Finite Group Databasebeta

← #13175371

Subgroups — #13175371

Order 1728.

1672 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C4⋊(C3⋊(C3⋊D3))C211yesmaximal, characteristic, complemented
2C3×C4×#634C211yesmaximal, characteristic, complemented
3C32×#872C211yesmaximal, characteristic, complemented
4#11723534C211yesmaximal, characteristic, complemented
5#11723516C211yesmaximal, characteristic
6#11723446C212yesmaximal, complemented
7#11723442C212yesmaximal, complemented
8#11723441C211yesmaximal, characteristic, complemented
9#11723425C211yesmaximal, characteristic, complemented
10#11723420C212yesmaximal, complemented
11#11723419C212yesmaximal, complemented
12#10596005—31nomaximal
13#10593878—34nomaximal
14C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
15C2×C4×C3⋊(C3⋊D3)C2212yes—
16C2×C3×#634C2211yescharacteristic
17C2×C32×C4×D3C2211yescharacteristic
18C2×#6293C2212yes—
19C2×#6291C2212yes—
20C33×C42C2211yescharacteristic
21#88859C2212yes—
22C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
23C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
24C3×#3750C2212yes—
25C3×C4×C3⋊Dic3C2212yes—
26C32×(C2×D3)⋊C4C2211yescharacteristic
27C32×(C2×D3)⋊C4C2211yescharacteristic
28C32×Dic3⋊C4C2211yescharacteristic
29C32×Dic3⋊C4C2211yescharacteristic
30C32×C4×Dic3C2211yescharacteristic
31#88806C2212yes—
32Dic3⋊(C3⋊Dic3)C2212yes—
33#88804C2212yes—
34#88804C2212yes—
35#88804C2212yes—
36(C2×D3)⋊(C3⋊Dic3)C2212yes—
37C32×#117—31no—
38#64139—31no—
39#64133—31no—
40#64107—32no—
41#64105—32no—
42#64094—32no—
43#64092—32no—
44#64084—31no—
45C4×C4⋊(C3⋊D3)—31no—
46C4×C4⋊(C3⋊D3)—34no—
47C4×C4⋊(C3⋊D3)—64no—
48#64078—31no—
49C4×#634D311yescharacteristic, complemented
50C3×#872D314yescomplemented
51C3×C4×Dic6—34no—
52#63918—34no—
53#63909—34no—
54#63868—38no—
55#63865—38no—
56#63864—34no—
57#63856—34no—
58#63853—38no—
59#63852—38no—
60C22×C3⋊(C3⋊D3)C2312yes—
61C22×C32×D3C2311yescharacteristic
62C2×C33×C4C2311yescharacteristic
63C2×C33×C4C2311yescharacteristic
64C2×C33×C4C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)C2312yes—
66C4⋊(C3⋊(C3⋊D3))—22no—
67C4×C3⋊(C3⋊D3)—42no—
68C2×C3×C3⋊Dic3C2312yes—
69C2×C3×C3⋊Dic3C2312yes—
70C3×#634—22no—
71C2×C32×Dic3C2311yescharacteristic
72C2×C32×Dic3C2311yescharacteristic
73C2×C32×Dic3C2311yescharacteristic
74C32×C4×D3—22no—
75#6293—42no—
76#6291—42no—
77#5582—91no—
78#5534—94no—
79C22×C32×C4—31no—
80C2×#638—32no—
81C2×C4⋊(C3⋊D3)—31no—
82C2×C4⋊(C3⋊D3)—34no—
83C2×C4⋊(C3⋊D3)—64no—
84C2×C4×C3⋊D3—32no—
85C2×C4×C3⋊D3—38no—
86C2×C4×C3⋊D3—68no—
87C2×#634—32no—
88C2×#634C2×D311yescharacteristic
89C2×C3×C4×D3C2×D314yes—
90C2×C3×Dic6—34no—
91C2×#627—38no—
92C2×#626—38no—
93C32×C4⋊C4—31no—
94C32×C4⋊C4—31no—
95C32×C22⋊C4—31no—
96C32×C22⋊C4—31no—
97C32×C42—24no—
98C32×C42—31no—
99C32×C42C2×D314yes—
100C32×C42C2×D311yescharacteristic
