Finite Group Databasebeta

← #13176519

Subgroups — #13176519

Order 1728.

1604 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4⋊C4×C3⋊(C3⋊D3)C211yesmaximal, characteristic
2C3×#64099C211yesmaximal, characteristic, complemented
3C32×#888C211yesmaximal, characteristic, complemented
4#11723535C211yesmaximal, characteristic, complemented
5#11723456C212yesmaximal
6#11723454C211yesmaximal, characteristic, complemented
7#11723452C211yesmaximal, characteristic
8#11723442C212yesmaximal
9#11723413C211yesmaximal, characteristic
10#11723402C212yesmaximal, complemented
11#11723393C212yesmaximal, complemented
12#10596200—31nomaximal
13#10594325—34nomaximal
14C2×C4×C3⋊(C3⋊D3)C2212yes—
15C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
16C2×C32×Dic6C2212yes—
17C2×#6293C2212yes—
18C33×C4⋊C4C2211yescharacteristic
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
20#88857C2212yes—
21C3×C4⋊(C3⋊Dic3)C2212yes—
22C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
23C3×#3750C2212yes—
24C3×C4×C3⋊Dic3C2211yescharacteristic
25C32×C4⋊Dic3C2211yescharacteristic
26C32×Dic3⋊C4C2212yes—
27C32×C4×Dic3C2211yescharacteristic
28#88807C2212yes—
29#88807C2212yes—
30#88806C2211yescharacteristic
31#88806C2211yescharacteristic
32#88806C2212yes—
33#88804C2212yes—
34#88804C2211yescharacteristic
35#88804C2211yescharacteristic
36Dic3×C3⋊Dic3C2212yes—
37C32×C4⋊Q8—31no—
38#64155—32no—
39#64153—32no—
40#64105—32no—
41#64101—31no—
42C4⋊C4×C3⋊D3—31no—
43C4⋊C4×C3⋊D3—34no—
44C4⋊C4×C3⋊D3—64no—
45#64099D311yescharacteristic, complemented
46#64098—32no—
47#64097—31no—
48#64081—31no—
49#64079—31no—
50C3×#890—34no—
51C3×#888D314yescomplemented
52#63918—34no—
53#63874—34no—
54#63871—38no—
55#63869—34no—
56#63865—38no—
57#63849—34no—
58#63847—38no—
59#63839—38no—
60C22×C3⋊(C3⋊D3)C2311yescharacteristic
61C2×C33×C4C2312yes—
62C2×C33×C4C2311yescharacteristic
63C2×C3⋊(C3⋊Dic3)C2312yes—
64C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
65C4×C3⋊(C3⋊D3)—42no—
66C4×C3⋊(C3⋊D3)Q814yescomplemented
67C2×C3×C3⋊Dic3C2311yescharacteristic
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C3×C3⋊Dic3C2312yes—
70C2×C32×Dic3C2312yes—
71C2×C32×Dic3C2311yescharacteristic
72C2×C32×Dic3C2311yescharacteristic
73C32×Dic6—42no—
74#6293—42no—
75#5729—94no—
76#5689—91no—
77C2×C32×Q8—32no—
78C2×C4×C3⋊D3—31no—
79C2×C4×C3⋊D3—34no—
80C2×C4×C3⋊D3—32no—
81C2×C4×C3⋊D3—38no—
82C2×C4×C3⋊D3—64no—
83C2×C4×C3⋊D3—68no—
84C2×#634—32no—
85C2×C3×Dic6C2×D318yes—
86C2×#627—38no—
87C32×C4⋊C4—24no—
88C32×C4⋊C4—31no—
89C32×C4⋊C4—32no—
90C32×C4⋊C4C2×D314yes—
91C32×C4⋊C4C2×D311yescharacteristic
92C32×C42—31no—
93#3752—31no—
