Finite Group Databasebeta

← C2×#11721104 #13150842

Subgroups — C2×#11721104 #13150842

Order 1728.

1433 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×#272C211yesmaximal, characteristic, complemented
2C22×#6200C211yesmaximal, characteristic
3C2×#3752⋊C3C211yesmaximal, characteristic, complemented
4C2×#3750⋊C3C211yesmaximal, characteristic, complemented
5C2×#88451C211yesmaximal, characteristic, complemented
6C2×#88450C211yesmaximal, characteristic, complemented
7C2×#88448C211yesmaximal, characteristic, complemented
8#11721104C218yesmaximal, direct product factor
9C2×#63905—31nomaximal
10C2×#63850—31nomaximal
11C23×#272C2211yescharacteristic
12C23×(C3⋊D3)⋊C3C2211yescharacteristic
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C22×#1084C2211yescharacteristic
15C2×C4×#272—24no—
16C2×C4×#272C2214yescomplemented
17C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×#6200—24no—
21C2×#6200C2214yes—
22#3752⋊C3C2214yescomplemented
23#3750⋊C3C2214yescomplemented
24#88451C2214yescomplemented
25#88450C2214yescomplemented
26#88448C2214yescomplemented
27C22×C3×C4×D3—31no—
28C22×C3×C4×D3—31no—
29C22×C3×C4×D3—61no—
30C22×#626—31no—
31C22×#625—31no—
32C2×#3752D311yescharacteristic, complemented
33C2×#3750D311yescharacteristic, complemented
34C2×C3×(C2×D3)⋊C4—31no—
35C2×C3×Dic3⋊C4—31no—
36C2×#3724—31no—
37C2×Dic3⋊Dic3—31no—
38C2×#3722—31no—
39C2×(C2×D3)⋊Dic3—31no—
40C2×(C2×D3)⋊Dic3—31no—
41C2×(C2×D3)⋊Dic3—31no—
42#63905—38no—
43#63850—38no—
44C23×H(3,𝔽3)C2311yescharacteristic
45C22×#272—22no—
46C22×#272—22no—
47C22×#272D414yes—
48C22×#272C2312yes—
49C22×(C3⋊D3)⋊C3—22no—
50C22×(C3⋊D3)⋊C3—22no—
51C22×(C3⋊D3)⋊C3—22no—
52C22×(C3⋊D3)⋊C3C2312yes—
53C2×C4×H(3,𝔽3)—22no—
54C2×C4×H(3,𝔽3)C2312yes—
55C2×#1084—22no—
56C2×#1084C2312yes—
57C4×#272—28no—
58C2×(C3⋊Dic3)⋊C3—22no—
59C2×(C3⋊Dic3)⋊C3—22no—
60C2×(C3⋊Dic3)⋊C3—22no—
61C2×(C3⋊Dic3)⋊C3C2312yes—
62C2×(C3⋊Dic3)⋊C3C2312yes—
63C2×(C3⋊Dic3)⋊C3C2312yes—
64#6200—28no—
65C2×#929—91no—
66C2×#894—91no—
67C2×#883—91no—
68C23×C3⋊D3C2×D311yescharacteristic
69C23×C3×D3—31no—
70C23×C3×D3—31no—
71C23×C3×D3—31no—
72C23×C3×D3—61no—
73C22×C32×C4—21no—
74C22×C32×C4C2×D311yescharacteristic
75C22×C32×C4C2×D311yescharacteristic
76C22×C3⋊Dic3C2×D311yescharacteristic
77C22×C3⋊Dic3C2×D311yescharacteristic
78C22×C3⋊Dic3C2×D311yescharacteristic
79C22×C3×Dic3—31no—
80C22×C3×Dic3—31no—
81C22×C3×Dic3—31no—
82C22×C3×Dic3—31no—
83C22×C3×Dic3—31no—
84C22×C3×Dic3—61no—
85C2×C3×C4×D3—34no—
86C2×C3×C4×D3—34no—
