Finite Group Databasebeta

← C2×#11723600 #13178096

Subgroups — C2×#11723600 #13178096

Order 1728.

2508 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×#67C211yesmaximal, characteristic
2C22×#6293C211yesmaximal, characteristic, complemented
3C2×#88864C211yesmaximal, characteristic, complemented
4C2×C3×#3757C211yesmaximal, characteristic, complemented
5C2×(C2×D3)⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
6C2×(C2×D3)⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
7#11723600C218yesmaximal, direct product factor
8C2×Dic3×C3⋊Dic3C211yesmaximal, characteristic, complemented
9C2×#64145—31nomaximal
10C2×#63959—34nomaximal
11C24×C33C2211yescharacteristic
12C23×C32×D3C2211yescharacteristic
13C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C22×C3×C3⋊Dic3C2211yescharacteristic
16C22×C3×C3⋊Dic3C2211yescharacteristic
17C2×C32×#67—24no—
18C2×C32×#67C2214yes—
19C22×C32×Dic3C2211yescharacteristic
20C2×#6293—24no—
21C2×#6293C2214yescomplemented
22#88864C2214yescomplemented
23C3×#3757C2214yescomplemented
24(C2×D3)⋊(C3⋊Dic3)C2214yescomplemented
25(C2×D3)⋊(C3⋊Dic3)C2214yescomplemented
26Dic3×C3⋊Dic3C2214yescomplemented
27C22×C32×D4—31no—
28C22×#634—31no—
29C22×C3×#67D314yes—
30C22×#627—34no—
31C2×#3757—31no—
32C2×#3757—31no—
33C2×#3757—31no—
34C2×#3757—34no—
35C2×#3757—64no—
36C2×#3757D311yescharacteristic, complemented
37#64145—38no—
38C2×C4×C3⋊Dic3—31no—
39C2×C3×#217—34no—
40C2×(C2×D3)⋊Dic3—34no—
41C2×(C2×D3)⋊Dic3—34no—
42#63959—332no—
43C2×Dic32—34no—
44C23×C33—22no—
45C23×C33—22no—
46C23×C33—22no—
47C23×C33C2312yes—
48C23×C33C2311yescharacteristic
49C22×C32×D3—22no—
50C22×C32×D3—22no—
51C22×C32×D3—22no—
52C22×C32×D3C2312yes—
53C2×C3⋊(C3⋊Dic3)—22no—
54C2×C3⋊(C3⋊Dic3)—22no—
55C2×C3⋊(C3⋊Dic3)C2312yes—
56C2×C3⋊(C3⋊Dic3)C2312yes—
57C2×C3×C3⋊Dic3—22no—
58C2×C3×C3⋊Dic3—22no—
59C2×C3×C3⋊Dic3C2312yes—
60C2×C3×C3⋊Dic3C2312yes—
61C32×#67—28no—
62C2×C32×Dic3D414yes—
63C2×C32×Dic3C2312yes—
64#6293—28no—
65C2×#936—94no—
66C2×#885—91no—
67C24×C32—24no—
68C24×C32—31no—
69C24×C32C2×D314yes—
70C24×C32C2×D311yescharacteristic
71C23×C3×D3C2×D314yes—
72C2×C32×D4—34no—
73C2×C32×D4—64no—
74C22×C32×C4—31no—
75C22×C3⋊Dic3—31no—
76C22×C3⋊Dic3—31no—
77C22×C3⋊Dic3—34no—
78C22×C3⋊Dic3—34no—
79C22×C3⋊Dic3—64no—
80C22×C3⋊Dic3—64no—
81C22×C3⋊Dic3C2×D311yescharacteristic
82C22×C3⋊Dic3C2×D311yescharacteristic
83C2×#634—34no—
84C2×#634—64no—
85C2×C3×#67—216no—
86C2×C3×#67C2×D3116yes—
87C22×C3×Dic3—34no—
88C22×C3×Dic3—34no—
89C22×C3×Dic3C2×D314yes—
90C2×#627—316no—
91C2×#627—616no—
92#3757—34no—
93#3757—316no—
94#3757—34no—
95#3757—34no—
96#3757—616no—
97#3757C2×D314yescomplemented
98C4×C3⋊Dic3—34no—
99C3×#217—316no—
100(C2×D3)⋊Dic3—316no—
101(C2×D3)⋊Dic3—316no—
102Dic32—316no—
103C22×C33—22no—
104C22×C33—22no—
105C22×C33—22no—
106C22×C33—44no—
