D3—128no—
259C2×C3×D3—124no—
260C2×C3×D3—124no—
261C32×C4—21no—
262C32×C4—22no—
263C32×C4—21no—
264C32×C4—22no—
265C32×C4—41no—
266C32×C4—41no—
267C32×C4—64no—
268C32×C4D3×D412yes—
269C32×C4D3×D411yescharacteristic
270C32×C4C23×D312yes—
271C32×C4C23×D311yescharacteristic
272C3⋊Dic3—32no—
273C3⋊Dic3—34no—
274C3⋊Dic3—64no—
275C3⋊Dic3—62no—
276C3⋊Dic3—122no—
277C3×Dic3—28no—
278C3×Dic3—24no—
279C3×Dic3—24no—
280C2×#41—272no—
281#135—278no—
282C2×Q16—91no—
283C2×SD16—92no—
284C2×D8—271no—
285C22×C8—271no—
286C33C2×#13511yescharacteristic, complemented, Sylow
287C23×C3—182no—
288C22×D3—64no—
289C22×D3—62no—
290C22×D3—62no—
291C22×D3—91no—
292C22×D3—92no—
293C22×D3—181no—
294C22×D3—182no—
295C22×D3—361no—
296C3×Q8—34no—
297C3×Q8—34no—
298C3×Q8—64no—
299C3×Q8—64no—
300C3×Q8—62no—
301C3×Q8—62no—
302C3×D4—94no—
303C3×D4—182no—
304C2×C3×C4—21no—
305C2×C3×C4—22no—
306C2×C3×C4—41no—
307C2×C3×C4—64no—
308C2×C3×C4—62no—
309C2×C3×C4—62no—
310C2×C3×C4C2×D3211yescharacteristic
311C2×C3×C4C2×D3212yes—
312#67—188no—
313C2×Dic3—22no—
314C2×Dic3—91no—
315C2×Dic3—92no—
316C2×Dic3—181no—
317C2×Dic3—182no—
318C2×Dic3—188no—
319C2×Dic3—361no—
320D12—34no—
321D12—34no—
322D12—64no—
323D12—64no—
324D12—62no—
325D12—62no—
326D12—188no—
327D12—188no—
328D12—1816no—
329C4×D3—94no—
330C4×D3—92no—
331C4×D3—92no—
332C4×D3—94no—
333C4×D3—182no—
334C4×D3—182no—
335C4×D3—184no—
336C4×D3—184no—
337C4×D3—188no—
338C4×D3—188no—
339C4×D3—1816no—
340C4×D3—362no—
341C4×D3—362no—
342Dic6—22no—
343Dic6C2×D3214yes—
344C3×C8—34no—
345C3×C8—62no—
346C3×C8—92no—
347C3⋊C8—34no—
348C3⋊C8—62no—
349C3⋊C8C2×D3212yes—
350C2×C32—21no—
351C2×C32—22no—
352C2×C32—22no—
353C2×C32—24no—
354C2×C32—42no—
355C2×C32—41no—
356C2×C32#91614yescomplemented
357C2×C32#91912yescomplemented
358C2×C32C2×D3×D412yes—
359C2×C32C2×D3×D411yescharacteristic
360C3⋊D3—44no—
361C3⋊D3—62no—
362C3⋊D3—64no—
363C3⋊D3—64no—
364C3⋊D3—124no—
365C3⋊D3—124no—
366C3×D3—128no—
367C3×D3—124no—
368C3×D3—124no—
369#41—278no—
370#41—544no—
371C2×Q8—32no—
372C2×D4—92no—
373C2×D4—542no—
374C22×C4—271no—
375C22×C4—542no—
376Q16—94no—
377SD16—98no—
378D8—274no—
379C2×C8—31no—
380C2×C8—91no—
381C2×C8—274no—
382C22×C3—22no—
383C22×C3—21no—
384C22×C3—41no—
385C22×C3—184no—
386C22×C3—364no—
387C22×C3#380211yescharacteristic
388C22×C3#381112yes—
389C2×D3—68no—
390C2×D3—64no—
391C2×D3—64no—
392C2×D3—92no—
393C2×D3—94no—
394C2×D3—128no—
395C2×D3—124no—
396C2×D3—124no—
397C2×D3—182no—
398C2×D3—184no—
399C2×D3—184no—
400C2×D3—182no—
