Finite Group Databasebeta

← C2×#11722846 #13170602

Subgroups — C2×#11722846 #13170602

Order 1728.

1346 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 1346 conjugacy classes of subgroups — too many to draw. Only its 212 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⋊D3×Dic6C211yesmaximal, characteristic
2C2×C3×#3797C211yesmaximal, characteristic
3C2×#88808C211yesmaximal, characteristic
4C2×#88799C212yesmaximal, complemented
5C2×C3×#3718C212yesmaximal, complemented
6#11722846C218yesmaximal, direct product factor
7C2×#63944—31nomaximal
8C2×D3×#216—32nomaximal
9C2×#6308C2211yescharacteristic
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C32×Dic6C2211yescharacteristic
12C2×C32×Dic6C2212yes—
13C2×#6293—21no—
14C2×Dic3×C3⋊D3—21no—
15C3⋊D3×Dic6—22no—
16C3⋊D3×Dic6C2214yes—
17C2×C3×#627—21no—
18C2×C3×#624—21no—
19C3×#3797—22no—
20C3×#3797C2214yes—
21C2×C3×C3⋊(C3⋊C8)C2212yes—
22#88808C2214yes—
23#88799C2218yescomplemented
24C3×#3718C2218yescomplemented
25C2×Q8×C3⋊D3—31no—
26C2×C3×D3×Q8—32no—
27C2×#3797D311yescharacteristic
28C2×D3×Dic6—32no—
29C2×D3×Dic6—61no—
30C2×#3756—32no—
31C2×C3×#216—32no—
32C2×C3×#216—32no—
33#63944—38no—
34D3×#216—316no—
35C2×#3719—32no—
36C2×#3719—32no—
37C2×#3719—62no—
38C2×#3718D312yescomplemented
39C2×#3710—31no—
40C2×D3×C3⋊C8—32no—
41C22×C3×C3⋊D3D411yescharacteristic
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)—21no—
44#6308—21no—
45#6308C2312yes—
46C2×C3×C3⋊Dic3D411yescharacteristic
47C3×C4×C3⋊D3D412yes—
48C3×C4×C3⋊D3C2312yes—
49C2×C32×Dic3—21no—
50C2×C32×Dic3—22no—
51C32×Dic6—21no—
52C32×Dic6—22no—
53C32×Dic6C2314yes—
54C32×Dic6C2312yes—
55#6293—24no—
56Dic3×C3⋊D3—24no—
57C3×#627—24no—
58C3×#624—24no—
59C3×C3⋊(C3⋊C8)C2314yes—
60C22×#216—91no—
61C2×D3×Q16—92no—
62C2×C32×Q8—31no—
63C2×C32×Q8—32no—
64Q8×C3⋊D3—34no—
65Q8×C3⋊D3—62no—
66C2×C4×C3⋊D3—61no—
67C2×C4×C3⋊D3C2×D311yescharacteristic
68C2×#634—32no—
69C2×#634—31no—
70C2×#634—61no—
71C2×#634—61no—
72C2×#634—62no—
73C2×#634—121no—
74C3×D3×Q8—38no—
75C3×D3×Q8—64no—
76C2×C3×C4×D3—32no—
77C2×C3×C4×D3—62no—
78C2×C3×C4×D3—61no—
79C2×C3×Dic6—21no—
80C2×C3×Dic6—22no—
81C2×C3×Dic6—62no—
82C2×C3×Dic6C2×D312yes—
83C2×C3×Dic6C2×D312yes—
84C2×C3×Dic6C2×D312yes—
85C2×#627—21no—
86C2×#627—62no—
87C2×#627—62no—
88C2×#624—21no—
89C2×D3×Dic3—62no—
90C2×D3×Dic3—62no—
91#3797—22no—
92#3797C2×D314yes—
93D3×Dic6—38no—
94D3×Dic6—64no—
95D3×Dic6—64no—
96D3×Dic6—64no—
97#3756—38no—
98C2×C3⋊(C3⋊C8)C2×D312yes—
99C3×#216—38no—
100C3×#216—38no—
101C2×C3×C3⋊C8—32no—
102C2×C3×C3⋊C8—32no—
103C2×C3×C3⋊C8—62no—
104#3719—38no—
105#3719—38no—
106#3719—68no—
107#3718C2×D318yescomplemented
108#3710—34no—
