Finite Group Databasebeta

← #13176809

Subgroups — #13176809

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#11723965C211yesmaximal, characteristic
2C3×#64135C211yesmaximal, characteristic
3C32×#924C211yesmaximal, characteristic
4#11723597C212yesmaximal
5#11723596C212yesmaximal
6#11723455C211yesmaximal, characteristic, complemented
7#11723447C211yesmaximal, characteristic, complemented
8#11723406C211yesmaximal, characteristic, complemented
9#11723402C212yesmaximal, complemented
10#11723401C212yesmaximal, complemented
11Dic6×C3⋊Dic3C211yesmaximal, characteristic, complemented
12#10596096—31nomaximal
13#10594824—34nomaximal
14C22×C33×C4C2211yescharacteristic
15C2×#6308C2211yescharacteristic
16C2×C32×Dic6C2211yescharacteristic
17#88864C2212yes—
18C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
19#88857C2212yes—
20C3×#3757C2211yescharacteristic
21C3×#3757C2211yescharacteristic
22C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
23C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
24C3×C4×C3⋊Dic3C2211yescharacteristic
25C3×C4×C3⋊Dic3C2211yescharacteristic
26C32×#217C2212yes—
27C32×C4⋊Dic3C2211yescharacteristic
28C32×Dic3⋊C4C2212yes—
29#88807C2212yes—
30#88807C2212yes—
31#88806C2212yes—
32#88806C2212yes—
33Dic3⋊(C3⋊Dic3)C2212yes—
34Dic3⋊(C3⋊Dic3)C2212yes—
35Dic3×C3⋊Dic3C2212yes—
36Dic3×C3⋊Dic3C2212yes—
37C32×C22⋊Q8—31no—
38Q8×C3⋊Dic3—31no—
39#64153—31no—
40#64135D311yescharacteristic
41#64133—31no—
42#64133—34no—
43#64133—64no—
44#64099—32no—
45#64099—31no—
46#64098—32no—
47#64096—31no—
48#64088—32no—
49#64088—32no—
50C3×#926—34no—
51C3×#924D314yes—
52#63956—38no—
53#63956—38no—
54#63869—34no—
55#63866—34no—
56#63847—38no—
57#63846—38no—
58#63844—34no—
59Dic3×Dic6—34no—
60C23×C33C2311yescharacteristic
61C2×C33×C4—22no—
62C2×C33×C4C2311yescharacteristic
63C2×C33×C4C2311yescharacteristic
64C2×C3⋊(C3⋊Dic3)C2312yes—
65C2×C3⋊(C3⋊Dic3)C2312yes—
66#6308—22no—
67C2×C3×C3⋊Dic3C2311yescharacteristic
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C3×C3⋊Dic3C2311yescharacteristic
70C2×C3×C3⋊Dic3C2311yescharacteristic
71C2×C32×Dic3C2312yes—
72C2×C32×Dic3C2312yes—
73C32×Dic6—22no—
74#5625—94no—
75#5518—91no—
76C2×C32×Q8—31no—
77C22×C32×C4—24no—
78C22×C32×C4C2×D314yes—
79C22×C32×C4C2×D311yescharacteristic
80C2×#634—31no—
81C2×#634—34no—
82C2×#634—64no—
83C2×C3×Dic6C2×D314yes—
84C32×C4⋊C4—31no—
85C32×C4⋊C4—32no—
86C32×C22⋊C4—32no—
87#3757—32no—
88#3757—38no—
89#3757—68no—
90#3757C2×D311yescharacteristic
91#3757C2×D311yescharacteristic
92C4⋊(C3⋊Dic3)—31no—
93C4⋊(C3⋊Dic3)—34no—
94C4⋊(C3⋊Dic3)—32no—
95C4⋊(C3⋊Dic3)—32no—
