Finite Group Databasebeta

← #13177668

Subgroups — #13177668

Order 1728.

1576 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723977C211yesmaximal, characteristic
2C3×D4×C3⋊Dic3C211yesmaximal, characteristic
3C32×#936C211yesmaximal, characteristic
4#11723641C212yesmaximal, complemented
5#11723597C212yesmaximal, complemented
6#11723596C212yesmaximal, complemented
7#11723595C212yesmaximal, complemented
8C4×#6293C211yesmaximal, characteristic, complemented
9#11723455C211yesmaximal, characteristic, complemented
10#11723399C211yesmaximal, characteristic, complemented
11#11723397C211yesmaximal, characteristic, complemented
12#10596142—31nomaximal
13#10595273—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C2×#6308C2211yescharacteristic
16C22×C3×C3⋊Dic3C2212yes—
17C2×C32×Dic6C2211yescharacteristic
18C2×#6293C2211yescharacteristic
19#88864C2212yes—
20#88864C2212yes—
21C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
22C3×#3757C2212yes—
23C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
24C3×C4×C3⋊Dic3C2211yescharacteristic
25C32×#217C2212yes—
26C32×#217C2212yes—
27C32×C4×Dic3C2211yescharacteristic
28#88807C2211yescharacteristic
29#88807C2212yes—
30#88807C2212yes—
31#88806C2211yescharacteristic
32#88806C2212yes—
33Dic3⋊(C3⋊Dic3)C2211yescharacteristic
34Dic3⋊(C3⋊Dic3)C2212yes—
35Dic3×C3⋊Dic3C2212yes—
36Dic3×C3⋊Dic3C2212yes—
37C32×#124—31no—
38Q8×C3⋊Dic3—31no—
39#64153—31no—
40#64145—31no—
41#64145—34no—
42#64145—64no—
43D4×C3⋊Dic3D311yescharacteristic
44#64088—32no—
45#64088—32no—
46#64087—32no—
47#64086—32no—
48C4×#634—31no—
49C4×#634—31no—
50C3×#936D314yes—
51C3×D4×Dic3—34no—
52#63964—38no—
53#63956—38no—
54#63956—38no—
55#63955—38no—
56C4×#627—34no—
57#63866—34no—
58#63844—34no—
59#63843—34no—
60C23×C33C2312yes—
61C33×D4—22no—
62C2×C33×C4C2311yescharacteristic
63C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
64C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)C2312yes—
66#6308—22no—
67C2×C3×C3⋊Dic3—24no—
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C3×C3⋊Dic3C2312yes—
70C2×C3×C3⋊Dic3C2311yescharacteristic
71C2×C32×Dic3C2311yescharacteristic
72C2×C32×Dic3C2312yes—
73C2×C32×Dic3C2311yescharacteristic
74C32×Dic6—22no—
75#6293—22no—
76#5671—94no—
77#5542—91no—
78C2×C32×Q8—31no—
79C2×C32×D4—24no—
80C2×C32×D4C2×D314yes—
81C2×C32×D4C2×D311yescharacteristic
82C22×C3⋊Dic3C2×D312yes—
83C2×#634—31no—
84C2×#634—34no—
85C2×#634—31no—
86C2×#634—64no—
87C22×C3×Dic3—38no—
88C2×C3×Dic6C2×D314yes—
89C2×#627—34no—
90C32×C22⋊C4—32no—
91C32×C22⋊C4—32no—
92C32×C42—31no—
93#3757—32no—
94#3757—32no—
95#3757—38no—
96#3757—38no—