109D3×C3⋊C8—38no—
110C22×C33C2×D411yescharacteristic
111C2×C3×C3⋊D3Q1614yescomplemented
112C2×C3×C3⋊D3C2×D412yes—
113C33×C4C2×D411yescharacteristic
114C33×C4C2411yescharacteristic
115C3⋊(C3⋊Dic3)—22no—
116C3×C3⋊Dic3C2×D412yes—
117C32×Dic3—22no—
118C32×Dic3—24no—
119C22×C3×Q8—91no—
120C2×D3×Q8—32no—
121C2×D3×Q8—92no—
122C2×D3×Q8—181no—
123C22×Dic6—91no—
124C2×C3×Q16—92no—
125C2×#216—32no—
126C2×#216—32no—
127C2×#216—92no—
128C2×#216—92no—
129C2×#216—98no—
130C2×#216—182no—
131C22×C3⋊C8—91no—
132D3×Q16—916no—
133C2×Dic12—92no—
134C2×D3×C8—92no—
135C22×C3⋊D3#6711yescharacteristic
136C22×C3×D3—32no—
137C22×C3×D3—61no—
138C32×Q8—32no—
139C32×Q8—34no—
140C32×Q8—61no—
141C32×Q8—62no—
142C2×C32×C4—22no—
143C2×C32×C4—21no—
144C2×C32×C4—41no—
145C2×C32×C4—61no—
146C2×C32×C4—62no—
147C2×C32×C4C22×D312yes—
148C2×C32×C4C22×D311yescharacteristic
149C2×C3⋊Dic3—61no—
150C2×C3⋊Dic3—62no—
151C2×C3⋊Dic3—62no—
152C2×C3⋊Dic3—122no—
153C2×C3⋊Dic3—122no—
154C2×C3⋊Dic3#6711yescharacteristic
155C4×C3⋊D3—64no—
156C4×C3⋊D3#6712yes—
157C4×C3⋊D3C22×D312yes—
158#634—32no—
159#634—34no—
160#634—62no—
161#634—64no—
162#634—61no—
163#634—64no—
164#634—62no—
165#634—62no—
166#634—122no—
167#634—122no—
168#634—122no—
169C2×C3×Dic3—22no—
170C2×C3×Dic3—22no—
171C2×C3×Dic3—22no—
172C2×C3×Dic3—22no—
173C2×C3×Dic3—32no—
174C2×C3×Dic3—42no—
175C2×C3×Dic3—61no—
176C3×C4×D3—34no—
177C3×C4×D3—34no—
178C3×C4×D3—62no—
179C3×C4×D3—62no—
180C3×C4×D3—68no—
181C3×Dic6—22no—
182C3×Dic6—22no—
183C3×Dic6—22no—
184C3×Dic6—22no—
185C3×Dic6—22no—
186C3×Dic6—24no—
187C3×Dic6—42no—
188C3×Dic6—68no—
189C3×Dic6C22×D314yes—
190C3×Dic6C22×D314yes—
191C3×Dic6C22×D314yes—
192#627—24no—
193#627—68no—
194#627—68no—
195#624—24no—
196D3×Dic3—68no—
197D3×Dic3—68no—
198C3⋊(C3⋊C8)C22×D314yes—
199C3×C3⋊C8—34no—
200C3×C3⋊C8—34no—
201C3×C3⋊C8—64no—
202C22×Q16—271noSylow
203C2×C33C2×Q1612yescomplemented
204C2×C33C22×D411yescharacteristic
205C3×C3⋊D3C2×Q1614yescomplemented
206C2×C3×Q8—32no—
207C2×C3×Q8—32no—
208C2×C3×Q8—32no—
209C2×C3×Q8—61no—
210C2×C3×Q8—62no—
211C2×C3×Q8—94no—
212C2×C3×Q8—181no—
213C2×C3×Q8—182no—
214C22×C3×C4—91no—
215C22×C3×C4—181no—
216C22×Dic3—181no—
217D3×Q8—38no—
218D3×Q8—64no—
219D3×Q8—98no—
220D3×Q8—184no—
221D3×Q8—184no—
222D3×Q8—184no—
223C2×C4×D3—32no—
224C2×C4×D3—61no—
225C2×C4×D3—62no—
226C2×C4×D3—182no—
227C2×C4×D3—182no—
228C2×Dic6—62no—
229C2×Dic6—92no—
230C2×Dic6—91no—
231C2×Dic6—94no—
232C2×Dic6—181no—
233C2×Dic6—182no—
234C2×Dic6—182no—
235C2×Dic6—181no—
236C2×Dic6—182no—
237C2×Dic6—182no—
238C2×Dic6—361no—
239C2×Dic6D3212yes—
240C2×Dic6D3211yescharacteristic
241C3×Q16—98no—
242C2×C3×C8—92no—
243#216—38no—
244#216—38no—