94#3752—31no—
95#3752—32no—
96C4⋊(C3⋊Dic3)—31no—
97C4⋊(C3⋊Dic3)—34no—
98C4⋊(C3⋊Dic3)—64no—
99C4⋊(C3⋊Dic3)C2×D311yescharacteristic
100C4⋊(C3⋊Dic3)C2×D312yes—
101#3750—32no—
102#3750—38no—
103#3750—32no—
104#3750—32no—
105#3750—31no—
106#3750—31no—
107#3750—32no—
108#3750—68no—
109#3750C2×D312yes—
110C4×C3⋊Dic3—32no—
111C4×C3⋊Dic3C2×D311yescharacteristic
112C3×C4⋊Dic3—38no—
113C3×C4⋊Dic3—34no—
114C3×C4⋊Dic3C2×D314yes—
115C3×Dic3⋊C4—38no—
116C3×Dic3⋊C4C2×D318yes—
117C3×C4×Dic3—34no—
118C3×C4×Dic3C2×D314yes—
119#3724—34no—
120#3724—34no—
121#3724—38no—
122Dic3⋊Dic3—38no—
123Dic3⋊Dic3—38no—
124#3722—38no—
125#3722—34no—
126#3722—34no—
127Dic32—38no—
128C22×C33C2411yescharacteristic
129C2×C3⋊(C3⋊D3)—21no—
130C2×C3⋊(C3⋊D3)—21no—
131C2×C3⋊(C3⋊D3)C2×Q812yes—
132C33×C4—22no—
133C33×C4C2×Q812yes—
134C3⋊(C3⋊Dic3)—22no—
135C3⋊(C3⋊Dic3)C2×Q812yes—
136C3×C3⋊Dic3—21no—
137C3×C3⋊Dic3—21no—
138C3×C3⋊Dic3—22no—
139C32×Dic3—22no—
140C32×Dic3—21no—
141C32×Dic3—21no—
142C3×C4⋊Q8—34no—
143C3×#125—91no—
144#946—98no—
145#944—98no—
146#897—92no—
147#896—98no—
148#896—92no—
149#892—94no—
150#892—91no—
151D3×C4⋊C4—94no—
152D3×C4⋊C4—91no—
153D3×C4⋊C4—184no—
154#890—34no—
155#890—91no—
156#889—98no—
157#889—92no—
158#888—94no—
159#888—92no—
160#888C3⋊D311yescharacteristic, complemented
161#872—94no—
162#872—91no—
163#870—94no—
164#869—91no—
165C22×C3⋊D3—31no—
166C22×C3⋊D3—34no—
167C22×C3⋊D3—64no—
168C32×Q8—122no—
169C2×C32×C4—24no—
170C2×C32×C4—28no—
171C2×C32×C4—32no—
172C2×C32×C4—31no—
173C2×C32×C4—31no—
174C2×C32×C4C22×D318yes—
175C2×C32×C4C22×D314yes—
176C2×C32×C4C22×D312yes—
177C2×C32×C4C22×D311yescharacteristic
178C2×C3⋊Dic3—31no—
179C2×C3⋊Dic3—32no—
180C2×C3⋊Dic3—34no—
181C2×C3⋊Dic3—38no—
182C2×C3⋊Dic3—64no—
183C2×C3⋊Dic3—68no—
184C2×C3⋊Dic3C22×D312yes—
185C2×C3⋊Dic3C22×D311yescharacteristic
186C2×C3⋊Dic3C22×D311yescharacteristic
187C4×C3⋊D3—34no—
188C4×C3⋊D3—316no—
189C4×C3⋊D3—616no—
190C4×C3⋊D3—122no—
191C4×C3⋊D3—128no—
192C4×C3⋊D3—1216no—
193#634—122no—
194C2×C3×Dic3—38no—
195C2×C3×Dic3—34no—
196C2×C3×Dic3—34no—
197C2×C3×Dic3C22×D318yes—
198C2×C3×Dic3C22×D314yes—
199C2×C3×Dic3C22×D314yes—
200C3×Dic6—48no—
201#627—128no—
202#544—271noSylow
203C2×C33C22×Q811yescharacteristic
204C2×C33H(5,𝔽2)11yescharacteristic
205C2×C332−1+411yescharacteristic
206C3⋊(C3⋊D3)—22no—
207C2×C3×Q8—38no—
208C2×C4×D3—94no—
209C2×C4×D3—91no—
210C2×C4×D3—98no—
211C2×C4×D3—92no—
212C2×C4×D3—184no—
213C2×C4×D3—188no—
214C2×Dic6—98no—
215C2×Dic6—92no—
216C2×Dic6C2×C3⋊D312yes—
217C3×C4⋊C4—24no—
218C3×C4⋊C4—34no—
219C3×C4⋊C4—38no—
220C3×C4⋊C4—92no—
221C3×C4⋊C4—92no—
222C3×C4⋊C4—91no—
223C3×C4⋊C4D3214yes—