101#3757—32no—
102#3752—32no—
103#3752—38no—
104#3752—32no—
105#3752—32no—
106#3752—32no—
107#3752—68no—
108C4⋊(C3⋊Dic3)—31no—
109C4⋊(C3⋊Dic3)—34no—
110C4⋊(C3⋊Dic3)—32no—
111C4⋊(C3⋊Dic3)—64no—
112C4⋊(C3⋊Dic3)C2×D311yescharacteristic
113#3750—32no—
114#3750C2×D312yes—
115C4×C3⋊Dic3C2×D312yes—
116C3×(C2×D3)⋊C4C2×D314yes—
117C3×(C2×D3)⋊C4C2×D314yes—
118C3×C4⋊Dic3—34no—
119C3×Dic3⋊C4—38no—
120C3×Dic3⋊C4C2×D314yes—
121C3×Dic3⋊C4C2×D314yes—
122C3×C4×Dic3—38no—
123C3×C4×Dic3C2×D314yes—
124#3724—38no—
125Dic3⋊Dic3—38no—
126#3722—38no—
127#3722—38no—
128#3722—38no—
129(C2×D3)⋊Dic3—38no—
130C22×C33C2411yescharacteristic
131C2×C3⋊(C3⋊D3)—22no—
132C2×C3⋊(C3⋊D3)—22no—
133C2×C3⋊(C3⋊D3)—22no—
134C2×C32×D3—21no—
135C2×C32×D3—21no—
136C2×C32×D3—21no—
137C33×C4—21no—
138C33×C4—21no—
139C33×C4#4112yes—
140C3⋊(C3⋊Dic3)—22no—
141C3×C3⋊Dic3—22no—
142C3×C3⋊Dic3—22no—
143C32×Dic3—21no—
144C32×Dic3—21no—
145C32×Dic3—21no—
146C3×C4×Q8—91no—
147C3×#117—34no—
148#947—91no—
149#946—91no—
150#930—94no—
151#924—94no—
152#898—98no—
153#896—98no—
154#896—92no—
155#895—91no—
156#894—92no—
157#888—91no—
158#885—98no—
159#883—98no—
160#876—92no—
161#875—94no—
162#875—92no—
163C4×D12—94no—
164C4×D12—91no—
165C4×D12—184no—
166#872C3⋊D311yescharacteristic, complemented
167#869—94no—
168C4×Dic6—34no—
169C23×C32—31no—
170C22×C3⋊D3—32no—
171C22×C3⋊D3—38no—
172C22×C3⋊D3—68no—
173C22×C3×D3C22×D314yes—
174C2×C32×C4—24no—
175C2×C32×C4—24no—
176C2×C32×C4—24no—
177C2×C32×C4—31no—
178C2×C32×C4—31no—
179C2×C32×C4—31no—
180C2×C32×C4—62no—
181C2×C32×C4C22×D314yes—
182C2×C32×C4C22×D314yes—
183C2×C32×C4C22×D314yes—
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D311yescharacteristic
187#638—122no—
188C2×C3⋊Dic3—32no—
189C2×C3⋊Dic3—38no—
190C2×C3⋊Dic3—68no—
191C2×C3⋊Dic3C22×D312yes—
192C2×C3⋊Dic3C22×D312yes—
193C4⋊(C3⋊D3)—616no—
194C4⋊(C3⋊D3)—68no—
195C4⋊(C3⋊D3)—62no—
196C4×C3⋊D3—122no—
197C4×C3⋊D3—128no—
198C4×C3⋊D3—1216no—
199#634—22no—
200#634—122no—
201C2×C3×Dic3—38no—
202C2×C3×Dic3—38no—
203C2×C3×Dic3C22×D314yes—
204C2×C3×Dic3C22×D314yes—
205C2×C3×Dic3C22×D314yes—
206C3×C4×D3—28no—
207C3×Dic6—68no—
208#627—128no—
209#626—128no—
210#542—271noSylow
211C2×C33C2×#4111yescharacteristic
212C2×C33H(5,𝔽2)11yescharacteristic
213C2×C332−1+411yescharacteristic
214C3⋊(C3⋊D3)—42no—
215C32×D3—41no—
216C2×C3×Q8—91no—
217C22×C3×C4—34no—
218C2×#67—98no—
219C2×D12—94no—
220C2×D12—91no—
221C2×D12—92no—
222C2×D12—184no—
223C2×C4×D3—98no—
224C2×C4×D3—92no—
225C2×C4×D3—188no—
226C2×C4×D3C2×C3⋊D311yescharacteristic
227C2×Dic6—34no—
228C2×Dic6—98no—
229C2×Dic6—92no—
230C3×C4⋊C4—34no—
231C3×C4⋊C4—34no—
232C3×C4⋊C4—91no—
233C3×C4⋊C4—92no—
234C3×C22⋊C4—34no—
235C3×C22⋊C4—34no—
236C3×C42—24no—
237C3×C42—34no—
238C3×C42—92no—
239C3×C42D3214yes—
240C3×C42C2×C3⋊D311yescharacteristic
241#217—98no—
242(C2×D3)⋊C4—98no—
243(C2×D3)⋊C4—98no—
