Finite Group Databasebeta

← #13176515

Subgroups — #13176515

Order 1728.

1460 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723931C211yesmaximal, characteristic, complemented
2C3×#64097C211yesmaximal, characteristic, complemented
3C32×D3×C4⋊C4C211yesmaximal, characteristic
4#11723536C211yesmaximal, characteristic, complemented
5#11723447C211yesmaximal, characteristic, complemented
6#11723425C212yesmaximal
7#11723424C211yesmaximal, characteristic
8#11723423C211yesmaximal, characteristic
9#11723415C212yesmaximal
10#11723400C212yesmaximal, complemented
11#11723394C212yesmaximal, complemented
12#10595971—31nomaximal
13#10594342—34nomaximal
14C2×C3×#634C2212yes—
15C2×C32×C4×D3C2212yes—
16C2×C32×C4×D3C2211yescharacteristic
17C2×#6293C2212yes—
18C33×C4⋊C4C2211yescharacteristic
19C4⋊(C3⋊(C3⋊Dic3))C2212yes—
20C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
21#88857C2212yes—
22C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
23C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
24C3×#3750C2212yes—
25C3×C4×C3⋊Dic3C2211yescharacteristic
26C32×C4⋊Dic3C2211yescharacteristic
27C32×Dic3⋊C4C2212yes—
28#88806C2212yes—
29#88806C2212yes—
30Dic3⋊(C3⋊Dic3)C2211yescharacteristic
31Dic3⋊(C3⋊Dic3)C2211yescharacteristic
32Dic3⋊(C3⋊Dic3)C2212yes—
33(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
34(C2×D3)⋊(C3⋊Dic3)C2212yes—
35(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
36Dic3×C3⋊Dic3C2212yes—
37C2×C32×C4⋊C4—31no—
38#64135—31no—
39#64135—31no—
40#64133—32no—
41#64133—32no—
42#64099—31no—
43#64099—32no—
44#64099—31no—
45#64099—34no—
46#64099—64no—
47#64097—32no—
48#64097—31no—
49#64097D311yescharacteristic, complemented
50C3×D3×C4⋊C4D314yes—
51C3×#888—34no—
52#63919—34no—
53#63869—34no—
54#63856—38no—
55#63855—34no—
56#63851—38no—
57#63845—38no—
58#63841—34no—
59#63839—38no—
60C22×C32×D3C2311yescharacteristic
61C2×C33×C4C2312yes—
62C2×C33×C4C2311yescharacteristic
63C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
64C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)C2312yes—
66C2×C3×C3⋊Dic3C2311yescharacteristic
67C2×C3×C3⋊Dic3C2311yescharacteristic
68C2×C3×C3⋊Dic3C2312yes—
69C3×#634—42no—
70C2×C32×Dic3C2312yes—
71C2×C32×Dic3C2311yescharacteristic
72C32×C4×D3—42no—
73C32×C4×D3Q814yescomplemented
74#6293—42no—
75#5729—91no—
76#5500—94no—
77C22×C32×C4—31no—
78C22×C32×C4—32no—
79C2×#634—32no—
80C2×#634C2×D312yes—
81C2×C3×C4×D3C2×D318yes—
82C2×C3×C4×D3C2×D314yes—
83C2×C3×Dic6—38no—
84C2×#627—38no—
85C32×C4⋊C4—24no—
86C32×C4⋊C4—32no—
87C32×C4⋊C4—31no—
88C32×C4⋊C4C2×D314yes—
89C32×C4⋊C4C2×D311yescharacteristic
90#3757—32no—
91#3757—31no—