87C2×C3×C4×D3—64no—
88C2×C3×C4×D3—64no—
89C2×C3×C4×D3—64no—
90C2×C3×C4×D3—68no—
91C2×#626—34no—
92C2×#626—64no—
93C2×#625—34no—
94C2×#625—64no—
95#3752C2×D314yescomplemented
96#3750C2×D314yescomplemented
97C3×(C2×D3)⋊C4—34no—
98C3×Dic3⋊C4—34no—
99#3724—34no—
100Dic3⋊Dic3—34no—
101#3722—34no—
102(C2×D3)⋊Dic3—34no—
103(C2×D3)⋊Dic3—34no—
104(C2×D3)⋊Dic3—34no—
105C22×H(3,𝔽3)C2×D412yes—
106C22×H(3,𝔽3)#4112yes—
107C22×H(3,𝔽3)#4112yes—
108C22×H(3,𝔽3)C2411yescharacteristic
109C2×#272—28no—
110C2×#272—22no—
111C2×#272—22no—
112C2×#272C2×D414yes—
113C2×(C3⋊D3)⋊C3—22no—
114C2×(C3⋊D3)⋊C3—22no—
115C2×(C3⋊D3)⋊C3—22no—
116C2×(C3⋊D3)⋊C3—44no—
117C4×H(3,𝔽3)—22no—
118#1084—22no—
119(C3⋊Dic3)⋊C3—22no—
120(C3⋊Dic3)⋊C3—22no—
121(C3⋊Dic3)⋊C3—22no—
122C23×C3×C4—91no—
123C22×#67—91no—
124C22×#67—91no—
125C22×D12—91no—
126C22×C4×D3—31no—
127C22×C4×D3—31no—
128C22×C4×D3—61no—
129C2×C3×C4⋊C4—91no—
130C2×C3×C22⋊C4—91no—
131C2×#217—91no—
132C2×#217—91no—
133#929—98no—
134C2×(C2×D3)⋊C4—31no—
135C2×(C2×D3)⋊C4—91no—
136C2×(C2×D3)⋊C4—91no—
137C2×(C2×D3)⋊C4—91no—
138C2×(C2×D3)⋊C4—91no—
139C2×(C2×D3)⋊C4—91no—
140C2×C4⋊Dic3—91no—
141C2×Dic3⋊C4—31no—
142C2×Dic3⋊C4—91no—
143C2×Dic3⋊C4—91no—
144C2×Dic3⋊C4—91no—
145#894—98no—
146#883—98no—
147C23×C32—21no—
148C23×C32C22×D311yescharacteristic
149C23×C32C22×D311yescharacteristic
150C22×C3⋊D3—22no—
151C22×C3⋊D3—22no—
152C22×C3⋊D3—22no—
153C22×C3⋊D3C22×D312yes—
154C22×C3×D3—34no—
155C22×C3×D3—34no—
156C22×C3×D3—32no—
157C22×C3×D3—32no—
158C22×C3×D3—32no—
159C22×C3×D3—64no—
160C22×C3×D3—62no—
161C22×C3×D3—64no—
162C22×C3×D3—64no—
163C22×C3×D3—62no—
164C22×C3×D3—62no—
165C22×C3×D3—62no—
166C22×C3×D3—62no—
167C22×C3×D3—62no—
168C22×C3×D3—62no—
169C22×C3×D3—62no—
170C2×C32×C4—24no—
171C2×C32×C4—22no—
172C2×C32×C4—22no—
173C2×C32×C4—22no—
174C2×C32×C4C22×D312yes—
175C2×C32×C4C22×D312yes—
176C2×C3⋊Dic3—22no—
177C2×C3⋊Dic3—22no—
178C2×C3⋊Dic3—22no—
179C2×C3⋊Dic3C22×D312yes—
180C2×C3⋊Dic3C22×D312yes—
181C2×C3⋊Dic3C22×D312yes—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—32no—
184C2×C3×Dic3—32no—
185C2×C3×Dic3—32no—
186C2×C3×Dic3—32no—
187C2×C3×Dic3—62no—
188C2×C3×Dic3—64no—
189C2×C3×Dic3—62no—
190C2×C3×Dic3—62no—
191C2×C3×Dic3—62no—
192C2×C3×Dic3—62no—
193C2×C3×Dic3—62no—
194C3×C4×D3—68no—
195C3×C4×D3—68no—
196C3×C4×D3—616no—
197#626—68no—
198#625—68no—
199C2×#123—271noSylow
200C2×H(3,𝔽3)#12314yescomplemented
201C2×H(3,𝔽3)C22×D411yescharacteristic
202C2×H(3,𝔽3)C2×#4111yescharacteristic
