Finite Group Databasebeta

← C2×#11721159 #13150845

Subgroups — C2×#11721159 #13150845

Order 1728.

1727 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×#272C211yesmaximal, characteristic, complemented
2C22×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
3C22×#6200C212yesmaximal, complemented
4C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
5C2×#88450C212yesmaximal, complemented
6#11721159C218yesmaximal, direct product factor
7C2×#63909—31nomaximal
8C2×#63908—31nomaximal
9C23×#272C2211yescharacteristic
10C23×(C3⋊D3)⋊C3C2212yes—
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C22×#1084C2211yescharacteristic
13C2×C4×#272—24no—
14C2×C4×#272C2214yescomplemented
15C22×(C3⋊Dic3)⋊C3C2212yes—
16C2×C4⋊((C3⋊D3)⋊C3)—24no—
17C2×C4⋊((C3⋊D3)⋊C3)C2214yescomplemented
18C2×#6200—28no—
19C2×#6200C2218yescomplemented
20C4⋊((C3⋊Dic3)⋊C3)C2214yescomplemented
21#88450C2218yescomplemented
22C22×C4⋊(C3⋊D3)D311yescharacteristic, complemented
23C22×C3×D12—31no—
24C22×C3×C4×D3—31no—
25C22×C3×C4×D3—31no—
26C22×C3×C4×D3—61no—
27C22×#626—32no—
28C22×#625—32no—
29C2×C4⋊(C3⋊Dic3)D311yescharacteristic, complemented
30C2×C3×C4⋊Dic3—31no—
31C2×#3722—32no—
32C2×(C2×D3)⋊Dic3—32no—
33#63909—38no—
34#63908—38no—
35C23×H(3,𝔽3)C2311yescharacteristic
36C22×#272—22no—
37C22×#272—22no—
38C22×#272D414yescomplemented
39C22×#272C2312yes—
40C22×(C3⋊D3)⋊C3—24no—
41C22×(C3⋊D3)⋊C3—24no—
42C22×(C3⋊D3)⋊C3—24no—
43C22×(C3⋊D3)⋊C3C2314yes—
44C2×C4×H(3,𝔽3)D414yescomplemented
45C2×C4×H(3,𝔽3)C2312yes—
46C2×#1084—22no—
47C2×#1084C2312yes—
48C4×#272—28no—
49C2×(C3⋊Dic3)⋊C3—24no—
50C2×(C3⋊Dic3)⋊C3C2314yes—
51C4⋊((C3⋊D3)⋊C3)—28no—
52#6200—216no—
53C2×#938—91no—
54C2×#930—91no—
55C2×#895—91no—
56C23×C3⋊D3C2×D312yes—
57C23×C3×D3—31no—
58C23×C3×D3—31no—
59C23×C3×D3—32no—
60C23×C3×D3—61no—
61C22×C32×C4—21no—
62C22×C32×C4C2×D311yescharacteristic
63C22×C32×C4C2×D311yescharacteristic
64C22×C3⋊Dic3C2×D312yes—
65C2×C4⋊(C3⋊D3)—24no—
66C2×C4⋊(C3⋊D3)C2×D314yescomplemented
67C22×C3×Dic3—32no—
68C22×C3×Dic3—31no—
69C22×C3×Dic3—31no—
70C22×C3×Dic3—61no—
71C2×C3×D12—34no—
72C2×C3×D12—64no—
73C2×C3×C4×D3—34no—
74C2×C3×C4×D3—34no—
75C2×C3×C4×D3—64no—
76C2×C3×C4×D3—68no—
77C2×C3×C4×D3—64no—
78C2×C3×C4×D3—64no—
79C2×#626—38no—
80C2×#626—68no—
81C2×#625—38no—
82C2×#625—68no—
83C4⋊(C3⋊Dic3)C2×D314yescomplemented
84C3×C4⋊Dic3—34no—
85#3722—38no—
86(C2×D3)⋊Dic3—38no—
87C22×H(3,𝔽3)C2×D412yes—
88C22×H(3,𝔽3)C2×D412yes—
89C22×H(3,𝔽3)#4112yes—
90C22×H(3,𝔽3)C2411yescharacteristic
91C2×#272—28no—
92C2×#272—22no—
93C2×#272—22no—
94C2×#272C2×D414yescomplemented
95C2×(C3⋊D3)⋊C3—24no—
96C2×(C3⋊D3)⋊C3—24no—
97C2×(C3⋊D3)⋊C3—24no—
98C2×(C3⋊D3)⋊C3—48no—
99C4×H(3,𝔽3)C2×D414yescomplemented
100#1084—22no—
101(C3⋊Dic3)⋊C3—24no—
102C22×C3×D4—91no—
103C23×C3×C4—91no—
104C22×#67—92no—
105C22×#67—92no—
106C22×D12—31no—
107C22×D12—91no—
108C22×D12—92no—