245#216—98no—
246#216—98no—
247#216—188no—
248C2×C3⋊C8—32no—
249C2×C3⋊C8—32no—
250C2×C3⋊C8—62no—
251C2×C3⋊C8—94no—
252Dic12—98no—
253D3×C8—98no—
254C22×C32—22no—
255C22×C32—21no—
256C22×C32—41no—
257C22×C32D3×D412yes—
258C22×C32C2×#6711yescharacteristic
259C2×C3⋊D3#21614yescomplemented
260C2×C3⋊D3C2×#6712yes—
261C2×C3×D3—38no—
262C2×C3×D3—34no—
263C2×C3×D3—64no—
264C2×C3×D3—62no—
265C32×C4—22no—
266C32×C4—21no—
267C32×C4—22no—
268C32×C4—21no—
269C32×C4—41no—
270C32×C4—41no—
271C32×C4—62no—
272C32×C4—64no—
273C32×C4D3×D412yes—
274C32×C4C2×#6711yescharacteristic
275C32×C4C23×D312yes—
276C32×C4C23×D311yescharacteristic
277C3⋊Dic3—62no—
278C3⋊Dic3—64no—
279C3⋊Dic3—64no—
280C3⋊Dic3—124no—
281C3⋊Dic3—124no—
282C3⋊Dic3C2×#6712yes—
283C3×Dic3—24no—
284C3×Dic3—24no—
285C3×Dic3—24no—
286C3×Dic3—24no—
287C3×Dic3—34no—
288C3×Dic3—44no—
289C3×Dic3—62no—
290C22×Q8—91no—
291C22×Q8—271no—
292C2×Q16—92no—
293C2×Q16—272no—
294C2×Q16—278no—
295C22×C8—271no—
296C33C22×Q1611yescharacteristic, complemented, Sylow
297C23×C3—91no—
298C22×D3—32no—
299C22×D3—61no—
300C3×Q8—34no—
301C3×Q8—34no—
302C3×Q8—34no—
303C3×Q8—62no—
304C3×Q8—64no—
305C3×Q8—62no—
306C3×Q8—62no—
307C3×Q8—62no—
308C3×Q8—62no—
309C3×Q8—122no—
310C3×Q8—184no—
311C2×C3×C4—22no—
312C2×C3×C4—21no—
313C2×C3×C4—41no—
314C2×C3×C4—62no—
315C2×C3×C4—62no—
316C2×C3×C4—62no—
317C2×C3×C4—62no—
318C2×C3×C4—91no—
319C2×C3×C4—92no—
320C2×C3×C4—92no—
321C2×C3×C4—122no—
322C2×C3×C4—184no—
323C2×C3×C4C2×D3212yes—
324C2×C3×C4C2×D3211yescharacteristic
325C2×Dic3—22no—
326C2×Dic3—21no—
327C2×Dic3—32no—
328C2×Dic3—61no—
329C2×Dic3—184no—
330C2×Dic3—182no—
331C2×Dic3—182no—
332C2×Dic3—181no—
333C2×Dic3—362no—
334C2×Dic3—362no—
335C4×D3—34no—
336C4×D3—34no—
337C4×D3—62no—
338C4×D3—62no—
339C4×D3—68no—
340C4×D3—188no—
341C4×D3—188no—
342Dic6—22no—
343Dic6—21no—
344Dic6—68no—
345Dic6—94no—
346Dic6—92no—
347Dic6—182no—
348Dic6—184no—
349Dic6—184no—
350Dic6—188no—
351Dic6—188no—
352Dic6—182no—
353Dic6—182no—
354Dic6—181no—
355Dic6—362no—
356Dic6—362no—
357Dic6—362no—
358Dic6C2×D3214yes—
359Dic6C2×D3212yes—
360C3×C8—94no—
361C3⋊C8—34no—
362C3⋊C8—34no—
363C3⋊C8—64no—
364C2×C32—22no—
365C2×C32—21no—
366C2×C32—24no—
367C2×C32—22no—
368C2×C32—42no—
369C2×C32—41no—
370C2×C32D3×Q1614yescomplemented
371C2×C32C2×#21612yescomplemented
372C2×C32C2×D3×D412yes—
373C2×C32C22×#6711yescharacteristic
374C3⋊D3C2×#21614yescomplemented
375C3×D3—38no—
376C3×D3—64no—
377C2×Q8—31no—
378C2×Q8—32no—
379C2×Q8—94no—
380C2×Q8—181no—
381C2×Q8—182no—
382C2×Q8—274no—
383C2×Q8—271no—
384C2×Q8—541no—
385C2×Q8—542no—
386C22×C4—91no—
387C22×C4—181no—
388C22×C4—541no—
389Q16—98no—
390Q16—278no—
391C2×C8—92no—