107C22×C33C2×D412yes—
108C22×C33#4112yes—
109C22×C33#4112yes—
110C22×C33C2411yescharacteristic
111C2×C32×D3—22no—
112C2×C32×D3—22no—
113C2×C32×D3—22no—
114C2×C32×D3—44no—
115C3⋊(C3⋊Dic3)—22no—
116C3⋊(C3⋊Dic3)—22no—
117C3×C3⋊Dic3—22no—
118C3×C3⋊Dic3—22no—
119C32×Dic3C2×D414yes—
120C22×C3×D4—34no—
121C22×#67C3⋊D311yescharacteristic
122C22×Dic6—94no—
123C22×Dic6—91no—
124C2×C3×C22⋊C4—91no—
125C2×#217—34no—
126C2×#217—94no—
127C2×#217—94no—
128C2×#217—94no—
129C2×#217—91no—
130C2×#217—184no—
131#936—932no—
132C2×(C2×D3)⋊C4—91no—
133C2×(C2×D3)⋊C4—91no—
134C2×C4×Dic3—94no—
135C2×C4×Dic3—91no—
136#885—98no—
137C23×C32—216no—
138C23×C32—22no—
139C23×C32—22no—
140C23×C32—28no—
141C23×C32—22no—
142C23×C32—24no—
143C23×C32—28no—
144C23×C32—28no—
145C23×C32—28no—
146C23×C32—32no—
147C23×C32—48no—
148C23×C32—48no—
149C23×C32—62no—
150C23×C32—62no—
151C23×C32—62no—
152C23×C32C22×D318yes—
153C23×C32C22×D314yes—
154C23×C32C22×D312yes—
155C23×C32C22×D311yescharacteristic
156C22×C3×D3—28no—
157C22×C3×D3—28no—
158C22×C3×D3—28no—
159C22×C3×D3C22×D318yes—
160C32×D4—68no—
161C2×C32×C4—34no—
162C2×C32×C4—32no—
163C2×C3⋊Dic3—22no—
164C2×C3⋊Dic3—22no—
165C2×C3⋊Dic3—32no—
166C2×C3⋊Dic3—32no—
167C2×C3⋊Dic3—38no—
168C2×C3⋊Dic3—38no—
169C2×C3⋊Dic3—68no—
170C2×C3⋊Dic3—68no—
171C2×C3⋊Dic3—62no—
172C2×C3⋊Dic3—62no—
173C2×C3⋊Dic3—68no—
174C2×C3⋊Dic3—68no—
175C2×C3⋊Dic3—128no—
176C2×C3⋊Dic3—128no—
177C2×C3⋊Dic3C22×D312yes—
178C2×C3⋊Dic3C22×D312yes—
179#634—68no—
180C3×#67—232no—
181C2×C3×Dic3—38no—
182C2×C3×Dic3—38no—
183C2×C3×Dic3—68no—
184C2×C3×Dic3—68no—
185C2×C3×Dic3#67116yes—
186C2×C3×Dic3C22×D318yes—
187#627—632no—
188C2×#124—271noSylow
189C2×C33—42no—
190C2×C33#12414yescomplemented
191C2×C33C22×D411yescharacteristic
192C2×C33C2×#4111yescharacteristic
193C2×C33C2×#4111yescharacteristic
194C32×D3—42no—
195C24×C3—24no—
196C24×C3—34no—
197C24×C3D3214yes—
198C24×C3C2×C3⋊D311yescharacteristic
199C23×D3C2×C3⋊D311yescharacteristic
200C2×C3×D4—316no—
201C2×C3×D4—616no—
202C22×C3×C4—34no—
203C22×C3×C4—91no—
204C22×C3×C4—91no—
205C2×#67—24no—
206C2×#67C2×C3⋊D314yes—
207C22×Dic3—34no—
208C22×Dic3—34no—
209C22×Dic3—94no—
210C22×Dic3—94no—
211C22×Dic3—91no—
212C22×Dic3—91no—
213C22×Dic3—184no—
214C22×Dic3—184no—
215C22×Dic3C2×C3⋊D311yescharacteristic
216C2×Dic6—916no—
217C2×Dic6—94no—
218C2×Dic6—1816no—
219C2×Dic6—184no—
220C3×C22⋊C4—94no—
221#217—316no—
222#217—916no—
223#217—916no—
224#217—916no—
225#217—94no—
226#217—1816no—
227(C2×D3)⋊C4—94no—
228(C2×D3)⋊C4—94no—
229C4×Dic3—916no—
230C4×Dic3—94no—
231C22×C32—216no—
232C22×C32—22no—
233C22×C32—22no—
234C22×C32—28no—
235C22×C32—22no—
236C22×C32—24no—
237C22×C32—28no—
238C22×C32—28no—
239C22×C32—28no—
240C22×C32—28no—
241C22×C32—28no—
242C22×C32—48no—
243C22×C32—48no—
244C22×C32—432no—
245C22×C32—44no—
