Finite Group Databasebeta

← #13175974

Subgroups — #13175974

Order 1728.

1448 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723938C211yesmaximal, characteristic
2C3×#64096C211yesmaximal, characteristic, complemented
3C32×#890C211yesmaximal, characteristic, complemented
4#11723452C211yesmaximal, characteristic
5#11723449C211yesmaximal, characteristic
6#11723447C212yesmaximal, complemented
7#11723446C212yesmaximal
8#11723443C211yesmaximal, characteristic, complemented
9#11723411C212yesmaximal
10#11723406C211yesmaximal, characteristic, complemented
11#11723400C212yesmaximal, complemented
12#10596169—31nomaximal
13#10594355—34nomaximal
14C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
15C2×#6308C2211yescharacteristic
16C2×C3×#634C2211yescharacteristic
17C33×C4⋊C4C2211yescharacteristic
18#88859C2212yes—
19C4⋊(C3⋊(C3⋊Dic3))C2212yes—
20C3×#3750C2212yes—
21C3×C4×C3⋊Dic3C2211yescharacteristic
22C3×C4×C3⋊Dic3C2212yes—
23C32×C4⋊Dic3C2211yescharacteristic
24C32×C4⋊Dic3C2212yes—
25C32×Dic3⋊C4C2212yes—
26C32×C4×Dic3C2211yescharacteristic
27#88806C2211yescharacteristic
28#88806C2212yes—
29#88806C2211yescharacteristic
30Dic3⋊(C3⋊Dic3)C2212yes—
31Dic3⋊(C3⋊Dic3)C2212yes—
32Dic3⋊(C3⋊Dic3)C2212yes—
33#88804C2211yescharacteristic
34#88804C2212yes—
35#88804C2211yescharacteristic
36Dic3×C3⋊Dic3C2212yes—
37C32×#125—31no—
38#64107—32no—
39#64107—32no—
40#64106—31no—
41#64106—31no—
42#64106—34no—
43#64106—64no—
44#64099—32no—
45#64099—32no—
46#64096—31no—
47#64096D311yescharacteristic, complemented
48#64081—31no—
49#64078—31no—
50C3×#890D314yescomplemented
51C3×#887—34no—
52#63874—34no—
53#63871—34no—
54#63869—38no—
55#63868—38no—
56#63866—34no—
57#63848—38no—
58#63845—38no—
59#63844—34no—
60C22×C3⋊(C3⋊D3)C2311yescharacteristic
61C2×C33×C4C2312yes—
62C2×C33×C4C2311yescharacteristic
63C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
64C2×C3⋊(C3⋊Dic3)C2312yes—
65C4×C3⋊(C3⋊D3)—22no—
66#6308—22no—
67C2×C3×C3⋊Dic3C2311yescharacteristic
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C3×C3⋊Dic3C2312yes—
70C3×#634—22no—
71C2×C32×Dic3C2312yes—
72C2×C32×Dic3C2311yescharacteristic
73C2×C32×Dic3C2311yescharacteristic
74#5698—94no—
75#5583—91no—
76C2×C4×C3⋊D3—31no—
77C2×C4×C3⋊D3—34no—
78C2×C4×C3⋊D3—64no—
79C2×#634—31no—
80C2×#634—34no—
81C2×#634—64no—
82C2×#634C2×D311yescharacteristic
83C2×C3×Dic6—34no—
84C32×C4⋊C4—24no—
85C32×C4⋊C4—31no—
86C32×C4⋊C4—32no—
87C32×C4⋊C4—32no—
88C32×C4⋊C4C2×D314yes—
89C32×C4⋊C4C2×D311yescharacteristic
90C32×C42—31no—
91#3752—32no—
92#3752—38no—
93#3752—31no—
94#3752—31no—
95#3752—32no—
96#3752—68no—
