Finite Group Databasebeta

← C2×#11721135 #13150699

Subgroups — C2×#11721135 #13150699

Order 1728.

1283 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6229C211yesmaximal, characteristic, complemented
2C22×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic
3C2×#3752⋊C3C211yesmaximal, characteristic
4C2×#3750⋊C3C211yesmaximal, characteristic, complemented
5C2×#88450C211yesmaximal, characteristic
6C2×#88449C211yesmaximal, characteristic, complemented
7C2×#88448C211yesmaximal, characteristic, complemented
8#11721135C218yesmaximal, direct product factor
9C2×#63888—31nomaximal
10C2×#63887—31nomaximal
11C23×(C3⋊D3)⋊C3C2211yescharacteristic
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C22×#1084C2211yescharacteristic
14C22×#1084C2211yescharacteristic
15C2×#6229—24no—
16C2×#6229C2214yescomplemented
17C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×((C3⋊D3)⋊C3)⋊C4—24no—
21C2×((C3⋊D3)⋊C3)⋊C4C2214yes—
22#3752⋊C3C2214yes—
23#3750⋊C3C2214yescomplemented
24#88450C2214yes—
25#88449C2214yescomplemented
26#88448C2214yescomplemented
27C22×C3×Dic6—31no—
28C22×C3×Dic6—31no—
29C22×C3×Dic6—61no—
30C22×#624—31no—
31C22×D3×Dic3—31no—
32C2×#3752D311yescharacteristic
33C2×#3750D311yescharacteristic, complemented
34C2×C3×(C2×D3)⋊C4—31no—
35C2×C3×Dic3⋊C4—31no—
36C2×#3724—31no—
37C2×Dic3⋊Dic3—31no—
38C2×Dic3⋊Dic3—31no—
39C2×Dic3⋊Dic3—31no—
40C2×#3722—31no—
41C2×(C2×D3)⋊Dic3—31no—
42#63888—38no—
43#63887—38no—
44C23×H(3,𝔽3)C2311yescharacteristic
45C22×(C3⋊D3)⋊C3—22no—
46C22×(C3⋊D3)⋊C3—22no—
47C22×(C3⋊D3)⋊C3Q814yescomplemented
48C22×(C3⋊D3)⋊C3C2312yes—
49C2×C4×H(3,𝔽3)—22no—
50C2×C4×H(3,𝔽3)C2312yes—
51C2×#1084—22no—
52C2×#1084D414yes—
53C2×#1084C2312yes—
54C2×#1084C2312yes—
55#6229—28no—
56C2×(C3⋊Dic3)⋊C3—22no—
57C2×(C3⋊Dic3)⋊C3—22no—
58C2×(C3⋊Dic3)⋊C3—22no—
59C2×(C3⋊Dic3)⋊C3C2312yes—
60C2×(C3⋊Dic3)⋊C3C2312yes—
61C2×(C3⋊Dic3)⋊C3C2312yes—
62((C3⋊D3)⋊C3)⋊C4—28no—
63C2×#946—91no—
64C2×#896—91no—
65C2×#878—91no—
66C23×C3⋊D3C2×D311yescharacteristic
67C23×C3×D3—31no—
68C22×C32×C4—21no—
69C22×C32×C4C2×D311yescharacteristic
70C22×C32×C4C2×D311yescharacteristic
71C22×C3⋊Dic3C2×D311yescharacteristic
72C22×C3⋊Dic3C2×D311yescharacteristic
73C22×C3⋊Dic3C2×D311yescharacteristic
74C22×C3×Dic3—31no—
75C22×C3×Dic3—31no—
76C22×C3×Dic3—31no—
77C22×C3×Dic3—31no—
78C22×C3×Dic3—31no—