401C2×D3—184no—
402C2×D3—362no—
403C2×D3—364no—
404C2×D3—364no—
405C3×C4—21no—
406C3×C4—22no—
407C3×C4—22no—
408C3×C4—21no—
409C3×C4—41no—
410C3×C4—41no—
411C3×C4—68no—
412C3×C4—64no—
413C3×C4—64no—
414C3×C4#380211yescharacteristic
415C3×C4#381112yes—
416C3×C4C22×D3211yescharacteristic
417C3×C4C22×D3212yes—
418Dic3—24no—
419Dic3—92no—
420Dic3—94no—
421Dic3—184no—
422Dic3—182no—
423Dic3—362no—
424C32—22no—
425C32—21no—
426C32—41no—
427C32C2×#91612yescomplemented
428C32C2×#91911yescharacteristic, complemented
429C23—182no—
430C23—271no—
431Q8—34no—
432Q8—62no—
433D4—94no—
434D4—182no—
435D4—548no—
436C2×C4—62no—
437C2×C4—271no—
438C2×C4—272no—
439C2×C4—272no—
440C2×C4—548no—
441C2×C4D3311yescharacteristic
442C8—32no—
443C8—92no—
444C2×C3—24no—
445C2×C3—22no—
446C2×C3—22no—
447C2×C3—21no—
448C2×C3—42no—
449C2×C3—41no—
450C2×C3—364no—
451C2×C3#6380912yescomplemented
452C2×C3#6395014yescomplemented
453C2×C3C2×#380211yescharacteristic
454C2×C3C2×#381112yes—
455D3—128no—
456D3—124no—
457D3—124no—
458D3—184no—
459D3—184no—
460D3—182no—
461D3—364no—
462D3—364no—
463C22—184no—
464C22—272no—
465C22—364no—
466C22—542no—
467C22#8895411yescharacteristic
468C4—64no—
469C4—272no—
470C4#8895411yescharacteristic
471C4C2×D3311yescharacteristic
472C3—22no—
473C3—21no—
474C3—41no—
475C3C2×#6380911yescharacteristic, complemented
476C3C2×#6395012yescomplemented
477C2—364no—
478C2—542no—
479C2#1172286312yesdirect product factor
480C2C2×#8895411yescharacteristic
subgroupG/H
centerC22#88954
commutator, cosocleC33×C4C24
FrattiniC4C2×D33
FittingC2×C33×C4C23
socleC22×C33C2×D4
hypercenterC2×C4D33
hypocenterC33C2×#135
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#89017C2semidirect
2. C2×#88810C2semidirect
3. C2×#88785C2semidirect
4. C2×C3×#3718C2semidirect
5. C2×C3×#3717C2semidirect
6. #11722863C2direct (N × H)
15. #89017C22semidirect
18. #88810C22semidirect
19. #88785C22semidirect
20. C3×#3718C22semidirect
21. C3×#3717C22semidirect
31. C2×#3718D3semidirect
32. C2×#3717D3semidirect
94. #3718C2×D3semidirect
95. #3717C2×D3semidirect
189. C2×C33#135semidirect
286. C33C2×#135semidirect
356. C2×C32#916semidirect
357. C2×C32#919semidirect
427. C32C2×#916semidirect
428. C32C2×#919semidirect
451. C2×C3#63809semidirect
452. C2×C3#63950semidirect
475. C3C2×#63809semidirect
476. C3C2×#63950semidirect
479. C2#11722863direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890172
C2×#888102
C2×#887852
C2×C3×#37182
C2×C3×#37172
#117228632
C2×#639503
C2×#638093