97#3757—68no—
98#3757—68no—
99#3757C2×D312yes—
100C4⋊(C3⋊Dic3)—31no—
101C4⋊(C3⋊Dic3)—32no—
102C4⋊(C3⋊Dic3)C2×D311yescharacteristic
103#3750—31no—
104#3750—32no—
105#3750—31no—
106#3750—32no—
107#3750—32no—
108C4×C3⋊Dic3—31no—
109C4×C3⋊Dic3—34no—
110C4×C3⋊Dic3—32no—
111C4×C3⋊Dic3—32no—
112C4×C3⋊Dic3—64no—
113C4×C3⋊Dic3C2×D311yescharacteristic
114C3×#217—38no—
115C3×#217C2×D318yes—
116C3×#217C2×D318yes—
117C3×C4⋊Dic3—34no—
118C3×C4×Dic3—34no—
119C3×C4×Dic3C2×D314yes—
120#3724—38no—
121#3724—34no—
122Dic3⋊Dic3—38no—
123Dic3⋊Dic3—38no—
124Dic3⋊Dic3—38no—
125Dic3⋊Dic3—34no—
126Dic3⋊Dic3—34no—
127Dic32—38no—
128Dic32—38no—
129C22×C33—22no—
130C22×C33—22no—
131C22×C33—22no—
132C22×C33C2411yescharacteristic
133C33×C4#4112yes—
134C3⋊(C3⋊Dic3)—21no—
135C3⋊(C3⋊Dic3)—22no—
136C3⋊(C3⋊Dic3)—21no—
137C3×C3⋊Dic3—22no—
138C3×C3⋊Dic3—21no—
139C3×C3⋊Dic3#4112yes—
140C32×Dic3—21no—
141C32×Dic3—22no—
142C32×Dic3—21no—
143C3×#124—34no—
144C3×C4×D4—91no—
145Q8×Dic3—94no—
146#944—94no—
147#936—94no—
148#936—91no—
149#936—184no—
150#936C3⋊D311yescharacteristic
151D4×Dic3—34no—
152#926—92no—
153#924—92no—
154#887—91no—
155#887—91no—
156#879—98no—
157#879—98no—
158#879—92no—
159#879—92no—
160#878—98no—
161#877—98no—
162#869—91no—
163C4×Dic6—94no—
164C4×Dic6—94no—
165C4×Dic6—91no—
166C23×C32—28no—
167C23×C32C22×D318yes—
168C23×C32C22×D312yes—
169C32×Q8—62no—
170C32×D4—22no—
171C32×D4—28no—
172C32×D4—48no—
173C2×C32×C4—24no—
174C2×C32×C4—31no—
175C2×C32×C4—32no—
176C2×C32×C4—31no—
177C2×C32×C4C22×D314yes—
178C2×C32×C4C22×D311yescharacteristic
179C2×C3⋊Dic3—24no—
180C2×C3⋊Dic3—31no—
181C2×C3⋊Dic3—32no—
182C2×C3⋊Dic3—31no—
183C2×C3⋊Dic3—34no—
184C2×C3⋊Dic3—38no—
185C2×C3⋊Dic3—34no—
186C2×C3⋊Dic3—64no—
187C2×C3⋊Dic3—64no—
188C2×C3⋊Dic3—68no—
189C2×C3⋊Dic3C22×D312yes—
190C2×C3⋊Dic3C22×D311yescharacteristic
191C2×C3⋊Dic3C22×D311yescharacteristic
192#634—62no—
193#634—62no—
194#634—68no—
195#634—128no—
196C2×C3×Dic3—34no—
197C2×C3×Dic3—38no—
198C2×C3×Dic3—34no—
199C2×C3×Dic3—616no—
200C2×C3×Dic3C22×D314yes—
201C2×C3×Dic3C22×D318yes—
202C2×C3×Dic3C22×D314yes—
203C3×Dic6—28no—
204#627—68no—
205#556—271noSylow
206C2×C33—42no—
207C2×C33C2×#4111yescharacteristic
208C2×C33C2×#4111yescharacteristic
209C2×C332−1+411yescharacteristic
210C2×C3×Q8—34no—
211C2×C3×D4—24no—
212C2×C3×D4D3214yes—
213C2×C3×D4C2×C3⋊D311yescharacteristic
214C22×C3×C4—92no—
215C22×Dic3—38no—
216C2×Dic6—94no—
217C2×Dic6—94no—
218C2×Dic6—91no—
219C2×Dic6—91no—
220C2×Dic6—184no—
221C2×Dic6C2×C3⋊D311yescharacteristic
222C3×C4⋊C4—91no—
223C3×C22⋊C4—38no—
224C3×C22⋊C4—38no—
225C3×C22⋊C4—92no—
226C3×C42—34no—
227C3×C42—91no—