203C2×H(3,𝔽3)C2×#4111yescharacteristic
204#272—24no—
205(C3⋊D3)⋊C3—42no—
206C24×C3—91no—
207C24×C3—91no—
208C23×D3—31no—
209C23×D3—31no—
210C23×D3—31no—
211C23×D3—61no—
212C23×D3—91no—
213C22×C3×C4—31no—
214C22×C3×C4—31no—
215C22×C3×C4—61no—
216C22×C3×C4—91no—
217C22×C3×C4—91no—
218C22×C3×C4—91no—
219C22×C3×C4—91no—
220C22×C3×C4—94no—
221C22×C3×C4—184no—
222C22×C3×C4D3211yescharacteristic
223C2×#67—94no—
224C2×#67—94no—
225C2×#67—184no—
226C2×#67—184no—
227C22×Dic3—31no—
228C22×Dic3—31no—
229C22×Dic3—31no—
230C22×Dic3—31no—
231C22×Dic3—31no—
232C22×Dic3—61no—
233C22×Dic3—91no—
234C22×Dic3—91no—
235C22×Dic3—91no—
236C2×D12—94no—
237C2×D12—184no—
238C2×C4×D3—34no—
239C2×C4×D3—34no—
240C2×C4×D3—64no—
241C2×C4×D3—64no—
242C2×C4×D3—64no—
243C2×C4×D3—68no—
244C3×C4⋊C4—94no—
245C3×C22⋊C4—94no—
246#217—94no—
247#217—94no—
248(C2×D3)⋊C4—34no—
249(C2×D3)⋊C4—94no—
250(C2×D3)⋊C4—94no—
251(C2×D3)⋊C4—94no—
252(C2×D3)⋊C4—94no—
253(C2×D3)⋊C4—94no—
254C4⋊Dic3—94no—
255Dic3⋊C4—34no—
256Dic3⋊C4—94no—
257Dic3⋊C4—94no—
258Dic3⋊C4—94no—
259C22×C32—21no—
260C22×C32—22no—
261C22×C32—22no—
262C22×C32—22no—
263C22×C32#23512yes—
264C22×C32#23512yes—
265C22×C32D3×D412yes—
266C22×C32D3×D412yes—
267C22×C32#23712yes—
268C22×C32#23912yes—
269C22×C32C23×D311yescharacteristic
270C22×C32C23×D311yescharacteristic
271C2×C3⋊D3—22no—
272C2×C3⋊D3—22no—
273C2×C3⋊D3—22no—
274C2×C3⋊D3—44no—
275C2×C3×D3—34no—
276C2×C3×D3—34no—
277C2×C3×D3—64no—
278C2×C3×D3—64no—
279C2×C3×D3—64no—
280C2×C3×D3—616no—
281C2×C3×D3—68no—
282C2×C3×D3—68no—
283C2×C3×D3—62no—
284C2×C3×D3—62no—
285C2×C3×D3—62no—
286C2×C3×D3—62no—
287C2×C3×D3—62no—
288C2×C3×D3—62no—
289C2×C3×D3—62no—
290C2×C3×D3—124no—
291C32×C4—24no—
292C32×C4—22no—
293C32×C4—22no—
294C3⋊Dic3—22no—
295C3⋊Dic3—22no—
296C3⋊Dic3—22no—
297C3×Dic3—64no—
298C3×Dic3—62no—
299C3×Dic3—62no—
300C3×Dic3—62no—
301C3×Dic3—62no—
302C3×Dic3—62no—
303C22×D4—271no—
304C23×C4—91no—
305#123—278no—
306C2×C4⋊C4—91no—
307C2×C4⋊C4—271no—
308C2×C22⋊C4—91no—
309C2×C22⋊C4—271no—
310C2×C22⋊C4—271no—
311H(3,𝔽3)C2×#12311yescharacteristic, complemented, Sylow
312C23×C3—31no—
313C23×C3—31no—
314C23×C3—61no—
315C23×C3—92no—
316C23×C3—92no—
317C23×C3—94no—
318C23×C3—182no—
319C23×C3—182no—
320C23×C3—182no—
321C23×C3—182no—
322C23×C3—182no—
323C23×C3C2×D3211yescharacteristic
324C22×D3—32no—
325C22×D3—32no—
326C22×D3—34no—
327C22×D3—34no—
328C22×D3—32no—
329C22×D3—62no—
330C22×D3—64no—
331C22×D3—62no—
332C22×D3—62no—
333C22×D3—62no—
334C22×D3—62no—
335C22×D3—64no—