109C22×C4×D3—31no—
110C22×C4×D3—31no—
111C22×C4×D3—61no—
112C2×C3×C4⋊C4—91no—
113C2×#217—92no—
114#938—98no—
115#930—98no—
116C2×(C2×D3)⋊C4—92no—
117C2×(C2×D3)⋊C4—92no—
118C2×C4⋊Dic3—31no—
119C2×C4⋊Dic3—91no—
120#895—98no—
121C23×C32—21no—
122C23×C32C22×D311yescharacteristic
123C23×C32C22×D311yescharacteristic
124C22×C3⋊D3—24no—
125C22×C3⋊D3—24no—
126C22×C3⋊D3—24no—
127C22×C3⋊D3C22×D314yes—
128C22×C3×D3—34no—
129C22×C3×D3—34no—
130C22×C3×D3—32no—
131C22×C3×D3—32no—
132C22×C3×D3—34no—
133C22×C3×D3—64no—
134C22×C3×D3—62no—
135C22×C3×D3—64no—
136C22×C3×D3—64no—
137C22×C3×D3—62no—
138C22×C3×D3—62no—
139C22×C3×D3—62no—
140C22×C3×D3—62no—
141C22×C3×D3—64no—
142C22×C3×D3—64no—
143C22×C3×D3—64no—
144C2×C32×C4—24no—
145C2×C32×C4—22no—
146C2×C32×C4D1214yescomplemented
147C2×C32×C4#6714yescomplemented
148C2×C32×C4C22×D312yes—
149C2×C32×C4C22×D312yes—
150C2×C3⋊Dic3—24no—
151C2×C3⋊Dic3C22×D314yes—
152C4⋊(C3⋊D3)—28no—
153C2×C3×Dic3—34no—
154C2×C3×Dic3—32no—
155C2×C3×Dic3—32no—
156C2×C3×Dic3—62no—
157C2×C3×Dic3—64no—
158C2×C3×Dic3—62no—
159C2×C3×Dic3—64no—
160C2×C3×Dic3—62no—
161C3×D12—68no—
162C3×C4×D3—616no—
163C3×C4×D3—68no—
164C3×C4×D3—68no—
165#626—616no—
166#625—616no—
167C2×#121—271noSylow
168C2×H(3,𝔽3)#12114yescomplemented
169C2×H(3,𝔽3)C22×D411yescharacteristic
170C2×H(3,𝔽3)C22×D411yescharacteristic
171C2×H(3,𝔽3)C2×#4111yescharacteristic
172#272—24no—
173(C3⋊D3)⋊C3—44no—
174C24×C3—92no—
175C24×C3—91no—
176C23×D3—31no—
177C23×D3—31no—
178C23×D3—32no—
179C23×D3—61no—
180C23×D3—92no—
181C2×C3×D4—94no—
182C2×C3×D4—184no—
183C22×C3×C4—31no—
184C22×C3×C4—31no—
185C22×C3×C4—61no—
186C22×C3×C4—92no—
187C22×C3×C4—91no—
188C22×C3×C4—94no—
189C22×C3×C4—184no—
190C22×C3×C4D3211yescharacteristic
191C2×#67—98no—
192C2×#67—98no—
193C2×#67—188no—
194C2×#67—188no—
195C22×Dic3—31no—
196C22×Dic3—31no—
197C22×Dic3—32no—
198C22×Dic3—61no—
199C22×Dic3—92no—
200C2×D12—34no—
201C2×D12—64no—
202C2×D12—94no—
203C2×D12—98no—
204C2×D12—184no—
205C2×D12—188no—
206C2×C4×D3—34no—
207C2×C4×D3—34no—
208C2×C4×D3—64no—
209C2×C4×D3—64no—
210C2×C4×D3—64no—
211C2×C4×D3—68no—
212C3×C4⋊C4—94no—
213#217—98no—
214(C2×D3)⋊C4—98no—
215(C2×D3)⋊C4—98no—
216C4⋊Dic3—34no—
217C4⋊Dic3—94no—
218C22×C32—21no—
219C22×C32—22no—
220C22×C32—22no—
221C22×C32—22no—
222C22×C32C2×D1212yes—
223C22×C32D3×D412yes—
224C22×C32D3×D412yes—
225C22×C32#23712yes—
226C22×C32#23912yes—
227C22×C32C2×#6712yes—
228C22×C32C23×D311yescharacteristic
229C22×C32C23×D311yescharacteristic
230C2×C3⋊D3—24no—
231C2×C3⋊D3—24no—
232C2×C3⋊D3—24no—
233C2×C3⋊D3—48no—
234C2×C3×D3—34no—
235C2×C3×D3—34no—
236C2×C3×D3—64no—
237C2×C3×D3—64no—
238C2×C3×D3—64no—
239C2×C3×D3—616no—
240C2×C3×D3—68no—
241C2×C3×D3—68no—
242C2×C3×D3—62no—
243C2×C3×D3—62no—
244C2×C3×D3—62no—
245C2×C3×D3—62no—
246C2×C3×D3—64no—
247C2×C3×D3—64no—
248C2×C3×D3—64no—
249C2×C3×D3—128no—
250C32×C4—24no—