92#3757—31no—
93C4⋊(C3⋊Dic3)—32no—
94C4⋊(C3⋊Dic3)—31no—
95C4⋊(C3⋊Dic3)—34no—
96C4⋊(C3⋊Dic3)—38no—
97C4⋊(C3⋊Dic3)—31no—
98C4⋊(C3⋊Dic3)—32no—
99C4⋊(C3⋊Dic3)—31no—
100C4⋊(C3⋊Dic3)—68no—
101C4⋊(C3⋊Dic3)—64no—
102C4⋊(C3⋊Dic3)C2×D311yescharacteristic
103#3750—32no—
104#3750—38no—
105#3750—32no—
106#3750—32no—
107#3750—68no—
108#3750C2×D312yes—
109C4×C3⋊Dic3—31no—
110C4×C3⋊Dic3—34no—
111C4×C3⋊Dic3—32no—
112C4×C3⋊Dic3—64no—
113C4×C3⋊Dic3C2×D311yescharacteristic
114C3×C4⋊Dic3—34no—
115C3×C4⋊Dic3C2×D314yes—
116C3×Dic3⋊C4—38no—
117C3×Dic3⋊C4C2×D318yes—
118C3×C4×Dic3—34no—
119#3724—38no—
120#3724—38no—
121Dic3⋊Dic3—34no—
122Dic3⋊Dic3—34no—
123Dic3⋊Dic3—38no—
124(C2×D3)⋊Dic3—38no—
125(C2×D3)⋊Dic3—34no—
126(C2×D3)⋊Dic3—34no—
127Dic32—38no—
128C22×C33C2411yescharacteristic
129C2×C32×D3—21no—
130C2×C32×D3—21no—
131C2×C32×D3C2×Q812yes—
132C33×C4—22no—
133C33×C4C2×Q812yes—
134C3⋊(C3⋊Dic3)—21no—
135C3⋊(C3⋊Dic3)—21no—
136C3⋊(C3⋊Dic3)—22no—
137C3×C3⋊Dic3—22no—
138C3×C3⋊Dic3—21no—
139C3×C3⋊Dic3—21no—
140C32×Dic3—22no—
141C32×Dic3C2×Q812yes—
142C3×C4⋊Q8—91no—
143C2×C3×C4⋊C4—34no—
144#946—92no—
145#944—92no—
146#926—94no—
147#926—94no—
148#924—98no—
149#924—98no—
150#896—92no—
151#892—91no—
152D3×C4⋊C4C3⋊D311yescharacteristic
153#890—94no—
154#890—98no—
155#890—94no—
156#890—91no—
157#890—184no—
158#889—92no—
159#888—34no—
160#888—94no—
161#888—98no—
162#888—91no—
163#872—91no—
164#870—91no—
165C23×C32—31no—
166C22×C3×D3C22×D314yes—
167C2×C32×C4—24no—
168C2×C32×C4—28no—
169C2×C32×C4—31no—
170C2×C32×C4—34no—
171C2×C32×C4—32no—
172C2×C32×C4—122no—
173C2×C32×C4C22×D318yes—
174C2×C32×C4C22×D314yes—
175C2×C32×C4C22×D312yes—
176C2×C32×C4C22×D311yescharacteristic
177C2×C3⋊Dic3—31no—
178C2×C3⋊Dic3—32no—
179C2×C3⋊Dic3—31no—
180C2×C3⋊Dic3—38no—
181C2×C3⋊Dic3—34no—
182C2×C3⋊Dic3—34no—
183C2×C3⋊Dic3—64no—
184C2×C3⋊Dic3—64no—
185C2×C3⋊Dic3—68no—
186C2×C3⋊Dic3C22×D312yes—
187C2×C3⋊Dic3C22×D311yescharacteristic
188C2×C3⋊Dic3C22×D311yescharacteristic
189#634—42no—
190#634—122no—
191C2×C3×Dic3—38no—
192C2×C3×Dic3—34no—
193C2×C3×Dic3—34no—
194C2×C3×Dic3C22×D318yes—
195C2×C3×Dic3C22×D314yes—
196C3×C4×D3—48no—
197C3×C4×D3Dic6116yescomplemented
198C3×Dic6—128no—
199#627—128no—
200#544—271noSylow
201C2×C33C22×Q811yescharacteristic
202C2×C33H(5,𝔽2)11yescharacteristic
203C2×C332−1+411yescharacteristic
204C32×D3—22no—
205C2×C3×Q8—92no—
206C22×C3×C4—34no—
207C22×C3×C4—38no—
208C2×C4×D3C2×C3⋊D312yes—
209C2×C4×D3C2×C3⋊D311yescharacteristic
210C2×Dic6—38no—
211C2×Dic6—98no—
212C2×Dic6—92no—
213C3×C4⋊C4—24no—
214C3×C4⋊C4—38no—
215C3×C4⋊C4—34no—
216C3×C4⋊C4—91no—
217C3×C4⋊C4—92no—
218C3×C4⋊C4D3214yes—