246C22×C32—416no—
247C22×C32—62no—
248C22×C32—62no—
249C22×C32—62no—
250C22×C32—124no—
251C22×C32#23512yes—
252C22×C32D3×D412yes—
253C22×C32#23718yes—
254C22×C32#23718yes—
255C22×C32#23712yes—
256C22×C32C2×#6718yes—
257C22×C32C23×D314yes—
258C22×C32C23×D311yescharacteristic
259C2×C3×D3—28no—
260C2×C3×D3—28no—
261C2×C3×D3—28no—
262C2×C3×D3—416no—
263C32×C4—34no—
264C3⋊Dic3—22no—
265C3⋊Dic3—22no—
266C3⋊Dic3—62no—
267C3⋊Dic3—62no—
268C3⋊Dic3—68no—
269C3⋊Dic3—68no—
270C3⋊Dic3—128no—
271C3⋊Dic3—128no—
272C3×Dic3—68no—
273C3×Dic3—68no—
274C3×Dic3C2×#67116yes—
275C22×Q8—271no—
276C22×D4—31no—
277#124—278no—
278C2×C22⋊C4—91no—
279C2×C22⋊C4—271no—
280C2×C22⋊C4—271no—
281C2×C22⋊C4—271no—
282C2×C42—271no—
283C33C2×#12411yescharacteristic, complemented, Sylow
284C23×C3—216no—
285C23×C3—28no—
286C23×C3—28no—
287C23×C3—28no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—28no—
292C23×C3—24no—
293C23×C3—38no—
294C23×C3—48no—
295C23×C3—48no—
296C23×C3—68no—
297C23×C3—68no—
298C23×C3—68no—
299C23×C3C2×D3218yes—
300C23×C3C2×D3214yes—
301C23×C3C22×C3⋊D312yes—
302C23×C3C22×C3⋊D311yescharacteristic
303C22×D3—22no—
304C22×D3—22no—
305C22×D3—22no—
306C22×D3C22×C3⋊D312yes—
307C3×D4—632no—
308C2×C3×C4—38no—
309C2×C3×C4—316no—
310C2×C3×C4—92no—
311C2×C3×C4—92no—
312C2×C3×C4—182no—
313C2×C3×C4—182no—
314#67—28no—
315C2×Dic3—38no—
316C2×Dic3—38no—
317C2×Dic3—68no—
318C2×Dic3—68no—
319C2×Dic3—98no—
320C2×Dic3—98no—
321C2×Dic3—92no—
322C2×Dic3—92no—
323C2×Dic3—188no—
324C2×Dic3—188no—
325C2×Dic3—188no—
326C2×Dic3—188no—
327C2×Dic3—182no—
328C2×Dic3—182no—
329C2×Dic3—368no—
330C2×Dic3—368no—
331C2×Dic3#63814yes—
332C2×Dic3C22×C3⋊D312yes—
333Dic6—1832no—
334Dic6—188no—
335C2×C32—24no—
336C2×C32—24no—
337C2×C32—24no—
338C2×C32—216no—
339C2×C32—416no—
340C2×C32—42no—
341C2×C32—48no—
342C2×C32—122no—
343C2×C32#88514yescomplemented
344C2×C32#936116yescomplemented
345C2×C32C2×#23511yescharacteristic
346C2×C32C2×D3×D411yescharacteristic
347C2×C32C2×#23714yes—
348C2×C32C2×#23714yes—
349C2×C32C2×#23711yescharacteristic
350C2×C32C22×#6714yes—
351C3×D3—48no—
352C24—31no—
353C24D3×C3⋊D311yescharacteristic
354C2×Q8—274no—
355C2×Q8—544no—
356C2×D4—34no—
357C2×D4—64no—
358C22×C4—31no—
359C22×C4—91no—
360C22×C4—91no—
361C22×C4—271no—
362C22×C4—271no—
363C22⋊C4—94no—
364C22⋊C4—274no—
365C22⋊C4—274no—
366C22⋊C4—274no—
367C42—274no—
368C22×C3—216no—
369C22×C3—28no—
370C22×C3—28no—
371C22×C3—28no—
372C22×C3—22no—
373C22×C3—22no—
374C22×C3—22no—
375C22×C3—24no—
376C22×C3—28no—
377C22×C3—28no—
378C22×C3—28no—
379C22×C3—48no—
380C22×C3—48no—
381C22×C3—432no—
382C22×C3—44no—
383C22×C3—416no—
384C22×C3—68no—
385C22×C3—68no—
386C22×C3—68no—
387C22×C3—1216no—
388C22×C3#380418yes—
389C22×C3#380518yes—
390C22×C3D3×#6718yes—
391C22×C3#383012yes—
392C22×C3#383012yes—
393C22×C3C2×#63812yes—