251C32×C4C2×D1214yescomplemented
252C32×C4C2×#6714yescomplemented
253C3⋊Dic3—24no—
254C3×Dic3—64no—
255C3×Dic3—62no—
256C3×Dic3—64no—
257C3×Dic3—62no—
258C22×D4—91no—
259C22×D4—272no—
260C23×C4—91no—
261#121—278no—
262C2×C4⋊C4—91no—
263C2×C22⋊C4—272no—
264H(3,𝔽3)C2×#12111yescharacteristic, complemented, Sylow
265C23×C3—31no—
266C23×C3—31no—
267C23×C3—61no—
268C23×C3—94no—
269C23×C3—92no—
270C23×C3—94no—
271C23×C3—184no—
272C23×C3—184no—
273C23×C3—184no—
274C23×C3—182no—
275C23×C3—182no—
276C23×C3C2×D3211yescharacteristic
277C22×D3—32no—
278C22×D3—32no—
279C22×D3—34no—
280C22×D3—34no—
281C22×D3—34no—
282C22×D3—62no—
283C22×D3—64no—
284C22×D3—62no—
285C22×D3—62no—
286C22×D3—62no—
287C22×D3—62no—
288C22×D3—64no—
289C22×D3—64no—
290C22×D3—64no—
291C22×D3—64no—
292C22×D3—64no—
293C22×D3—94no—
294C22×D3—184no—
295C22×D3—184no—
296C22×D3—184no—
297C3×D4—188no—
298C2×C3×C4—34no—
299C2×C3×C4—34no—
300C2×C3×C4—32no—
301C2×C3×C4—32no—
302C2×C3×C4—64no—
303C2×C3×C4—62no—
304C2×C3×C4—94no—
305C2×C3×C4—92no—
306C2×C3×C4—184no—
307C2×C3×C4—182no—
308C2×C3×C4—188no—
309C2×C3×C4#62614yes—
310C2×C3×C4C2×D3212yes—
311#67—1816no—
312#67—1816no—
313C2×Dic3—32no—
314C2×Dic3—32no—
315C2×Dic3—34no—
316C2×Dic3—62no—
317C2×Dic3—62no—
318C2×Dic3—62no—
319C2×Dic3—64no—
320C2×Dic3—64no—
321C2×Dic3—94no—
322C2×Dic3—184no—
323D12—68no—
324D12—188no—
325D12—1816no—
326C4×D3—68no—
327C4×D3—68no—
328C4×D3—616no—
329C2×C32—21no—
330C2×C32—21no—
331C2×C32—21no—
332C2×C32—24no—
333C2×C32#89514yescomplemented
334C2×C32#93814yescomplemented
335C2×C32C22×D1211yescharacteristic
336C2×C32C2×D3×D411yescharacteristic
337C2×C32C2×D3×D411yescharacteristic
338C2×C32C2×#23711yescharacteristic
339C2×C32C2×#23911yescharacteristic
340C2×C32C22×#6711yescharacteristic
341C3⋊D3—44no—
342C3×D3—68no—
343C3×D3—64no—
344C3×D3—64no—
345C3×D3—124no—
346C24—91no—
347C24—92no—
348C2×D4—94no—
349C2×D4—184no—
350C2×D4—278no—
351C2×D4—548no—
352C22×C4—91no—
353C22×C4—92no—
354C22×C4—94no—
355C22×C4—184no—
356C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
357C4⋊C4—94no—
358C22⋊C4—278no—
359C22×C3—31no—
360C22×C3—31no—
361C22×C3—32no—
362C22×C3—32no—
363C22×C3—32no—
364C22×C3—32no—
365C22×C3—32no—
366C22×C3—32no—
367C22×C3—61no—
368C22×C3—62no—
369C22×C3—62no—
370C22×C3—62no—
371C22×C3—94no—
372C22×C3—184no—
373C22×C3—184no—
374C22×C3—184no—
375C22×C3—182no—
376C22×C3—182no—
377C22×C3—188no—
378C22×C3—368no—
379C22×C3#379612yes—
380C22×C3#380212yes—
381C22×C3C2×#62612yes—
382C22×C3C22×D3211yescharacteristic
383C2×D3—34no—
384C2×D3—34no—
385C2×D3—64no—
386C2×D3—62no—
387C2×D3—62no—
388C2×D3—68no—
389C2×D3—68no—
390C2×D3—62no—
391C2×D3—62no—
392C2×D3—64no—
393C2×D3—616no—
394C2×D3—64no—
395C2×D3—64no—
396C2×D3—64no—
397C2×D3—64no—
398C2×D3—128no—
399C2×D3—184no—
400C2×D3—184no—
401C2×D3—184no—
402C2×D3—368no—
403C3×C4—34no—