336C22×D3—64no—
337C22×D3—62no—
338C22×D3—62no—
339C22×D3—62no—
340C22×D3—92no—
341C22×D3—182no—
342C22×D3—182no—
343C22×D3—182no—
344C2×C3×C4—22no—
345C2×C3×C4—32no—
346C2×C3×C4—32no—
347C2×C3×C4—62no—
348C2×C3×C4—62no—
349C2×C3×C4—64no—
350C2×C3×C4—62no—
351C2×C3×C4—92no—
352C2×C3×C4—92no—
353C2×C3×C4—92no—
354C2×C3×C4—92no—
355C2×C3×C4—188no—
356C2×C3×C4—182no—
357C2×C3×C4—182no—
358C2×C3×C4—182no—
359C2×C3×C4—182no—
360C2×C3×C4C2×D3212yes—
361#67—188no—
362#67—188no—
363C2×Dic3—32no—
364C2×Dic3—32no—
365C2×Dic3—32no—
366C2×Dic3—32no—
367C2×Dic3—32no—
368C2×Dic3—62no—
369C2×Dic3—62no—
370C2×Dic3—62no—
371C2×Dic3—64no—
372C2×Dic3—62no—
373C2×Dic3—62no—
374C2×Dic3—62no—
375C2×Dic3—92no—
376C2×Dic3—92no—
377C2×Dic3—92no—
378C2×Dic3—182no—
379C2×Dic3—182no—
380C2×Dic3—182no—
381D12—188no—
382C4×D3—68no—
383C4×D3—68no—
384C4×D3—616no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—24no—
389C2×C32#88314yescomplemented
390C2×C32#89414yescomplemented
391C2×C32C2×#23511yescharacteristic
392C2×C32C2×#23511yescharacteristic
393C2×C32C2×D3×D411yescharacteristic
394C2×C32C2×D3×D411yescharacteristic
395C2×C32C2×#23711yescharacteristic
396C2×C32C2×#23911yescharacteristic
397C3⋊D3—42no—
398C3×D3—68no—
399C3×D3—64no—
400C3×D3—64no—
401C3×D3—122no—
402C24—91no—
403C24—91no—
404C2×D4—274no—
405C2×D4—544no—
406C22×C4—91no—
407C22×C4—91no—
408C22×C4—91no—
409C22×C4—91no—
410C22×C4—94no—
411C22×C4—184no—
412C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
413C4⋊C4—94no—
414C4⋊C4—274no—
415C22⋊C4—94no—
416C22⋊C4—274no—
417C22⋊C4—274no—
418C22×C3—31no—
419C22×C3—31no—
420C22×C3—32no—
421C22×C3—32no—
422C22×C3—32no—
423C22×C3—32no—
424C22×C3—32no—
425C22×C3—32no—
426C22×C3—61no—
427C22×C3—62no—
428C22×C3—62no—
429C22×C3—62no—
430C22×C3—94no—
431C22×C3—182no—
432C22×C3—182no—
433C22×C3—182no—
434C22×C3—182no—
435C22×C3—182no—
436C22×C3—188no—
437C22×C3—364no—
438C22×C3#379612yes—
439C22×C3#379812yes—
440C22×C3#381112yes—
441C22×C3C22×D3211yescharacteristic
442C2×D3—34no—
443C2×D3—34no—
444C2×D3—64no—
445C2×D3—68no—
446C2×D3—68no—
447C2×D3—62no—
448C2×D3—62no—
449C2×D3—62no—
450C2×D3—62no—
451C2×D3—616no—
452C2×D3—64no—
453C2×D3—64no—
454C2×D3—62no—
455C2×D3—62no—
456C2×D3—62no—
457C2×D3—124no—
458C2×D3—182no—
459C2×D3—182no—
460C2×D3—182no—
461C2×D3—364no—
462C3×C4—22no—
463C3×C4—62no—
464C3×C4—62no—
465C3×C4—64no—
466C3×C4—182no—
467C3×C4—182no—
468C3×C4—182no—
469C3×C4—182no—
470Dic3—62no—
471Dic3—62no—
472Dic3—64no—