96C4⋊(C3⋊Dic3)—64no—
97C4⋊(C3⋊Dic3)C2×D311yescharacteristic
98C4⋊(C3⋊Dic3)C2×D311yescharacteristic
99#3750—32no—
100#3750—38no—
101#3750—32no—
102#3750—32no—
103#3750—32no—
104#3750—32no—
105#3750—68no—
106C4×C3⋊Dic3—32no—
107C4×C3⋊Dic3—32no—
108C4×C3⋊Dic3C2×D311yescharacteristic
109C4×C3⋊Dic3C2×D311yescharacteristic
110C3×#217—34no—
111C3×#217—34no—
112C3×#217C2×D318yes—
113C3×C4⋊Dic3—34no—
114C3×C4⋊Dic3—34no—
115C3×C4⋊Dic3C2×D314yes—
116C3×Dic3⋊C4C2×D318yes—
117C3×C4×Dic3—34no—
118C3×C4×Dic3—34no—
119#3724—38no—
120#3724—38no—
121Dic3⋊Dic3—38no—
122Dic3⋊Dic3—38no—
123Dic3⋊Dic3—38no—
124Dic3⋊Dic3—38no—
125Dic32—38no—
126Dic32—38no—
127C22×C33—21no—
128C22×C33—21no—
129C22×C33—21no—
130C22×C33C2411yescharacteristic
131C33×C4—21no—
132C33×C4#4112yes—
133C3⋊(C3⋊Dic3)—22no—
134C3⋊(C3⋊Dic3)—22no—
135C3×C3⋊Dic3—21no—
136C3×C3⋊Dic3—21no—
137C3×C3⋊Dic3—21no—
138C3×C3⋊Dic3—21no—
139C32×Dic3—22no—
140C32×Dic3—22no—
141C3×C22⋊Q8—34no—
142C3×#117—91no—
143Q8×Dic3—94no—
144#944—94no—
145#926—34no—
146#924—94no—
147#924—91no—
148#924—184no—
149#924C3⋊D311yescharacteristic
150#890—94no—
151#890—98no—
152#889—98no—
153#889—92no—
154#889—92no—
155#887—94no—
156#879—98no—
157#879—98no—
158#879—92no—
159#879—92no—
160#870—91no—
161#869—91no—
162C4×Dic6—91no—
163C4×Dic6—91no—
164C23×C32—24no—
165C23×C32C22×D314yes—
166C23×C32C22×D311yescharacteristic
167C32×Q8—62no—
168C2×C32×C4—22no—
169C2×C32×C4—24no—
170C2×C32×C4—24no—
171C2×C32×C4—28no—
172C2×C32×C4—32no—
173C2×C32×C4—32no—
174C2×C32×C4—48no—
175C2×C32×C4C22×D314yes—
176C2×C32×C4C22×D314yes—
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C22×D311yescharacteristic
179C2×C3⋊Dic3—32no—
180C2×C3⋊Dic3—32no—
181C2×C3⋊Dic3—38no—
182C2×C3⋊Dic3—38no—
183C2×C3⋊Dic3—68no—
184C2×C3⋊Dic3—68no—
185C2×C3⋊Dic3C22×D311yescharacteristic
186C2×C3⋊Dic3C22×D311yescharacteristic
187C2×C3⋊Dic3C22×D311yescharacteristic
188C2×C3⋊Dic3C22×D311yescharacteristic
189#634—62no—
190#634—68no—
191#634—128no—
192C2×C3×Dic3—34no—
193C2×C3×Dic3—34no—
194C2×C3×Dic3—34no—
195C2×C3×Dic3—34no—
196C2×C3×Dic3C22×D318yes—
197C2×C3×Dic3C22×D318yes—
198C3×Dic6—28no—
199#541—271noSylow
200C2×C33—41no—
201C2×C33C2×#4111yescharacteristic
202C2×C332−1+411yescharacteristic
203C2×C332−1+411yescharacteristic
204C2×C3×Q8—34no—
205C22×C3×C4—24no—
206C22×C3×C4D3214yes—
207C22×C3×C4C2×C3⋊D311yescharacteristic
208C2×Dic6—94no—
209C2×Dic6—91no—
210C2×Dic6—184no—
211C2×Dic6C2×C3⋊D311yescharacteristic
212C3×C4⋊C4—34no—
213C3×C4⋊C4—38no—
214C3×C4⋊C4—91no—
215C3×C4⋊C4—91no—
216C3×C22⋊C4—38no—
217C3×C22⋊C4—91no—
218C3×C22⋊C4—91no—
219C3×C42—91no—