79C22×C3×Dic3—31no—
80C22×C3×Dic3—31no—
81C22×C3×Dic3—61no—
82C22×C3×Dic3—61no—
83C2×C3×Dic6—34no—
84C2×C3×Dic6—34no—
85C2×C3×Dic6—64no—
86C2×C3×Dic6—64no—
87C2×C3×Dic6—64no—
88C2×C3×Dic6—68no—
89C2×#624—34no—
90C2×#624—64no—
91C2×D3×Dic3—34no—
92C2×D3×Dic3—64no—
93#3752C2×D314yes—
94#3750C2×D314yescomplemented
95C3×(C2×D3)⋊C4—34no—
96C3×Dic3⋊C4—34no—
97#3724—34no—
98Dic3⋊Dic3—34no—
99Dic3⋊Dic3—34no—
100Dic3⋊Dic3—34no—
101#3722—34no—
102(C2×D3)⋊Dic3—34no—
103C22×H(3,𝔽3)C2×D412yes—
104C22×H(3,𝔽3)C2×Q812yes—
105C22×H(3,𝔽3)#4112yes—
106C22×H(3,𝔽3)C2411yescharacteristic
107C2×(C3⋊D3)⋊C3—28no—
108C2×(C3⋊D3)⋊C3—22no—
109C2×(C3⋊D3)⋊C3—22no—
110C2×(C3⋊D3)⋊C3C2×Q814yescomplemented
111C4×H(3,𝔽3)—22no—
112#1084—22no—
113#1084C2×D414yes—
114(C3⋊Dic3)⋊C3—22no—
115(C3⋊Dic3)⋊C3—22no—
116(C3⋊Dic3)⋊C3—22no—
117C22×C3×Q8—91no—
118C23×Dic3—91no—
119C22×C4×D3—91no—
120C22×C4×D3—91no—
121C22×Dic6—31no—
122C22×Dic6—31no—
123C22×Dic6—61no—
124C2×C3×C4⋊C4—91no—
125C2×C3×C22⋊C4—91no—
126C2×#217—91no—
127#946—98no—
128C2×(C2×D3)⋊C4—31no—
129C2×(C2×D3)⋊C4—91no—
130C2×(C2×D3)⋊C4—91no—
131C2×(C2×D3)⋊C4—91no—
132C2×C4⋊Dic3—91no—
133C2×C4⋊Dic3—91no—
134C2×Dic3⋊C4—31no—
135C2×Dic3⋊C4—91no—
136C2×Dic3⋊C4—91no—
137C2×Dic3⋊C4—91no—
138C2×Dic3⋊C4—91no—
139C2×Dic3⋊C4—91no—
140#896—98no—
141#878—98no—
142C23×C32—21no—
143C23×C32C22×D311yescharacteristic
144C23×C32C22×D311yescharacteristic
145C22×C3⋊D3—22no—
146C22×C3⋊D3—22no—
147C22×C3⋊D3Dic614yescomplemented
148C22×C3⋊D3C22×D312yes—
149C22×C3×D3—34no—
150C22×C3×D3—32no—
151C22×C3×D3—62no—
152C22×C3×D3—62no—
153C2×C32×C4—24no—
154C2×C32×C4—22no—
155C2×C32×C4—22no—
156C2×C32×C4—22no—
157C2×C32×C4C22×D312yes—
158C2×C32×C4C22×D312yes—
159C2×C3⋊Dic3—22no—
160C2×C3⋊Dic3—22no—
161C2×C3⋊Dic3—22no—
162C2×C3⋊Dic3C22×D312yes—
163C2×C3⋊Dic3C22×D312yes—
164C2×C3⋊Dic3C22×D312yes—
165C2×C3×Dic3—34no—
166C2×C3×Dic3—34no—
167C2×C3×Dic3—32no—
168C2×C3×Dic3—32no—
169C2×C3×Dic3—32no—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3—32no—
174C2×C3×Dic3—64no—
175C2×C3×Dic3—62no—
176C2×C3×Dic3—62no—
177C2×C3×Dic3—64no—
178C2×C3×Dic3—62no—
179C2×C3×Dic3—62no—
180C2×C3×Dic3—62no—
181C2×C3×Dic3—62no—
182C2×C3×Dic3—62no—
183C3×Dic6—68no—
184C3×Dic6—68no—
185C3×Dic6—616no—
186#624—68no—
187D3×Dic3—68no—
188C2×C22⋊Q8—271noSylow
189C2×H(3,𝔽3)C22⋊Q814yescomplemented