473Dic3—62no—
474Dic3—62no—
475Dic3—62no—
476Dic3—182no—
477Dic3—182no—
478Dic3—182no—
479C32—21no—
480C32C2×#88311yescharacteristic, complemented
481C32C2×#89411yescharacteristic, complemented
482C23—92no—
483C23—94no—
484C23—92no—
485C23—182no—
486C23—182no—
487C23—182no—
488C23—182no—
489C23—182no—
490C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
491D4—548no—
492C2×C4—22no—
493C2×C4—92no—
494C2×C4—92no—
495C2×C4—92no—
496C2×C4—92no—
497C2×C4—188no—
498C2×C4—182no—
499C2×C4—182no—
500C2×C4—182no—
501C2×C4—182no—
502C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
503C2×C3—31no—
504C2×C3—31no—
505C2×C3—34no—
506C2×C3—34no—
507C2×C3—31no—
508C2×C3—31no—
509C2×C3—31no—
510C2×C3—31no—
511C2×C3—61no—
512C2×C3—64no—
513C2×C3—61no—
514C2×C3—61no—
515C2×C3—184no—
516C2×C3—362no—
517C2×C3#6385314yes—
518C2×C3C2×#379611yescharacteristic
519C2×C3C2×#379811yescharacteristic
520C2×C3C2×#381111yescharacteristic
521D3—64no—
522D3—64no—
523D3—68no—
524D3—122no—
525D3—362no—
526C22—94no—
527C22—188no—
528C22—182no—
529C22—182no—
530C22—182no—
531C22—182no—
532C22—182no—
533C22—364no—
534C22#8864912yes—
535C22#8865012yes—
536C22#8867712yes—
537C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
538C4—22no—
539C4—182no—
540C4—182no—
541C4—182no—
542C4—182no—
543C3—31no—
544C3—31no—
545C3—61no—
546C3C2×#6385311yescharacteristic
547C2—184no—
548C2—362no—
549C2#1172110414yesdirect product factor
550C2C2×#8864911yescharacteristic
551C2C2×#8865011yescharacteristic
552C2C2×#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 radicalC22×#6200C2
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#123
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×C4×#272C2semidirect
3. C2×#3752⋊C3C2semidirect
4. C2×#3750⋊C3C2semidirect
5. C2×#88451C2semidirect
6. C2×#88450C2semidirect
7. C2×#88448C2semidirect
8. #11721104C2direct (N × H)
16. C2×C4×#272C22semidirect
22. #3752⋊C3C22semidirect
23. #3750⋊C3C22semidirect
24. #88451C22semidirect
25. #88450C22semidirect
26. #88448C22semidirect
32. C2×#3752D3semidirect
33. C2×#3750D3semidirect
95. #3752C2×D3semidirect
96. #3750C2×D3semidirect
200. C2×H(3,𝔽3)#123semidirect
311. H(3,𝔽3)C2×#123semidirect
389. C2×C32#883semidirect
390. C2×C32#894semidirect
480. C32C2×#883semidirect
481. C32C2×#894semidirect
549. C2#11721104direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×#2722
C22×#62002
C2×#3752⋊C32
C2×#3750⋊C32
C2×#884512
C2×#884502
C2×#884482
#117211042
C2×#639053
C2×#638503