97C4⋊(C3⋊Dic3)—32no—
98C4⋊(C3⋊Dic3)—38no—
99C4⋊(C3⋊Dic3)—32no—
100C4⋊(C3⋊Dic3)—32no—
101C4⋊(C3⋊Dic3)—32no—
102C4⋊(C3⋊Dic3)—68no—
103#3750—31no—
104#3750—31no—
105#3750—32no—
106#3750C2×D312yes—
107C4×C3⋊Dic3—32no—
108C4×C3⋊Dic3C2×D311yescharacteristic
109C4×C3⋊Dic3C2×D312yes—
110C3×C4⋊Dic3C2×D314yes—
111C3×C4⋊Dic3C2×D318yes—
112C3×Dic3⋊C4—38no—
113C3×Dic3⋊C4C2×D318yes—
114C3×C4×Dic3—34no—
115C3×C4×Dic3—38no—
116C3×C4×Dic3C2×D314yes—
117#3724—38no—
118#3724—34no—
119#3724—34no—
120Dic3⋊Dic3—38no—
121Dic3⋊Dic3—38no—
122Dic3⋊Dic3—38no—
123#3722—34no—
124#3722—38no—
125#3722—34no—
126Dic32—38no—
127C22×C33C2411yescharacteristic
128C2×C3⋊(C3⋊D3)—21no—
129C2×C3⋊(C3⋊D3)—21no—
130C2×C3⋊(C3⋊D3)—21no—
131C33×C4—22no—
132C33×C4#4112yes—
133C3⋊(C3⋊Dic3)—22no—
134C3⋊(C3⋊Dic3)—21no—
135C3×C3⋊Dic3—22no—
136C3×C3⋊Dic3—21no—
137C3×C3⋊Dic3—21no—
138C32×Dic3—22no—
139C32×Dic3—21no—
140C32×Dic3—21no—
141C3×#125—34no—
142C3×C4×Q8—91no—
143#946—91no—
144#944—91no—
145#898—98no—
146#898—98no—
147#898—92no—
148#897—94no—
149#897—94no—
150#897—91no—
151#897—184no—
152#890—98no—
153#890—98no—
154#890C3⋊D311yescharacteristic, complemented
155#889—92no—
156#887—34no—
157#887—94no—
158#876—92no—
159#872—94no—
160#872—91no—
161#870—92no—
162#869—94no—
163C4×Dic6—91no—
164C22×C3⋊D3—31no—
165C22×C3⋊D3—34no—
166C22×C3⋊D3—64no—
167C2×C32×C4—24no—
168C2×C32×C4—28no—
169C2×C32×C4—32no—
170C2×C32×C4—31no—
171C2×C32×C4—31no—
172C2×C32×C4C22×D318yes—
173C2×C32×C4C22×D314yes—
174C2×C32×C4C22×D312yes—
175C2×C32×C4C22×D311yescharacteristic
176C2×C3⋊Dic3—32no—
177C2×C3⋊Dic3—31no—
178C2×C3⋊Dic3—38no—
179C2×C3⋊Dic3—34no—
180C2×C3⋊Dic3—64no—
181C2×C3⋊Dic3—68no—
182C2×C3⋊Dic3C22×D311yescharacteristic
183C2×C3⋊Dic3C22×D311yescharacteristic
184C2×C3⋊Dic3C22×D312yes—
185C4×C3⋊D3—62no—
186C4×C3⋊D3—68no—
187C4×C3⋊D3—128no—
188#634—22no—
189#634—62no—
190#634—68no—
191#634—128no—
192C2×C3×Dic3—34no—
193C2×C3×Dic3—38no—
194C2×C3×Dic3—34no—
195C2×C3×Dic3C22×D318yes—
196C2×C3×Dic3C22×D314yes—
197C2×C3×Dic3C22×D314yes—
198C3×Dic6—68no—
199#541—271noSylow
200C2×C33C2×#4111yescharacteristic
201C2×C332−1+411yescharacteristic
202C2×C332−1+411yescharacteristic
203C3⋊(C3⋊D3)—41no—
204C2×C3×Q8—91no—
205C2×C4×D3—94no—
206C2×C4×D3—91no—
207C2×C4×D3—184no—
208C2×Dic6—34no—
209C2×Dic6—94no—
210C2×Dic6—91no—
211C2×Dic6—184no—
212C3×C4⋊C4—24no—
213C3×C4⋊C4—34no—
214C3×C4⋊C4—38no—
215C3×C4⋊C4—38no—
216C3×C4⋊C4—92no—
217C3×C4⋊C4D3214yes—
218C3×C4⋊C4C2×C3⋊D311yescharacteristic
219C3×C42—34no—
220C3×C42—91no—
221C3×C42—92no—
222(C2×D3)⋊C4—94no—
223(C2×D3)⋊C4—94no—
224(C2×D3)⋊C4—98no—