219C3×C4⋊C4C2×C3⋊D311yescharacteristic
220C3×C42—91no—
221#217—94no—
222#217—94no—
223#217—98no—
224(C2×D3)⋊C4—91no—
225(C2×D3)⋊C4—91no—
226(C2×D3)⋊C4—92no—
227C4⋊Dic3—34no—
228C4⋊Dic3—94no—
229C4⋊Dic3—98no—
230C4⋊Dic3—94no—
231C4⋊Dic3—98no—
232C4⋊Dic3—94no—
233C4⋊Dic3—91no—
234C4⋊Dic3—92no—
235C4⋊Dic3—184no—
236C4⋊Dic3—188no—
237C4⋊Dic3C2×C3⋊D311yescharacteristic
238Dic3⋊C4—38no—
239Dic3⋊C4—98no—
240Dic3⋊C4—98no—
241Dic3⋊C4—98no—
242Dic3⋊C4—92no—
243Dic3⋊C4—92no—
244Dic3⋊C4—92no—
245Dic3⋊C4—91no—
246Dic3⋊C4—91no—
247Dic3⋊C4—92no—
248Dic3⋊C4—188no—
249Dic3⋊C4C2×C3⋊D312yes—
250C4×Dic3—34no—
251C4×Dic3—98no—
252C4×Dic3—94no—
253C4×Dic3—91no—
254C4×Dic3—92no—
255C4×Dic3—184no—
256C22×C32—24no—
257C22×C32—32no—
258C22×C32—61no—
259C22×C32—61no—
260C22×C32C23×D314yes—
261C22×C32C23×D311yescharacteristic
262C2×C3×D3—24no—
263C2×C3×D3—24no—
264C2×C3×D3C2×Dic618yes—
265C32×C4—22no—
266C32×C4—28no—
267C32×C4—28no—
268C32×C4—32no—
269C32×C4—48no—
270C32×C4—62no—
271C32×C4C2×Dic618yes—
272C32×C4D3×Q812yes—
273C3⋊Dic3—21no—
274C3⋊Dic3—22no—
275C3⋊Dic3—21no—
276C3⋊Dic3—61no—
277C3⋊Dic3—61no—
278C3⋊Dic3—62no—
279C3⋊Dic3—64no—
280C3⋊Dic3—68no—
281C3⋊Dic3—64no—
282C3⋊Dic3—124no—
283C3⋊Dic3—124no—
284C3⋊Dic3—128no—
285C3×Dic3—28no—
286C3×Dic3—68no—
287C3×Dic3—64no—
288C3×Dic3—64no—
289C3×Dic3C2×Dic618yes—
290C4⋊Q8—91no—
291C4⋊Q8—271no—
292C4⋊Q8—272no—
293#125—271no—
294#125—272no—
295#125—271no—
296C22⋊Q8—272no—
297C22⋊Q8—272no—
298#117—271no—
299#117—271no—
300C2×C4⋊C4—31no—
301C33#54411yescharacteristic, complemented, Sylow
302C23×C3—34no—
303C22×D3C22×C3⋊D311yescharacteristic
304C3×Q8—362no—
305C2×C3×C4—24no—
306C2×C3×C4—28no—
307C2×C3×C4—34no—
308C2×C3×C4—316no—
309C2×C3×C4—38no—
310C2×C3×C4—92no—
311C2×C3×C4—91no—
312C2×C3×C4—91no—
313C2×C3×C4—128no—
314C2×C3×C4C2×D3218yes—
315C2×C3×C4C2×D3214yes—
316C2×C3×C4C22×C3⋊D312yes—
317C2×C3×C4C22×C3⋊D311yescharacteristic
318C2×Dic3—34no—
319C2×Dic3—34no—
320C2×Dic3—38no—
321C2×Dic3—98no—
322C2×Dic3—94no—
323C2×Dic3—94no—
324C2×Dic3—92no—
325C2×Dic3—91no—
326C2×Dic3—91no—
327C2×Dic3—184no—
328C2×Dic3—184no—
329C2×Dic3—188no—
330C2×Dic3C22×C3⋊D312yes—
331C2×Dic3C22×C3⋊D311yescharacteristic
332C4×D3—42no—
333C4×D3#63414yescomplemented
334Dic6—128no—
335Dic6—362no—
336Dic6—368no—
337C2×C32—24no—
338C2×C32—24no—
339C2×C32—24no—
340C2×C32—62no—
341C2×C32C22×Dic614yes—
342C2×C32#100414yes—
343C2×C32#100411yescharacteristic
344C2×C32C2×D3×Q811yescharacteristic
345C2×C32#100714yes—
346C2×C32#100711yescharacteristic
347C3×D3—28no—
348C2×Q8—92no—
349C2×Q8—272no—
350C22×C4—31no—
351C22×C4—32no—
352C4⋊C4—32no—
353C4⋊C4—31no—
354C4⋊C4—91no—
355C4⋊C4—92no—