220C3×C42—91no—
221#217—34no—
222#217—34no—
223#217—98no—
224#217—92no—
225#217—188no—
226#217C2×C3⋊D312yes—
227C4⋊Dic3—34no—
228C4⋊Dic3—34no—
229C4⋊Dic3—98no—
230C4⋊Dic3—98no—
231C4⋊Dic3—94no—
232C4⋊Dic3—91no—
233C4⋊Dic3—92no—
234C4⋊Dic3—92no—
235C4⋊Dic3—184no—
236C4⋊Dic3C2×C3⋊D311yescharacteristic
237Dic3⋊C4—98no—
238Dic3⋊C4—98no—
239Dic3⋊C4—98no—
240Dic3⋊C4—98no—
241Dic3⋊C4—98no—
242Dic3⋊C4—92no—
243Dic3⋊C4—92no—
244Dic3⋊C4—92no—
245Dic3⋊C4—92no—
246Dic3⋊C4—92no—
247Dic3⋊C4—188no—
248Dic3⋊C4C2×C3⋊D312yes—
249C4×Dic3—34no—
250C4×Dic3—34no—
251C4×Dic3—98no—
252C4×Dic3—98no—
253C4×Dic3—92no—
254C4×Dic3—92no—
255C22×C32—28no—
256C22×C32—21no—
257C22×C32—24no—
258C22×C32—21no—
259C22×C32—21no—
260C22×C32—24no—
261C22×C32—24no—
262C22×C32—24no—
263C22×C32—44no—
264C22×C32—44no—
265C22×C32C23×D314yes—
266C22×C32C23×D311yescharacteristic
267C32×C4—21no—
268C32×C4—28no—
269C32×C4—24no—
270C32×C4—44no—
271C32×C4—62no—
272C32×C4—62no—
273C32×C4#23512yes—
274C32×C4#23718yes—
275C3⋊Dic3—21no—
276C3⋊Dic3—21no—
277C3⋊Dic3—21no—
278C3⋊Dic3—21no—
279C3⋊Dic3—62no—
280C3⋊Dic3—62no—
281C3⋊Dic3—68no—
282C3⋊Dic3—68no—
283C3⋊Dic3—128no—
284C3⋊Dic3—128no—
285C3×Dic3—28no—
286C3×Dic3—28no—
287C3×Dic3—64no—
288C3×Dic3—64no—
289C3×Dic3—64no—
290C3×Dic3—64no—
291C4⋊Q8—271no—
292#126—272no—
293#126—272no—
294#125—271no—
295#125—272no—
296#125—272no—
297C22⋊Q8—31no—
298C22⋊Q8—271no—
299C4×Q8—271no—
300C4×Q8—271no—
301#117—91no—
302C33#54111yescharacteristic, complemented, Sylow
303C23×C3—24no—
304C23×C3C2×D3214yes—
305C23×C3C22×C3⋊D311yescharacteristic
306C3×Q8—68no—
307C2×C3×C4—28no—
308C2×C3×C4—22no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—38no—
312C2×C3×C4—38no—
313C2×C3×C4—48no—
314C2×C3×C4—91no—
315C2×C3×C4—91no—
316C2×C3×C4—91no—
317C2×C3×C4—91no—
318C2×C3×C4C2×D3214yes—
319C2×C3×C4C2×D3214yes—
320C2×C3×C4C22×C3⋊D311yescharacteristic
321C2×C3×C4C22×C3⋊D311yescharacteristic
322C2×Dic3—34no—
323C2×Dic3—34no—
324C2×Dic3—34no—
325C2×Dic3—34no—
326C2×Dic3—98no—
327C2×Dic3—98no—
328C2×Dic3—92no—
329C2×Dic3—92no—
330C2×Dic3—188no—
331C2×Dic3—188no—
332C2×Dic3C22×C3⋊D312yes—
333C2×Dic3C22×C3⋊D312yes—
334Dic6—22no—
335Dic6—188no—
336Dic6—182no—
337Dic6—368no—
338C2×C32—24no—
339C2×C32—24no—
340C2×C32—24no—
341C2×C32—48no—
342C2×C32—41no—
343C2×C32—44no—
344C2×C32C2×#23511yescharacteristic
345C2×C32C2×#23714yes—
346C2×C32#100714yes—
347C2×C32#101014yes—
348C2×C32#101011yescharacteristic
349C2×C32#101011yescharacteristic
350C2×Q8—31no—
351C2×Q8—271no—
352C22×C4D3×C3⋊D311yescharacteristic
353C4⋊C4—31no—
354C4⋊C4—32no—
355C4⋊C4—91no—
356C4⋊C4—91no—