190C2×H(3,𝔽3)C22×D411yescharacteristic
191C2×H(3,𝔽3)C22×Q811yescharacteristic
192C2×H(3,𝔽3)C2×#4111yescharacteristic
193(C3⋊D3)⋊C3—24no—
194C24×C3—91no—
195C23×D3—31no—
196C23×D3—91no—
197C2×C3×Q8—94no—
198C2×C3×Q8—184no—
199C22×C3×C4—31no—
200C22×C3×C4—31no—
201C22×C3×C4—61no—
202C22×C3×C4—91no—
203C22×C3×C4—91no—
204C22×C3×C4—91no—
205C22×C3×C4—91no—
206C22×C3×C4—91no—
207C22×C3×C4D3211yescharacteristic
208C22×Dic3—31no—
209C22×Dic3—31no—
210C22×Dic3—31no—
211C22×Dic3—31no—
212C22×Dic3—31no—
213C22×Dic3—31no—
214C22×Dic3—31no—
215C22×Dic3—61no—
216C22×Dic3—61no—
217C22×Dic3—91no—
218C22×Dic3—91no—
219C22×Dic3—91no—
220C22×Dic3—94no—
221C22×Dic3—184no—
222C2×C4×D3—94no—
223C2×C4×D3—94no—
224C2×C4×D3—184no—
225C2×C4×D3—184no—
226C2×Dic6—34no—
227C2×Dic6—34no—
228C2×Dic6—64no—
229C2×Dic6—64no—
230C2×Dic6—64no—
231C2×Dic6—68no—
232C3×C4⋊C4—94no—
233C3×C22⋊C4—94no—
234#217—94no—
235(C2×D3)⋊C4—34no—
236(C2×D3)⋊C4—94no—
237(C2×D3)⋊C4—94no—
238(C2×D3)⋊C4—94no—
239C4⋊Dic3—94no—
240C4⋊Dic3—94no—
241Dic3⋊C4—34no—
242Dic3⋊C4—94no—
243Dic3⋊C4—94no—
244Dic3⋊C4—94no—
245Dic3⋊C4—94no—
246Dic3⋊C4—94no—
247C22×C32—21no—
248C22×C32—22no—
249C22×C32—22no—
250C22×C32—22no—
251C22×C32C2×Dic612yes—
252C22×C32#23512yes—
253C22×C32D3×D412yes—
254C22×C32D3×D412yes—
255C22×C32#23712yes—
256C22×C32D3×Q812yes—
257C22×C32C23×D311yescharacteristic
258C22×C32C23×D311yescharacteristic
259C2×C3⋊D3—28no—
260C2×C3⋊D3—22no—
261C2×C3⋊D3—22no—
262C2×C3⋊D3C2×Dic614yescomplemented
263C2×C3×D3—34no—
264C2×C3×D3—68no—
265C2×C3×D3—62no—
266C2×C3×D3—62no—
267C32×C4—24no—
268C32×C4—22no—
269C32×C4—22no—
270C3⋊Dic3—22no—
271C3⋊Dic3—22no—
272C3⋊Dic3—22no—
273C3×Dic3—34no—
274C3×Dic3—34no—
275C3×Dic3—64no—
276C3×Dic3—64no—
277C3×Dic3—62no—
278C3×Dic3—62no—
279C3×Dic3—62no—
280C3×Dic3—62no—
281C3×Dic3—62no—
282C22×Q8—91no—
283C23×C4—271no—
284C22⋊Q8—278no—
285C2×C4⋊C4—91no—
286C2×C4⋊C4—271no—
287C2×C4⋊C4—271no—
288C2×C22⋊C4—91no—
289C2×C22⋊C4—271no—
290H(3,𝔽3)C2×C22⋊Q811yescharacteristic, complemented, Sylow
291C23×C3—31no—
292C23×C3—31no—
293C23×C3—61no—
294C23×C3—92no—
295C23×C3—94no—
296C23×C3—182no—
297C23×C3—182no—
298C23×C3C2×D3211yescharacteristic
299C22×D3—32no—
300C22×D3—34no—
301C22×D3—62no—
302C22×D3—62no—
303C22×D3—94no—
304C22×D3—92no—
305C22×D3—182no—
306C22×D3—182no—
307C3×Q8—188no—
308C2×C3×C4—22no—