394C22×C3C22×D3214yes—
395C22×C3C23×C3⋊D311yescharacteristic
396C2×D3—22no—
397C2×D3—22no—
398C2×D3—22no—
399C2×D3—44no—
400C3×C4—316no—
401C3×C4—182no—
402C3×C4—182no—
403Dic3—68no—
404Dic3—68no—
405Dic3—188no—
406Dic3—188no—
407Dic3—182no—
408Dic3—182no—
409Dic3—368no—
410Dic3—368no—
411Dic3C2×#63814yes—
412C32—24no—
413C32C2×#88511yescharacteristic, complemented
414C32C2×#93614yescomplemented
415C23—22no—
416C23—22no—
417C23—22no—
418C23—32no—
419C23—62no—
420C23—62no—
421C23—62no—
422C23C2×D3×C3⋊D312yes—
423C23C2×D3×C3⋊D311yescharacteristic
424Q8—548no—
425D4—68no—
426C2×C4—32no—
427C2×C4—34no—
428C2×C4—92no—
429C2×C4—92no—
430C2×C4—182no—
431C2×C4—182no—
432C2×C4—272no—
433C2×C4—272no—
434C2×C4—542no—
435C2×C4—542no—
436C2×C3—24no—
437C2×C3—24no—
438C2×C3—24no—
439C2×C3—216no—
440C2×C3—416no—
441C2×C3—48no—
442C2×C3—42no—
443C2×C3—128no—
444C2×C3#63959116yescomplemented
445C2×C3#6414514yescomplemented
446C2×C3C2×#380414yes—
447C2×C3C2×#380514yes—
448C2×C3C2×D3×#6714yes—
449C2×C3C2×#383011yescharacteristic
450C2×C3C2×#383011yescharacteristic
451C2×C3C22×#63811yescharacteristic
452D3—42no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—44no—
457C22—62no—
458C22—62no—
459C22—62no—
460C22—124no—
461C22#8902812yes—
462C22#8902912yes—
463C22D3×#63812yes—
464C22C22×D3×C3⋊D311yescharacteristic
465C4—34no—
466C4—182no—
467C4—182no—
468C4—542no—
469C4—542no—
470C3—24no—
471C3C2×#6395914yescomplemented
472C3C2×#6414511yescharacteristic, complemented
473C2—42no—
474C2—122no—
475C2#1172360014yesdirect product factor
476C2C2×#8902811yescharacteristic
477C2C2×#8902911yescharacteristic
478C2C2×D3×#63811yescharacteristic
subgroupG/H
centerC23C2×D3×C3⋊D3
commutator, cosocleC22×C33C24
FrattiniC22C22×D3×C3⋊D3
FittingC24×C33C22
socleC23×C33C23
prime radicalC22×C32×#67C2
hypercenterC24D3×C3⋊D3
hypocenterC33C2×#124
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C22×#6293C2semidirect
3. C2×#88864C2semidirect
4. C2×C3×#3757C2semidirect
5. C2×(C2×D3)⋊(C3⋊Dic3)C2semidirect
6. C2×(C2×D3)⋊(C3⋊Dic3)C2semidirect
7. #11723600C2direct (N × H)
8. C2×Dic3×C3⋊Dic3C2semidirect
21. C2×#6293C22semidirect
22. #88864C22semidirect
23. C3×#3757C22semidirect
24. (C2×D3)⋊(C3⋊Dic3)C22semidirect
25. (C2×D3)⋊(C3⋊Dic3)C22semidirect
26. Dic3×C3⋊Dic3C22semidirect
36. C2×#3757D3semidirect
97. #3757C2×D3semidirect
190. C2×C33#124semidirect
283. C33C2×#124semidirect
343. C2×C32#885semidirect
344. C2×C32#936semidirect
413. C32C2×#885semidirect
414. C32C2×#936semidirect
444. C2×C3#63959semidirect
445. C2×C3#64145semidirect
471. C3C2×#63959semidirect
472. C3C2×#64145semidirect
475. C2#11723600direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×#672
C22×#62932
C2×#888642
C2×C3×#37572
C2×(C2×D3)⋊(C3⋊Dic3)2
#117236002
C2×Dic3×C3⋊Dic32
C2×#641453
C2×#639593