228#217—38no—
229#217—98no—
230#217—98no—
231#217—92no—
232#217—92no—
233#217—188no—
234#217—188no—
235#217C2×C3⋊D312yes—
236#217C2×C3⋊D312yes—
237C4⋊Dic3—34no—
238C4⋊Dic3—94no—
239C4⋊Dic3—98no—
240C4⋊Dic3—91no—
241C4⋊Dic3—92no—
242C4⋊Dic3—92no—
243Dic3⋊C4—94no—
244Dic3⋊C4—98no—
245Dic3⋊C4—94no—
246Dic3⋊C4—98no—
247Dic3⋊C4—98no—
248Dic3⋊C4—91no—
249Dic3⋊C4—91no—
250Dic3⋊C4—92no—
251Dic3⋊C4—92no—
252C4×Dic3—34no—
253C4×Dic3—98no—
254C4×Dic3—98no—
255C4×Dic3—94no—
256C4×Dic3—91no—
257C4×Dic3—92no—
258C4×Dic3—92no—
259C4×Dic3—184no—
260C4×Dic3C2×C3⋊D311yescharacteristic
261C22×C32—22no—
262C22×C32—22no—
263C22×C32—22no—
264C22×C32—24no—
265C22×C32—28no—
266C22×C32—28no—
267C22×C32—28no—
268C22×C32—48no—
269C22×C32—48no—
270C22×C32—48no—
271C22×C32C23×D314yes—
272C22×C32C23×D311yescharacteristic
273C32×C4—28no—
274C32×C4—62no—
275C32×C4—61no—
276C32×C4—61no—
277C32×C4#23512yes—
278C32×C4#23718yes—
279C3⋊Dic3—22no—
280C3⋊Dic3—21no—
281C3⋊Dic3—62no—
282C3⋊Dic3—61no—
283C3⋊Dic3—61no—
284C3⋊Dic3—68no—
285C3⋊Dic3—64no—
286C3⋊Dic3—64no—
287C3⋊Dic3—128no—
288C3⋊Dic3—124no—
289C3⋊Dic3—124no—
290C3⋊Dic3#23712yes—
291C3×Dic3—28no—
292C3×Dic3—24no—
293C3×Dic3—24no—
294C3×Dic3—38no—
295C3×Dic3—68no—
296C3×Dic3—64no—
297C4⋊Q8—271no—
298#126—272no—
299#126—272no—
300#124—31no—
301#124—271no—
302C22⋊Q8—272no—
303C4×Q8—271no—
304C4×Q8—271no—
305C4×Q8—271no—
306C4×D4—91no—
307#117—272no—
308C33#55611yescharacteristic, complemented, Sylow
309C23×C3—28no—
310C23×C3C2×D3218yes—
311C23×C3C22×C3⋊D312yes—
312C3×Q8—68no—
313C3×D4—28no—
314C3×D4—22no—
315C3×D4—48no—
316C2×C3×C4—24no—
317C2×C3×C4—34no—
318C2×C3×C4—38no—
319C2×C3×C4—34no—
320C2×C3×C4—91no—
321C2×C3×C4—92no—
322C2×C3×C4—91no—
323C2×C3×C4—184no—
324C2×C3×C4C2×D3214yes—
325C2×C3×C4C22×C3⋊D311yescharacteristic
326C2×Dic3—34no—
327C2×Dic3—34no—
328C2×Dic3—38no—
329C2×Dic3—616no—
330C2×Dic3—94no—
331C2×Dic3—98no—
332C2×Dic3—94no—
333C2×Dic3—91no—
334C2×Dic3—91no—
335C2×Dic3—92no—
336C2×Dic3—184no—
337C2×Dic3—184no—
338C2×Dic3—188no—
339C2×Dic3C22×C3⋊D311yescharacteristic
340C2×Dic3C22×C3⋊D312yes—
341C2×Dic3C22×C3⋊D311yescharacteristic
342Dic6—22no—
343Dic6—182no—
344Dic6—188no—
345Dic6—188no—
346Dic6—182no—
347Dic6—368no—
348C2×C32—24no—
349C2×C32—24no—
350C2×C32—24no—
351C2×C32—42no—
352C2×C32—48no—
353C2×C32—88no—
354C2×C32C2×#23511yescharacteristic
355C2×C32C2×#23714yes—
356C2×C32C2×#23711yescharacteristic
357C2×C32D3×#4114yes—
358C2×C32#101014yes—
359C2×C32#101011yescharacteristic
360C2×Q8—31no—
361C2×Q8—271no—
362C2×Q8—271no—
363C2×D4D3×C3⋊D311yescharacteristic
364C22×C4—92no—
365C4⋊C4—91no—
366C4⋊C4—271no—