392C2×C8—274no—
393C22×C3—21no—
394C22×C3—22no—
395C22×C3—41no—
396C22×C3—92no—
397C22×C3—94no—
398C22×C3D3×#6712yes—
399C22×C3#381111yescharacteristic
400C2×D3—34no—
401C2×D3—38no—
402C2×D3—62no—
403C2×D3—64no—
404C3×C4—22no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—22no—
408C3×C4—41no—
409C3×C4—41no—
410C3×C4—64no—
411C3×C4—64no—
412C3×C4—64no—
413C3×C4—64no—
414C3×C4—92no—
415C3×C4—124no—
416C3×C4D3×#6712yes—
417C3×C4#381111yescharacteristic
418C3×C4C22×D3212yes—
419C3×C4C22×D3211yescharacteristic
420Dic3—24no—
421Dic3—22no—
422Dic3—34no—
423Dic3—62no—
424Dic3—184no—
425Dic3—184no—
426Dic3—182no—
427Dic3—364no—
428Dic3—364no—
429C32—21no—
430C32—22no—
431C32—41no—
432C32C2×D3×Q1612yescomplemented
433C32C22×#21611yescharacteristic, complemented
434C23—91no—
435Q8—32no—
436Q8—34no—
437Q8—62no—
438Q8—61no—
439Q8—184no—
440Q8—272no—
441Q8—544no—
442Q8—541no—
443C2×C4—62no—
444C2×C4—61no—
445C2×C4—91no—
446C2×C4—92no—
447C2×C4—92no—
448C2×C4—184no—
449C2×C4—544no—
450C2×C4—541no—
451C2×C4D3311yescharacteristic
452C8—94no—
453C2×C3—22no—
454C2×C3—24no—
455C2×C3—21no—
456C2×C3—22no—
457C2×C3—42no—
458C2×C3—41no—
459C2×C3—94no—
460C2×C3D3×#21614yescomplemented
461C2×C3#6394412yescomplemented
462C2×C3C2×D3×#6712yes—
463C2×C3C2×#381111yescharacteristic
464D3—38no—
465D3—64no—
466C22—94no—
467C22—92no—
468C22#8895211yescharacteristic
469C4—64no—
470C4—62no—
471C4—92no—
472C4—542no—
473C4#8895211yescharacteristic
474C4C2×D3311yescharacteristic
475C3—21no—
476C3—22no—
477C3—41no—
478C3C2×D3×#21612yescomplemented
479C3C2×#6394411yescharacteristic, complemented
480C2—94no—
481C2#1172284612yesdirect product factor
482C2C2×#8895211yescharacteristic
subgroupG/H
centerC22#88952
commutator, cosocleC33×C4C24
FrattiniC4C2×D33
FittingC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC2×C4D33
hypocenterC33C22×Q16
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. C2×#88799C2semidirect
5. C2×C3×#3718C2semidirect
6. #11722846C2direct (N × H)
23. #88799C22semidirect
24. C3×#3718C22semidirect
38. C2×#3718D3semidirect
107. #3718C2×D3semidirect
111. C2×C3×C3⋊D3Q16semidirect
203. C2×C33C2×Q16semidirect
205. C3×C3⋊D3C2×Q16semidirect
259. C2×C3⋊D3#216semidirect
296. C33C22×Q16semidirect
370. C2×C32D3×Q16semidirect
371. C2×C32C2×#216semidirect
374. C3⋊D3C2×#216semidirect
432. C32C2×D3×Q16semidirect
433. C32C22×#216semidirect
460. C2×C3D3×#216semidirect
461. C2×C3#63944semidirect
478. C3C2×D3×#216semidirect
479. C3C2×#63944semidirect
481. C2#11722846direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×Dic62
C2×C3×#37972
C2×#888082
C2×#887992
C2×C3×#37182
#117228462
C2×#639443
C2×D3×#2163