224C3×C4⋊C4C2×C3⋊D311yescharacteristic
225C3×C42—34no—
226C3×C42—91no—
227(C2×D3)⋊C4—94no—
228(C2×D3)⋊C4—94no—
229(C2×D3)⋊C4—98no—
230(C2×D3)⋊C4—91no—
231(C2×D3)⋊C4—91no—
232(C2×D3)⋊C4—92no—
233C4⋊Dic3—34no—
234C4⋊Dic3—38no—
235C4⋊Dic3—94no—
236C4⋊Dic3—91no—
237C4⋊Dic3—92no—
238C4⋊Dic3—92no—
239C4⋊Dic3—184no—
240C4⋊Dic3C2×C3⋊D311yescharacteristic
241Dic3⋊C4—38no—
242Dic3⋊C4—98no—
243Dic3⋊C4—98no—
244Dic3⋊C4—94no—
245Dic3⋊C4—94no—
246Dic3⋊C4—98no—
247Dic3⋊C4—98no—
248Dic3⋊C4—92no—
249Dic3⋊C4—91no—
250Dic3⋊C4—91no—
251Dic3⋊C4—92no—
252Dic3⋊C4—188no—
253Dic3⋊C4C2×C3⋊D312yes—
254C4×Dic3—34no—
255C4×Dic3—98no—
256C4×Dic3—92no—
257C4×Dic3C2×C3⋊D311yescharacteristic
258C22×C32—24no—
259C22×C32C23×D314yes—
260C22×C32C23×D311yescharacteristic
261C2×C3⋊D3—32no—
262C2×C3⋊D3—38no—
263C2×C3⋊D3—68no—
264C2×C3⋊D3—68no—
265C2×C3⋊D3—68no—
266C2×C3⋊D3—64no—
267C2×C3⋊D3—64no—
268C2×C3⋊D3—61no—
269C2×C3⋊D3—61no—
270C32×C4—22no—
271C32×C4—28no—
272C32×C4—28no—
273C32×C4—48no—
274C32×C4—62no—
275C32×C4—61no—
276C32×C4—61no—
277C32×C4D3×Q818yes—
278C32×C4D3×Q812yes—
279C3⋊Dic3—21no—
280C3⋊Dic3—22no—
281C3⋊Dic3—21no—
282C3⋊Dic3—32no—
283C3⋊Dic3—38no—
284C3⋊Dic3—68no—
285C3⋊Dic3—62no—
286C3⋊Dic3—68no—
287C3⋊Dic3—128no—
288C3×Dic3—28no—
289C3×Dic3—24no—
290C3×Dic3—24no—
291C3×Dic3—64no—
292C3×Dic3—68no—
293C3×Dic3—64no—
294C4⋊Q8—31no—
295C4⋊Q8—271no—
296C4⋊Q8—272no—
297#125—91no—
298#125—271no—
299#125—272no—
300C22⋊Q8—272no—
301C22⋊Q8—272no—
302#117—271no—
303#117—271no—
304C2×C4⋊C4—271no—
305C33#54411yescharacteristic, complemented, Sylow
306C22×D3—94no—
307C22×D3—91no—
308C22×D3—184no—
309C3×Q8—128no—
310C2×C3×C4—24no—
311C2×C3×C4—28no—
312C2×C3×C4—38no—
313C2×C3×C4—34no—
314C2×C3×C4—34no—
315C2×C3×C4—92no—
316C2×C3×C4—91no—
317C2×C3×C4—91no—
318C2×C3×C4C2×D3218yes—
319C2×C3×C4C2×D3214yes—
320C2×C3×C4C22×C3⋊D312yes—
321C2×C3×C4C22×C3⋊D311yescharacteristic
322C2×Dic3—34no—
323C2×Dic3—34no—
324C2×Dic3—38no—
325C2×Dic3—94no—
326C2×Dic3—91no—
327C2×Dic3—98no—
328C2×Dic3—92no—
329C2×Dic3—184no—
330C2×Dic3—188no—
331C2×Dic3C22×C3⋊D312yes—
332C2×Dic3C22×C3⋊D311yescharacteristic
333C2×Dic3C22×C3⋊D311yescharacteristic
334C4×D3—916no—
335C4×D3—94no—
336C4×D3—1816no—
337C4×D3—368no—
338C4×D3—362no—
339C4×D3—3616no—
340Dic6—42no—
341Dic6—362no—
342Dic6—368no—
343C2×C32—24no—
344C2×C32—24no—
345C2×C32—24no—
346C2×C32#100414yes—
347C2×C32C2×D3×Q814yes—
348C2×C32C2×D3×Q811yescharacteristic
349C2×C32#100714yes—
350C2×C32#100911yescharacteristic
351C2×C32#101011yescharacteristic
352C3⋊D3—616no—
353C3⋊D3—62no—
354C3⋊D3—68no—
355C2×Q8—32no—
356C2×Q8—272no—
357C22×C4—271no—
358C22×C4—272no—