357C4⋊C4—272no—
358C4⋊C4—272no—
359C4⋊C4—272no—
360C4⋊C4—272no—
361C4⋊C4—272no—
362C4⋊C4—272no—
363C4⋊C4—271no—
364C4⋊C4—272no—
365C22⋊C4—32no—
366C22⋊C4—91no—
367C22⋊C4—91no—
368C22⋊C4—272no—
369C42—91no—
370C42—91no—
371C42—272no—
372C42—272no—
373C22×C3—28no—
374C22×C3—24no—
375C22×C3—24no—
376C22×C3—24no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—24no—
381C22×C3—44no—
382C22×C3—44no—
383C22×C3C22×D3214yes—
384C22×C3C23×C3⋊D311yescharacteristic
385C3×C4—24no—
386C3×C4—21no—
387C3×C4—28no—
388C3×C4—44no—
389C3×C4—68no—
390C3×C4—68no—
391C3×C4—181no—
392C3×C4—181no—
393C3×C4—181no—
394C3×C4—181no—
395C3×C4#380418yes—
396C3×C4#383012yes—
397Dic3—22no—
398Dic3—22no—
399Dic3—64no—
400Dic3—64no—
401Dic3—64no—
402Dic3—64no—
403Dic3—188no—
404Dic3—188no—
405Dic3—182no—
406Dic3—182no—
407Dic3—368no—
408Dic3—368no—
409C32—24no—
410C32#551811yescharacteristic, complemented
411C32#562514yescomplemented
412C23C2×D3×C3⋊D311yescharacteristic
413Q8—62no—
414Q8—542no—
415C2×C4—22no—
416C2×C4—32no—
417C2×C4—32no—
418C2×C4—91no—
419C2×C4—91no—
420C2×C4—91no—
421C2×C4—91no—
422C2×C4—272no—
423C2×C4—272no—
424C2×C4C2×D3×C3⋊D311yescharacteristic
425C2×C4C2×D3×C3⋊D311yescharacteristic
426C2×C3—24no—
427C2×C3—24no—
428C2×C3—24no—
429C2×C3—48no—
430C2×C3—41no—
431C2×C3—44no—
432C2×C3#6429714yes—
433C2×C3#6429814yes—
434C2×C3C2×#380414yes—
435C2×C3C2×#383011yescharacteristic
436C2×C3#6436411yescharacteristic
437C2×C3#6436711yescharacteristic
438C22—21no—
439C22—21no—
440C22—21no—
441C22C22×D3×C3⋊D311yescharacteristic
442C4—21no—
443C4—62no—
444C4—62no—
445C4—181no—
446C4—181no—
447C4—181no—
448C4—181no—
449C4—542no—
450C4—542no—
451C4#8902812yes—
452C3—24no—
453C3#1059482414yescomplemented
454C3#1059609611yescharacteristic, complemented
455C2—41no—
456C2#1172456311yescharacteristic
457C2#1172456611yescharacteristic
458C2C2×#8902811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC22×C33×C4C22
prime radicalC23×C33C23
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#541
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
6. #11723455C2semidirect
7. #11723447C2semidirect
8. #11723406C2semidirect
9. #11723402C2semidirect
10. #11723401C2semidirect
11. Dic6×C3⋊Dic3C2semidirect
302. C33#541semidirect
410. C32#5518semidirect
411. C32#5625semidirect
453. C3#10594824semidirect
454. C3#10596096semidirect

Maximal subgroups

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

subgroupindex
#117239652
C3×#641352
C32×#9242
#117235972
#117235962
#117234552
#117234472
#117234062
#117234022
#117234012
Dic6×C3⋊Dic32
#105960963
#105948243