225(C2×D3)⋊C4—98no—
226(C2×D3)⋊C4—92no—
227(C2×D3)⋊C4—91no—
228(C2×D3)⋊C4—91no—
229(C2×D3)⋊C4—92no—
230(C2×D3)⋊C4—188no—
231C4⋊Dic3—98no—
232C4⋊Dic3—98no—
233C4⋊Dic3—98no—
234C4⋊Dic3—98no—
235C4⋊Dic3—92no—
236C4⋊Dic3—188no—
237C4⋊Dic3C2×C3⋊D311yescharacteristic
238C4⋊Dic3C2×C3⋊D312yes—
239Dic3⋊C4—38no—
240Dic3⋊C4—94no—
241Dic3⋊C4—94no—
242Dic3⋊C4—98no—
243Dic3⋊C4—92no—
244Dic3⋊C4—91no—
245Dic3⋊C4—92no—
246Dic3⋊C4—92no—
247Dic3⋊C4—91no—
248Dic3⋊C4—92no—
249Dic3⋊C4C2×C3⋊D312yes—
250C4×Dic3—34no—
251C4×Dic3—38no—
252C4×Dic3—98no—
253C4×Dic3—92no—
254C4×Dic3C2×C3⋊D311yescharacteristic
255C22×C32—24no—
256C22×C32C23×D314yes—
257C22×C32C23×D311yescharacteristic
258C2×C3⋊D3—68no—
259C2×C3⋊D3—61no—
260C2×C3⋊D3—61no—
261C2×C3⋊D3—64no—
262C2×C3⋊D3—61no—
263C2×C3⋊D3—64no—
264C2×C3⋊D3—64no—
265C2×C3⋊D3—124no—
266C2×C3⋊D3—124no—
267C32×C4—22no—
268C32×C4—28no—
269C32×C4—28no—
270C32×C4—48no—
271C32×C4—62no—
272C32×C4—61no—
273C32×C4—61no—
274C32×C4#23512yes—
275C32×C4#23918yes—
276C3⋊Dic3—22no—
277C3⋊Dic3—21no—
278C3⋊Dic3—21no—
279C3⋊Dic3—62no—
280C3⋊Dic3—61no—
281C3⋊Dic3—68no—
282C3⋊Dic3—64no—
283C3⋊Dic3—128no—
284C3⋊Dic3—124no—
285C3×Dic3—24no—
286C3×Dic3—28no—
287C3×Dic3—24no—
288C3×Dic3—68no—
289C3×Dic3—64no—
290C3×Dic3—64no—
291C4⋊Q8—271no—
292#126—272no—
293#126—272no—
294#125—31no—
295#125—272no—
296#125—272no—
297C22⋊Q8—271no—
298C22⋊Q8—271no—
299C4×Q8—91no—
300C4×Q8—271no—
301#117—271no—
302C33#54111yescharacteristic, complemented, Sylow
303C22×D3—94no—
304C22×D3—91no—
305C22×D3—184no—
306C3×Q8—182no—
307C2×C3×C4—24no—
308C2×C3×C4—28no—
309C2×C3×C4—38no—
310C2×C3×C4—34no—
311C2×C3×C4—34no—
312C2×C3×C4—91no—
313C2×C3×C4—91no—
314C2×C3×C4—92no—
315C2×C3×C4C2×D3218yes—
316C2×C3×C4C2×D3214yes—
317C2×C3×C4C22×C3⋊D312yes—
318C2×C3×C4C22×C3⋊D311yescharacteristic
319C2×Dic3—34no—
320C2×Dic3—34no—
321C2×Dic3—38no—
322C2×Dic3—98no—
323C2×Dic3—94no—
324C2×Dic3—92no—
325C2×Dic3—91no—
326C2×Dic3—184no—
327C2×Dic3—188no—
328C2×Dic3C22×C3⋊D312yes—
329C2×Dic3C22×C3⋊D311yescharacteristic
330C2×Dic3C22×C3⋊D311yescharacteristic
331C4×D3—188no—
332C4×D3—182no—
333C4×D3—368no—
334Dic6—68no—
335Dic6—188no—
336Dic6—182no—
337Dic6—368no—
338C2×C32—24no—
339C2×C32—24no—
340C2×C32—24no—
341C2×C32C2×#23511yescharacteristic
342C2×C32C2×#23914yes—
343C2×C32#100711yescharacteristic
344C2×C32#101014yes—
345C2×C32#101014yes—
346C2×C32#101011yescharacteristic
347C3⋊D3—128no—
348C3⋊D3—124no—
349C3⋊D3—121no—
350C2×Q8—91no—
351C2×Q8—271no—
352C22×C4—271no—
353C4⋊C4—31no—
354C4⋊C4—32no—
355C4⋊C4—32no—
356C4⋊C4—92no—
357C4⋊C4—271no—
358C4⋊C4—271no—
359C4⋊C4—272no—