309C2×C3×C4—32no—
310C2×C3×C4—32no—
311C2×C3×C4—62no—
312C2×C3×C4—62no—
313C2×C3×C4—64no—
314C2×C3×C4—62no—
315C2×C3×C4—92no—
316C2×C3×C4—94no—
317C2×C3×C4—92no—
318C2×C3×C4—92no—
319C2×C3×C4—92no—
320C2×C3×C4—92no—
321C2×C3×C4—182no—
322C2×C3×C4—182no—
323C2×C3×C4—182no—
324C2×C3×C4—182no—
325C2×C3×C4C2×D3212yes—
326C2×Dic3—32no—
327C2×Dic3—32no—
328C2×Dic3—34no—
329C2×Dic3—34no—
330C2×Dic3—32no—
331C2×Dic3—32no—
332C2×Dic3—32no—
333C2×Dic3—32no—
334C2×Dic3—32no—
335C2×Dic3—62no—
336C2×Dic3—64no—
337C2×Dic3—62no—
338C2×Dic3—62no—
339C2×Dic3—62no—
340C2×Dic3—64no—
341C2×Dic3—62no—
342C2×Dic3—62no—
343C2×Dic3—62no—
344C2×Dic3—92no—
345C2×Dic3—92no—
346C2×Dic3—92no—
347C2×Dic3—182no—
348C2×Dic3—182no—
349C2×Dic3—182no—
350C2×Dic3—188no—
351C4×D3—188no—
352C4×D3—188no—
353Dic6—68no—
354Dic6—68no—
355Dic6—616no—
356C2×C32—21no—
357C2×C32—21no—
358C2×C32—21no—
359C2×C32—24no—
360C2×C32#87814yescomplemented
361C2×C32#89614yescomplemented
362C2×C32C22×Dic611yescharacteristic
363C2×C32C2×#23511yescharacteristic
364C2×C32C2×D3×D411yescharacteristic
365C2×C32C2×D3×D411yescharacteristic
366C2×C32C2×#23711yescharacteristic
367C2×C32C2×D3×Q811yescharacteristic
368C3⋊D3—24no—
369C3×D3—64no—
370C24—91no—
371C2×Q8—94no—
372C2×Q8—184no—
373C22×C4—91no—
374C22×C4—91no—
375C22×C4—91no—
376C22×C4—91no—
377C22×C4—91no—
378C22×C4—274no—
379C22×C4—544no—
380C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
381C4⋊C4—94no—
382C4⋊C4—274no—
383C4⋊C4—274no—
384C22⋊C4—94no—
385C22⋊C4—274no—
386C22×C3—31no—
387C22×C3—31no—
388C22×C3—32no—
389C22×C3—32no—
390C22×C3—32no—
391C22×C3—32no—
392C22×C3—32no—
393C22×C3—32no—
394C22×C3—61no—
395C22×C3—62no—
396C22×C3—62no—
397C22×C3—62no—
398C22×C3—94no—
399C22×C3—182no—
400C22×C3—188no—
401C22×C3—182no—
402C22×C3D3×Dic612yes—
403C22×C3#379512yes—
404C22×C3#381112yes—
405C22×C3C22×D3211yescharacteristic
406C2×D3—34no—
407C2×D3—62no—
408C2×D3—62no—
409C2×D3—68no—
410C2×D3—94no—
411C2×D3—182no—
412C2×D3—188no—
413C2×D3—182no—
414C3×C4—22no—
415C3×C4—62no—
416C3×C4—62no—
417C3×C4—64no—
418C3×C4—94no—
419C3×C4—182no—
420C3×C4—182no—
421C3×C4—182no—
422C3×C4—182no—
423Dic3—34no—
424Dic3—34no—
425Dic3—64no—
426Dic3—62no—
427Dic3—62no—
428Dic3—64no—
429Dic3—62no—
430Dic3—62no—
431Dic3—62no—
432Dic3—182no—
433Dic3—182no—
434Dic3—182no—
435C32—21no—