356C4⋊C4—271no—
357C4⋊C4—271no—
358C4⋊C4—272no—
359C4⋊C4—272no—
360C4⋊C4—272no—
361C4⋊C4—271no—
362C4⋊C4—272no—
363C4⋊C4—272no—
364C4⋊C4D3×C3⋊D311yescharacteristic
365C22⋊C4—271no—
366C22⋊C4—271no—
367C22⋊C4—272no—
368C42—91no—
369C42—272no—
370C42—271no—
371C22×C3—24no—
372C22×C3—38no—
373C22×C3—64no—
374C22×C3—64no—
375C22×C3C22×D3214yes—
376C22×C3C23×C3⋊D311yescharacteristic
377C2×D3—21no—
378C2×D3—21no—
379C2×D3C2×#63412yes—
380C3×C4—28no—
381C3×C4—22no—
382C3×C4—28no—
383C3×C4—38no—
384C3×C4—48no—
385C3×C4—68no—
386C3×C4—182no—
387C3×C4—181no—
388C3×C4—181no—
389C3×C4D3×Dic618yes—
390C3×C4C2×#63412yes—
391Dic3—22no—
392Dic3—64no—
393Dic3—68no—
394Dic3—64no—
395Dic3—184no—
396Dic3—188no—
397Dic3—184no—
398Dic3—181no—
399Dic3—182no—
400Dic3—181no—
401Dic3—364no—
402Dic3—364no—
403Dic3—368no—
404Dic3C2×#63412yes—
405C32—24no—
406C32#550014yescomplemented
407C32#572911yescharacteristic, complemented
408C23—31no—
409Q8—362no—
410Q8—1082no—
411C2×C4—31no—
412C2×C4—34no—
413C2×C4—32no—
414C2×C4—91no—
415C2×C4—91no—
416C2×C4—92no—
417C2×C4—122no—
418C2×C4—272no—
419C2×C4—271no—
420C2×C4—271no—
421C2×C4C2×D3×C3⋊D312yes—
422C2×C4C2×D3×C3⋊D311yescharacteristic
423C2×C3—24no—
424C2×C3—24no—
425C2×C3—24no—
426C2×C3—68no—
427C2×C3C2×D3×Dic614yes—
428C2×C3#6431314yes—
429C2×C3#6431614yes—
430C2×C3C22×#63411yescharacteristic
431C2×C3#6436111yescharacteristic
432C2×C3#6436411yescharacteristic
433D3—22no—
434C22—32no—
435C22—61no—
436C22—61no—
437C22C22×D3×C3⋊D311yescharacteristic
438C4—22no—
439C4—32no—
440C4—62no—
441C4—181no—
442C4—182no—
443C4—181no—
444C4—541no—
445C4—542no—
446C4—541no—
447C4D3×#63412yes—
448C3—24no—
449C3#1059434214yescomplemented
450C3#1059597111yescharacteristic, complemented
451C2—62no—
452C2C2×D3×#63411yescharacteristic
453C2#1172458711yescharacteristic
454C2#1172459111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
prime radicalC22×C32×D3C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#544
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11723931C2semidirect
2. C3×#64097C2semidirect
4. #11723536C2semidirect
5. #11723447C2semidirect
10. #11723400C2semidirect
11. #11723394C2semidirect
49. #64097D3semidirect
73. C32×C4×D3Q8semidirect
197. C3×C4×D3Dic6semidirect
301. C33#544semidirect
333. C4×D3#634semidirect
406. C32#5500semidirect
407. C32#5729semidirect
449. C3#10594342semidirect
450. C3#10595971semidirect

Maximal subgroups

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

subgroupindex
#117239312
C3×#640972
C32×D3×C4⋊C42
#117235362
#117234472
#117234252
#117234242
#117234232
#117234152
#117234002
#117233942
#105959713
#105943423