367C4⋊C4—272no—
368C4⋊C4—272no—
369C4⋊C4—272no—
370C4⋊C4—271no—
371C4⋊C4—271no—
372C4⋊C4—272no—
373C22⋊C4—32no—
374C22⋊C4—32no—
375C22⋊C4—92no—
376C22⋊C4—272no—
377C22⋊C4—272no—
378C42—31no—
379C42—91no—
380C42—272no—
381C42—272no—
382C42—271no—
383C22×C3—28no—
384C22×C3—28no—
385C22×C3—28no—
386C22×C3—22no—
387C22×C3—22no—
388C22×C3—22no—
389C22×C3—24no—
390C22×C3—48no—
391C22×C3—48no—
392C22×C3—48no—
393C22×C3C22×D3214yes—
394C22×C3C23×C3⋊D311yescharacteristic
395C3×C4—28no—
396C3×C4—64no—
397C3×C4—64no—
398C3×C4—68no—
399C3×C4—92no—
400C3×C4—182no—
401C3×C4—181no—
402C3×C4#380418yes—
403C3×C4#383012yes—
404Dic3—22no—
405Dic3—21no—
406Dic3—21no—
407Dic3—38no—
408Dic3—64no—
409Dic3—68no—
410Dic3—184no—
411Dic3—184no—
412Dic3—188no—
413Dic3—182no—
414Dic3—181no—
415Dic3—181no—
416Dic3—368no—
417Dic3—364no—
418Dic3—364no—
419C32—24no—
420C32#554211yescharacteristic, complemented
421C32#567114yescomplemented
422C23C2×D3×C3⋊D312yes—
423Q8—62no—
424Q8—542no—
425Q8—542no—
426D4—22no—
427C2×C4—31no—
428C2×C4—31no—
429C2×C4—32no—
430C2×C4—91no—
431C2×C4—91no—
432C2×C4—92no—
433C2×C4—184no—
434C2×C4—271no—
435C2×C4—271no—
436C2×C4—272no—
437C2×C4C2×D3×C3⋊D311yescharacteristic
438C2×C3—24no—
439C2×C3—24no—
440C2×C3—24no—
441C2×C3—42no—
442C2×C3—48no—
443C2×C3—88no—
444C2×C3#6430914yes—
445C2×C3D3×#23714yes—
446C2×C3C2×#380414yes—
447C2×C3C2×#383011yescharacteristic
448C2×C3#41×C3⋊D311yescharacteristic
449C2×C3#6436711yescharacteristic
450C22—22no—
451C22—22no—
452C22—22no—
453C22C22×D3×C3⋊D311yescharacteristic
454C4—61no—
455C4—61no—
456C4—62no—
457C4—92no—
458C4—181no—
459C4—182no—
460C4—541no—
461C4—541no—
462C4—542no—
463C4#8902812yes—
464C3—24no—
465C3#1059527314yescomplemented
466C3#1059614211yescharacteristic, complemented
467C2—42no—
468C2#1172458211yescharacteristic
469C2C3⋊D3×#23711yescharacteristic
470C2C2×#8902811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
Fitting, prime radicalC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#556
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
4. #11723641C2semidirect
5. #11723597C2semidirect
6. #11723596C2semidirect
7. #11723595C2semidirect
8. C4×#6293C2semidirect
9. #11723455C2semidirect
10. #11723399C2semidirect
11. #11723397C2semidirect
308. C33#556semidirect
420. C32#5542semidirect
421. C32#5671semidirect
465. C3#10595273semidirect
466. C3#10596142semidirect

Maximal subgroups

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

subgroupindex
#117239772
C3×D4×C3⋊Dic32
C32×#9362
#117236412
#117235972
#117235962
#117235952
C4×#62932
#117234552
#117233992
#117233972
#105961423
#105952733