436C32C2×#87811yescharacteristic, complemented
437C32C2×#89611yescharacteristic, complemented
438C23—92no—
439C23—94no—
440C23—182no—
441C23—182no—
442C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
443Q8—188no—
444C2×C4—22no—
445C2×C4—92no—
446C2×C4—94no—
447C2×C4—92no—
448C2×C4—92no—
449C2×C4—92no—
450C2×C4—92no—
451C2×C4—182no—
452C2×C4—182no—
453C2×C4—182no—
454C2×C4—182no—
455C2×C4—548no—
456C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
457C2×C3—31no—
458C2×C3—31no—
459C2×C3—34no—
460C2×C3—34no—
461C2×C3—31no—
462C2×C3—31no—
463C2×C3—31no—
464C2×C3—31no—
465C2×C3—61no—
466C2×C3—64no—
467C2×C3—61no—
468C2×C3—61no—
469C2×C3—184no—
470C2×C3#6389614yes—
471C2×C3C2×D3×Dic611yescharacteristic
472C2×C3C2×#379511yescharacteristic
473C2×C3C2×#381111yescharacteristic
474D3—64no—
475D3—184no—
476C22—94no—
477C22—182no—
478C22—188no—
479C22—182no—
480C22((C3⋊D3)⋊C3)⋊Q812yes—
481C22#8864812yes—
482C22#8867712yes—
483C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
484C4—22no—
485C4—94no—
486C4—182no—
487C4—182no—
488C4—182no—
489C4—182no—
490C3—31no—
491C3—31no—
492C3—61no—
493C3C2×#6389611yescharacteristic
494C2—184no—
495C2#1172113514yesdirect product factor
496C2C2×((C3⋊D3)⋊C3)⋊Q811yescharacteristic
497C2C2×#8864811yescharacteristic
498C2C2×#8867711yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
FrattiniC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC23×C3C2×D32
prime radicalC23×(C3⋊D3)⋊C3C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×C22⋊Q8
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×#6229C2semidirect
4. C2×#3750⋊C3C2semidirect
6. C2×#88449C2semidirect
7. C2×#88448C2semidirect
8. #11721135C2direct (N × H)
16. C2×#6229C22semidirect
23. #3750⋊C3C22semidirect
25. #88449C22semidirect
26. #88448C22semidirect
33. C2×#3750D3semidirect
47. C22×(C3⋊D3)⋊C3Q8semidirect
94. #3750C2×D3semidirect
110. C2×(C3⋊D3)⋊C3C2×Q8semidirect
147. C22×C3⋊D3Dic6semidirect
189. C2×H(3,𝔽3)C22⋊Q8semidirect
262. C2×C3⋊D3C2×Dic6semidirect
290. H(3,𝔽3)C2×C22⋊Q8semidirect
360. C2×C32#878semidirect
361. C2×C32#896semidirect
436. C32C2×#878semidirect
437. C32C2×#896semidirect
495. C2#11721135direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62292
C22×((C3⋊D3)⋊C3)⋊C42
C2×#3752⋊C32
C2×#3750⋊C32
C2×#884502
C2×#884492
C2×#884482
#117211352
C2×#638883
C2×#638873