404C3×C4—34no—
405C3×C4—64no—
406C3×C4—184no—
407C3×C4—182no—
408C3×C4C2×#62614yes—
409Dic3—62no—
410Dic3—62no—
411Dic3—64no—
412Dic3—64no—
413Dic3—184no—
414C32—21no—
415C32C2×#89511yescharacteristic, complemented
416C32C2×#93811yescharacteristic, complemented
417C23—92no—
418C23—94no—
419C23—94no—
420C23—182no—
421C23—182no—
422C23—184no—
423C23—184no—
424C23—184no—
425C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
426D4—188no—
427D4—5416no—
428C2×C4—92no—
429C2×C4—94no—
430C2×C4—184no—
431C2×C4—182no—
432C2×C4—188no—
433C2×C4#620014yescomplemented
434C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
435C2×C3—31no—
436C2×C3—31no—
437C2×C3—34no—
438C2×C3—34no—
439C2×C3—31no—
440C2×C3—31no—
441C2×C3—31no—
442C2×C3—31no—
443C2×C3—61no—
444C2×C3—64no—
445C2×C3—61no—
446C2×C3—61no—
447C2×C3—184no—
448C2×C3—364no—
449C2×C3#6391614yes—
450C2×C3C2×#379611yescharacteristic
451C2×C3C2×#380211yescharacteristic
452C2×C3C22×#62611yescharacteristic
453D3—64no—
454D3—64no—
455D3—68no—
456D3—124no—
457D3—364no—
458C22—94no—
459C22—182no—
460C22—188no—
461C22—182no—
462C22—184no—
463C22—184no—
464C22—184no—
465C22—368no—
466C22#8864912yes—
467C22#8865512yes—
468C22C2×#620012yes—
469C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
470C4—182no—
471C4—184no—
472C4C2×#620014yescomplemented
473C3—31no—
474C3—31no—
475C3—61no—
476C3C2×#6391611yescharacteristic
477C2—184no—
478C2—364no—
479C2#1172115914yesdirect product factor
480C2C2×#8864911yescharacteristic
481C2C2×#8865511yescharacteristic
482C2C22×#620011yescharacteristic
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
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#121
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
2. C22×C4⋊((C3⋊D3)⋊C3)C2semidirect
3. C22×#6200C2semidirect
4. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
5. C2×#88450C2semidirect
6. #11721159C2direct (N × H)
14. C2×C4×#272C22semidirect
17. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
19. C2×#6200C22semidirect
20. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
21. #88450C22semidirect
22. C22×C4⋊(C3⋊D3)D3semidirect
29. C2×C4⋊(C3⋊Dic3)D3semidirect
38. C22×#272D4semidirect
44. C2×C4×H(3,𝔽3)D4semidirect
66. C2×C4⋊(C3⋊D3)C2×D3semidirect
83. C4⋊(C3⋊Dic3)C2×D3semidirect
94. C2×#272C2×D4semidirect
99. C4×H(3,𝔽3)C2×D4semidirect
146. C2×C32×C4D12semidirect
147. C2×C32×C4#67semidirect
168. C2×H(3,𝔽3)#121semidirect
251. C32×C4C2×D12semidirect
252. C32×C4C2×#67semidirect
264. H(3,𝔽3)C2×#121semidirect
333. C2×C32#895semidirect
334. C2×C32#938semidirect
415. C32C2×#895semidirect
416. C32C2×#938semidirect
433. C2×C4#6200semidirect
472. C4C2×#6200semidirect
479. C2#11721159direct (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×C4⋊((C3⋊D3)⋊C3)2
C22×#62002
C2×C4⋊((C3⋊Dic3)⋊C3)2
C2×#884502
#117211592
C2×#639093
C2×#639083