359C4⋊C4—31no—
360C4⋊C4—32no—
361C4⋊C4—91no—
362C4⋊C4—92no—
363C4⋊C4—92no—
364C4⋊C4—271no—
365C4⋊C4—271no—
366C4⋊C4—272no—
367C4⋊C4—272no—
368C4⋊C4—272no—
369C4⋊C4—272no—
370C4⋊C4—271no—
371C4⋊C4D3×C3⋊D311yescharacteristic
372C22⋊C4—271no—
373C22⋊C4—271no—
374C22⋊C4—272no—
375C42—31no—
376C42—91no—
377C42—272no—
378C22×C3—24no—
379C22×C3C22×D3214yes—
380C22×C3C23×C3⋊D311yescharacteristic
381C2×D3—98no—
382C2×D3—92no—
383C2×D3—188no—
384C2×D3—188no—
385C2×D3—188no—
386C2×D3—184no—
387C2×D3—184no—
388C2×D3—181no—
389C2×D3—181no—
390C3×C4—28no—
391C3×C4—22no—
392C3×C4—28no—
393C3×C4—48no—
394C3×C4—68no—
395C3×C4—64no—
396C3×C4—64no—
397C3×C4—181no—
398C3×C4—182no—
399C3×C4—181no—
400C3×C4#379718yes—
401C3×C4Q8×C3⋊D312yes—
402Dic3—21no—
403Dic3—22no—
404Dic3—21no—
405Dic3—64no—
406Dic3—64no—
407Dic3—68no—
408Dic3—98no—
409Dic3—92no—
410Dic3—188no—
411Dic3—188no—
412Dic3—182no—
413Dic3—368no—
414C32—24no—
415C32#568911yescharacteristic, complemented
416C32#572914yescomplemented
417C23—271no—
418Q8—122no—
419Q8—1082no—
420C2×C4—31no—
421C2×C4—31no—
422C2×C4—32no—
423C2×C4—91no—
424C2×C4—91no—
425C2×C4—92no—
426C2×C4—271no—
427C2×C4—274no—
428C2×C4—272no—
429C2×C4—1082no—
430C2×C4C2×D3×C3⋊D312yes—
431C2×C4C2×D3×C3⋊D311yescharacteristic
432C2×C3—24no—
433C2×C3—24no—
434C2×C3—24no—
435C2×C3C2×#379714yes—
436C2×C3#6431414yes—
437C2×C3#6431514yes—
438C2×C3#6436111yescharacteristic
439C2×C3C2×Q8×C3⋊D311yescharacteristic
440C2×C3#6436411yescharacteristic
441D3—1816no—
442D3—188no—
443D3—182no—
444C22—272no—
445C22—541no—
446C22—541no—
447C22C22×D3×C3⋊D311yescharacteristic
448C4—22no—
449C4—61no—
450C4—62no—
451C4—61no—
452C4—181no—
453C4—181no—
454C4—182no—
455C4—272no—
456C4—542no—
457C4#8901912yes—
458C3—24no—
459C3#1059432514yescomplemented
460C3#1059620011yescharacteristic, complemented
461C2—542no—
462C2C2×#8901911yescharacteristic
463C2#1172458811yescharacteristic
464C2#1172459211yescharacteristic
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#544
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#64099C2semidirect
3. C32×#888C2semidirect
4. #11723535C2semidirect
6. #11723454C2semidirect
10. #11723402C2semidirect
11. #11723393C2semidirect
45. #64099D3semidirect
51. C3×#888D3semidirect
66. C4×C3⋊(C3⋊D3)Q8semidirect
160. #888C3⋊D3semidirect
305. C33#544semidirect
415. C32#5689semidirect
416. C32#5729semidirect
459. C3#10594325semidirect
460. C3#10596200semidirect

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×#640992
C32×#8882
#117235352
#117234562
#117234542
#117234522
#117234422
#117234132
#117234022
#117233932
#105962003
#105943253