Finite Group Databasebeta

← #13176510

Subgroups — #13176510

Order 1728.

1744 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723934C211yesmaximal, characteristic
2C3×#64099C211yesmaximal, characteristic, complemented
3C32×#890C211yesmaximal, characteristic, complemented
4#11723536C211yesmaximal, characteristic, complemented
5C4×#6293C211yesmaximal, characteristic, complemented
6#11723503C212yesmaximal, complemented
7#11723495C212yesmaximal, complemented
8#11723494C211yesmaximal, characteristic
9#11723485C212yesmaximal, complemented
10#11723484C211yesmaximal, characteristic
11#11723442C212yesmaximal, complemented
12#10596161—31nomaximal
13#10594484—34nomaximal
14C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
15C2×C4×C3⋊(C3⋊D3)C2212yes—
16C2×#6293C2211yescharacteristic
17C2×#6293C2212yes—
18C2×#6289C2212yes—
19C2×#6289C2212yes—
20C33×C4⋊C4C2211yescharacteristic
21#88859C2212yes—
22C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
23C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
24C3×C4⋊(C3⋊Dic3)C2212yes—
25C3×#3750C2212yes—
26C3×C4×C3⋊Dic3C2211yescharacteristic
27C32×C4⋊Dic3C2211yescharacteristic
28C32×C4⋊Dic3C2212yes—
29C32×Dic3⋊C4C2212yes—
30C32×C4×Dic3C2211yescharacteristic
31#88807C2211yescharacteristic
32#88806C2211yescharacteristic
33#88806C2212yes—
34Dic3⋊(C3⋊Dic3)C2211yescharacteristic
35#88804C2212yes—
36#88804C2212yes—
37C32×#125—31no—
38#64106—32no—
39#64105—32no—
40#64105—32no—
41#64102—31no—
42#64102—31no—
43#64102—34no—
44#64102—64no—
45C4⋊C4×C3⋊D3—32no—
46#64099D311yescharacteristic, complemented
47#64097—31no—
48C4×C4⋊(C3⋊D3)—31no—
49C4×#634—31no—
50C3×#890—34no—
51C3×#890D314yescomplemented
52#63919—34no—
53C4×#627—34no—
54#63900—38no—
55#63895—38no—
56#63895—38no—
57#63894—34no—
58#63894—34no—
59#63865—38no—
60C22×C3⋊(C3⋊D3)C2312yes—
61C2×C33×C4C2312yes—
62C2×C33×C4C2311yescharacteristic
63C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
64C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
65C4⋊(C3⋊(C3⋊D3))Q814yescomplemented
66C4×C3⋊(C3⋊D3)—24no—
67C2×C3×C3⋊Dic3C2311yescharacteristic
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C3×C3⋊Dic3C2312yes—
70C2×C32×Dic3C2311yescharacteristic
71C2×C32×Dic3C2312yes—
72C2×C32×Dic3C2311yescharacteristic
73#6293—22no—
74#6293—42no—
75#6289—42no—
76#6289—42no—
77#5690—94no—
78#5690—91no—
79C2×C4⋊(C3⋊D3)—31no—
80C2×C4⋊(C3⋊D3)—34no—
81C2×C4⋊(C3⋊D3)—64no—
82C2×C4×C3⋊D3—32no—
83C2×C4×C3⋊D3—38no—
84C2×C4×C3⋊D3—32no—
85C2×C4×C3⋊D3—32no—
86C2×C4×C3⋊D3—68no—
87C2×#634—31no—
88C2×#634—32no—
89C2×#627—34no—
90C2×#627—38no—
91C2×#624—38no—
92C2×#624—38no—
93C32×C4⋊C4—24no—
94C32×C4⋊C4—31no—
95C32×C4⋊C4—32no—
96C32×C4⋊C4—32no—
97C32×C4⋊C4C2×D314yes—
98C32×C4⋊C4C2×D311yescharacteristic
99C32×C42—31no—
100#3752—32no—
101#3752—38no—
102#3752—32no—
103#3752—32no—
104#3752—68no—
105C4⋊(C3⋊Dic3)—31no—
106C4⋊(C3⋊Dic3)C2×D311yescharacteristic
107C4⋊(C3⋊Dic3)C2×D312yes—
108#3750—31no—
109#3750—31no—
110#3750—32no—
111#3750C2×D312yes—
112C4×C3⋊Dic3—31no—
113C4×C3⋊Dic3—34no—
114C4×C3⋊Dic3—64no—
115C4×C3⋊Dic3C2×D311yescharacteristic
116C3×C4⋊Dic3—34no—
117C3×C4⋊Dic3—38no—
118C3×C4⋊Dic3C2×D314yes—
119C3×C4⋊Dic3C2×D318yes—
120C3×Dic3⋊C4—38no—
121C3×Dic3⋊C4C2×D318yes—
122C3×C4×Dic3—34no—
123C3×C4×Dic3C2×D314yes—
124#3724—38no—