360C4⋊C4—272no—
361C4⋊C4—272no—
362C4⋊C4—272no—
363C4⋊C4—272no—
364C4⋊C4D3×C3⋊D311yescharacteristic
365C22⋊C4—271no—
366C22⋊C4—271no—
367C22⋊C4—272no—
368C22⋊C4—272no—
369C42—31no—
370C42—91no—
371C42—92no—
372C42—272no—
373C22×C3—24no—
374C22×C3C22×D3214yes—
375C22×C3C23×C3⋊D311yescharacteristic
376C2×D3—188no—
377C2×D3—184no—
378C2×D3—184no—
379C2×D3—184no—
380C2×D3—181no—
381C2×D3—181no—
382C2×D3—181no—
383C2×D3—364no—
384C2×D3—364no—
385C3×C4—28no—
386C3×C4—22no—
387C3×C4—28no—
388C3×C4—48no—
389C3×C4—64no—
390C3×C4—68no—
391C3×C4—64no—
392C3×C4—182no—
393C3×C4—181no—
394C3×C4—181no—
395C3×C4#379918yes—
396C3×C4#383212yes—
397Dic3—21no—
398Dic3—22no—
399Dic3—21no—
400Dic3—68no—
401Dic3—64no—
402Dic3—64no—
403Dic3—188no—
404Dic3—184no—
405Dic3—182no—
406Dic3—181no—
407Dic3—368no—
408Dic3—364no—
409C32—24no—
410C32#558311yescharacteristic, complemented
411C32#569814yescomplemented
412C23—271no—
413Q8—182no—
414Q8—542no—
415C2×C4—31no—
416C2×C4—31no—
417C2×C4—32no—
418C2×C4—91no—
419C2×C4—91no—
420C2×C4—92no—
421C2×C4—272no—
422C2×C4—271no—
423C2×C4—542no—
424C2×C4C2×D3×C3⋊D312yes—
425C2×C4C2×D3×C3⋊D311yescharacteristic
426C2×C3—24no—
427C2×C3—24no—
428C2×C3—24no—
429C2×C3C2×#379914yes—
430C2×C3#6430914yes—
431C2×C3#6431514yes—
432C2×C3C2×#383211yescharacteristic
433C2×C3#6436711yescharacteristic
434C2×C3#6436711yescharacteristic
435D3—368no—
436D3—364no—
437D3—361no—
438C22—541no—
439C22—541no—
440C22—541no—
441C22C22×D3×C3⋊D311yescharacteristic
442C4—22no—
443C4—61no—
444C4—61no—
445C4—62no—
446C4—182no—
447C4—181no—
448C4—181no—
449C4—541no—
450C4—542no—
451C4#8901812yes—
452C3—24no—
453C3#1059435514yescomplemented
454C3#1059616911yescharacteristic, complemented
455C2—1081no—
456C2C2×#8901811yescharacteristic
457C2#1172458211yescharacteristic
458C2#1172459011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
prime radicalC22×C3⋊(C3⋊D3)C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#541
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#64096C2semidirect
3. C32×#890C2semidirect
6. #11723447C2semidirect
8. #11723443C2semidirect
10. #11723406C2semidirect
11. #11723400C2semidirect
47. #64096D3semidirect
50. C3×#890D3semidirect
154. #890C3⋊D3semidirect
302. C33#541semidirect
410. C32#5583semidirect
411. C32#5698semidirect
453. C3#10594355semidirect
454. C3#10596169semidirect

Maximal subgroups

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

subgroupindex
#117239382
C3×#640962
C32×#8902
#117234522
#117234492
#117234472
#117234462
#117234432
#117234112
#117234062
#117234002
#105961693
#105943553