244(C2×D3)⋊C4—98no—
245(C2×D3)⋊C4—92no—
246(C2×D3)⋊C4—98no—
247(C2×D3)⋊C4—92no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4—92no—
250(C2×D3)⋊C4—92no—
251(C2×D3)⋊C4—188no—
252(C2×D3)⋊C4C2×C3⋊D311yescharacteristic
253(C2×D3)⋊C4C2×C3⋊D311yescharacteristic
254C4⋊Dic3—34no—
255C4⋊Dic3—98no—
256C4⋊Dic3—94no—
257C4⋊Dic3—91no—
258C4⋊Dic3—184no—
259Dic3⋊C4—38no—
260Dic3⋊C4—98no—
261Dic3⋊C4—92no—
262Dic3⋊C4—92no—
263Dic3⋊C4C2×C3⋊D311yescharacteristic
264Dic3⋊C4C2×C3⋊D311yescharacteristic
265C4×Dic3—38no—
266C4×Dic3C2×C3⋊D311yescharacteristic
267C22×C32—24no—
268C22×C32—61no—
269C22×C32—61no—
270C22×C32—61no—
271C22×C32C23×D314yes—
272C22×C32C23×D311yescharacteristic
273C2×C3⋊D3—616no—
274C2×C3⋊D3—68no—
275C2×C3⋊D3—62no—
276C2×C3⋊D3—62no—
277C2×C3⋊D3—62no—
278C2×C3⋊D3—68no—
279C2×C3⋊D3—68no—
280C2×C3⋊D3—128no—
281C2×C3⋊D3—128no—
282C2×C3×D3—24no—
283C2×C3×D3—24no—
284C2×C3×D3—24no—
285C32×C4—28no—
286C32×C4—21no—
287C32×C4—24no—
288C32×C4—21no—
289C32×C4—28no—
290C32×C4—24no—
291C32×C4—44no—
292C32×C4—61no—
293C32×C4—61no—
294C32×C4—61no—
295C32×C4#23518yes—
296C32×C4#23512yes—
297C3⋊Dic3—22no—
298C3⋊Dic3—22no—
299C3⋊Dic3—62no—
300C3⋊Dic3—68no—
301C3⋊Dic3—128no—
302C3×Dic3—24no—
303C3×Dic3—24no—
304C3×Dic3—24no—
305C3×Dic3—68no—
306C3×Dic3—68no—
307C4⋊Q8—271no—
308#126—272no—
309#124—271no—
310#124—272no—
311#123—272no—
312C22⋊Q8—272no—
313C22⋊Q8—271no—
314#121—271no—
315C4×Q8—91no—
316C4×D4—271no—
317#117—31no—
318C33#54211yescharacteristic, complemented, Sylow
319C23×C3—34no—
320C22×D3—98no—
321C22×D3—92no—
322C22×D3—188no—
323C22×D3C22×C3⋊D311yescharacteristic
324C3×Q8—182no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—34no—
329C2×C3×C4—34no—
330C2×C3×C4—34no—
331C2×C3×C4—68no—
332C2×C3×C4—92no—
333C2×C3×C4—92no—
334C2×C3×C4C2×D3214yes—
335C2×C3×C4C2×D3214yes—
336C2×C3×C4C2×D3214yes—
337C2×C3×C4C22×C3⋊D311yescharacteristic
338C2×C3×C4C22×C3⋊D311yescharacteristic
339C2×C3×C4C22×C3⋊D311yescharacteristic
340#67—368no—
341C2×Dic3—38no—
342C2×Dic3—38no—
343C2×Dic3—98no—
344C2×Dic3—92no—
345C2×Dic3—188no—
346C2×Dic3C22×C3⋊D311yescharacteristic
347C2×Dic3C22×C3⋊D311yescharacteristic
348C2×Dic3C22×C3⋊D311yescharacteristic
349D12—1816no—
350D12—188no—
351D12—182no—
352D12—362no—
353C4×D3—22no—
354C4×D3—368no—
355C4×D3—362no—
356C4×D3—3616no—
357Dic6—68no—
358Dic6—362no—
359Dic6—368no—
360C2×C32—24no—
361C2×C32—24no—
362C2×C32—24no—
363C2×C32—121no—
364C2×C32C2×#23514yes—
365C2×C32C2×#23511yescharacteristic
366C2×C32#100711yescharacteristic
367C2×C32#100914yes—
368C2×C32#100911yescharacteristic
369C2×C32#101014yes—
370C3⋊D3—1216no—
371C3⋊D3—128no—
372C3⋊D3—122no—
373C3×D3—44no—
374C2×Q8—91no—
375C2×Q8—272no—
376C2×D4—272no—
377C2×D4—271no—
378C22×C4—31no—
379C22×C4—272no—
380C4⋊C4—31no—
381C4⋊C4—31no—
382C4⋊C4—91no—
383C4⋊C4—92no—
384C4⋊C4—272no—
385C4⋊C4—272no—
386C4⋊C4—271no—
387C22⋊C4—31no—
388C22⋊C4—31no—
389C22⋊C4—272no—
390C22⋊C4—272no—
391C22⋊C4—272no—
392C22⋊C4—272no—
393C22⋊C4—272no—
394C42—31no—
395C42—92no—
396C42D3×C3⋊D311yescharacteristic
397C22×C3—24no—
398C22×C3—64no—
399C22×C3—64no—
400C22×C3—64no—
401C22×C3C22×D3214yes—
402C22×C3C23×C3⋊D311yescharacteristic
403C2×D3—21no—
404C2×D3—21no—
405C2×D3—21no—
406C2×D3—1816no—
407C2×D3—188no—
408C2×D3—188no—
409C2×D3—188no—
410C2×D3—182no—
411C2×D3—182no—
412C2×D3—182no—
413C2×D3—368no—
414C2×D3—368no—
415C3×C4—28no—
416C3×C4—24no—
417C3×C4—24no—
418C3×C4—21no—
419C3×C4—21no—
420C3×C4—28no—
421C3×C4—44no—
422C3×C4—64no—
423C3×C4—64no—
424C3×C4—64no—
425C3×C4—182no—
426C3×C4—182no—
427C3×C4#379818yes—
428C3×C4#382812yes—
429Dic3—21no—
430Dic3—21no—
431Dic3—21no—
432Dic3—68no—
433Dic3—68no—
434Dic3—188no—
435Dic3—182no—
436Dic3—368no—
437C32—24no—
438C32#553414yescomplemented
439C32#558211yescharacteristic, complemented
440C23—31no—
441C23—272no—
442Q8—182no—
443Q8—1082no—
444D4—542no—
445D4—1082no—
446C2×C4—31no—
447C2×C4—31no—
448C2×C4—31no—
449C2×C4—62no—
450C2×C4—92no—
451C2×C4—92no—
452C2×C4—272no—
453C2×C4—1082no—
454C2×C4C2×D3×C3⋊D311yescharacteristic
455C2×C4C2×D3×C3⋊D311yescharacteristic
456C2×C4C2×D3×C3⋊D311yescharacteristic
457C2×C3—24no—
458C2×C3—24no—
459C2×C3—24no—
460C2×C3—124no—
461C2×C3#6429714yes—
462C2×C3C2×#379814yes—
463C2×C3#6430814yes—
464C2×C3C2×#382811yescharacteristic
465C2×C3#6436611yescharacteristic
466C2×C3#6436711yescharacteristic
467D3—41no—
468D3—3616no—
469D3—368no—
470D3—362no—
471C22—61no—
472C22—61no—
473C22—61no—
474C22—542no—
475C22—542no—
476C22—542no—
477C22C22×D3×C3⋊D311yescharacteristic
478C4—21no—
479C4—21no—
480C4—61no—
481C4—61no—
482C4—61no—
483C4—182no—
484C4—182no—
485C4—542no—
486C4#8902012yes—
487C3—24no—
488C3#1059387814yescomplemented
489C3#1059600511yescharacteristic, complemented
490C2—121no—
491C2—1082no—
492C2#1172456211yescharacteristic
493C2C2×#8902011yescharacteristic
494C2#1172458111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
prime radical#11723516C2
hypercenterC42D3×C3⋊D3
hypocenterC33#542
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×C4⋊(C3⋊(C3⋊D3))C2semidirect
2. C3×C4×#634C2semidirect
3. C32×#872C2semidirect
4. #11723534C2semidirect
6. #11723446C2semidirect
7. #11723442C2semidirect
8. #11723441C2semidirect
9. #11723425C2semidirect
10. #11723420C2semidirect
11. #11723419C2semidirect
49. C4×#634D3semidirect
50. C3×#872D3semidirect
166. #872C3⋊D3semidirect
318. C33#542semidirect
438. C32#5534semidirect
439. C32#5582semidirect
488. C3#10593878semidirect
489. C3#10596005semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊(C3⋊(C3⋊D3))2
C3×C4×#6342
C32×#8722
#117235342
#117235162
#117234462
#117234422
#117234412
#117234252
#117234202
#117234192
#105960053
#105938783