125#3724—34no—
126Dic3⋊Dic3—34no—
127Dic3⋊Dic3—34no—
128#3722—38no—
129#3722—38no—
130C22×C33C2411yescharacteristic
131C2×C3⋊(C3⋊D3)—22no—
132C2×C3⋊(C3⋊D3)—22no—
133C2×C3⋊(C3⋊D3)C2×Q814yes—
134C33×C4—22no—
135C33×C4C2×Q812yes—
136C3⋊(C3⋊Dic3)—21no—
137C3⋊(C3⋊Dic3)#4112yes—
138C3×C3⋊Dic3—21no—
139C3×C3⋊Dic3—21no—
140C3×C3⋊Dic3—22no—
141C32×Dic3—21no—
142C32×Dic3—21no—
143C32×Dic3—22no—
144C3×#125—34no—
145C3×#125—91no—
146#897—98no—
147#897—92no—
148#896—98no—
149#896—98no—
150#896—92no—
151#896—92no—
152#893—94no—
153#893—94no—
154#893—91no—
155#893—91no—
156#893—184no—
157D3×C4⋊C4—98no—
158D3×C4⋊C4—92no—
159#890—34no—
160#890C3⋊D311yescharacteristic, complemented
161#888—94no—
162#888—91no—
163C4×D12—94no—
164C4×D12—91no—
165C4×Dic6—94no—
166C4×Dic6—91no—
167C22×C3⋊D3—32no—
168C22×C3⋊D3—38no—
169C22×C3⋊D3—68no—
170C2×C32×C4—24no—
171C2×C32×C4—28no—
172C2×C32×C4—31no—
173C2×C32×C4—32no—
174C2×C32×C4—31no—
175C2×C32×C4C22×D318yes—
176C2×C32×C4C22×D314yes—
177C2×C32×C4C22×D312yes—
178C2×C32×C4C22×D311yescharacteristic
179C2×C3⋊Dic3—31no—
180C2×C3⋊Dic3—31no—
181C2×C3⋊Dic3—34no—
182C2×C3⋊Dic3—34no—
183C2×C3⋊Dic3—64no—
184C2×C3⋊Dic3—64no—
185C2×C3⋊Dic3C22×D311yescharacteristic
186C2×C3⋊Dic3C22×D311yescharacteristic
187C2×C3⋊Dic3C22×D312yes—
188C4⋊(C3⋊D3)—34no—
189C4⋊(C3⋊D3)—316no—
190C4⋊(C3⋊D3)—616no—
191C4×C3⋊D3—64no—
192C4×C3⋊D3—616no—
193C4×C3⋊D3—1216no—
194C4×C3⋊D3—122no—
195C4×C3⋊D3—122no—
196#634—62no—
197#634—122no—
198C2×C3×Dic3—34no—
199C2×C3×Dic3—38no—
200C2×C3×Dic3—34no—
201C2×C3×Dic3C22×D314yes—
202C2×C3×Dic3C22×D318yes—
203C2×C3×Dic3C22×D314yes—
204#627—68no—
205#627—128no—
206#624—128no—
207#624—128no—
208#554—271noSylow
209C2×C33C22×Q811yescharacteristic
210C2×C33C2×#4111yescharacteristic
211C2×C33H(5,𝔽2)11yescharacteristic
212C3⋊(C3⋊D3)—24no—
213C2×D12—94no—
214C2×D12—91no—
215C2×D12—184no—
216C2×C4×D3—98no—
217C2×C4×D3—92no—
218C2×C4×D3—98no—
219C2×C4×D3—98no—
220C2×C4×D3—92no—
221C2×C4×D3—92no—
222C2×C4×D3—188no—
223C2×Dic6—94no—
224C2×Dic6—91no—
225C2×Dic6—98no—
226C2×Dic6—92no—
227C3×C4⋊C4—24no—
228C3×C4⋊C4—34no—
229C3×C4⋊C4—38no—
230C3×C4⋊C4—38no—
231C3×C4⋊C4—91no—
232C3×C4⋊C4—92no—
233C3×C4⋊C4—92no—
234C3×C4⋊C4D3214yes—
235C3×C4⋊C4C2×C3⋊D311yescharacteristic
236C3×C42—34no—
237C3×C42—91no—
238(C2×D3)⋊C4—98no—
239(C2×D3)⋊C4—98no—
240(C2×D3)⋊C4—98no—
241(C2×D3)⋊C4—92no—
242(C2×D3)⋊C4—92no—
243(C2×D3)⋊C4—92no—
244(C2×D3)⋊C4—188no—
245C4⋊Dic3—34no—
246C4⋊Dic3—38no—
247C4⋊Dic3—94no—
248C4⋊Dic3—91no—
249C4⋊Dic3C2×C3⋊D311yescharacteristic
250C4⋊Dic3C2×C3⋊D312yes—
251Dic3⋊C4—38no—
252Dic3⋊C4—94no—
253Dic3⋊C4—94no—
254Dic3⋊C4—98no—
255Dic3⋊C4—91no—
256Dic3⋊C4—91no—
257Dic3⋊C4—92no—
258Dic3⋊C4C2×C3⋊D312yes—
259C4×Dic3—34no—
260C4×Dic3—94no—
261C4×Dic3—91no—
262C4×Dic3—184no—
263C4×Dic3C2×C3⋊D311yescharacteristic
264C22×C32—24no—
265C22×C32C23×D314yes—
266C22×C32C23×D311yescharacteristic
267C2×C3⋊D3—34no—
268C2×C3⋊D3—316no—
269C2×C3⋊D3—616no—
270C2×C3⋊D3—62no—
271C2×C3⋊D3—62no—
272C2×C3⋊D3—68no—
273C2×C3⋊D3—68no—
274C2×C3⋊D3—128no—
275C2×C3⋊D3—128no—
276C32×C4—22no—
277C32×C4—28no—
278C32×C4—28no—
279C32×C4—48no—
280C32×C4—61no—
281C32×C4—62no—
282C32×C4—61no—
283C32×C4D3×Q818yes—
284C32×C4D3×Q812yes—
285C3⋊Dic3—21no—
286C3⋊Dic3—22no—
287C3⋊Dic3—21no—
288C3⋊Dic3—32no—
289C3⋊Dic3—38no—
290C3⋊Dic3—68no—
291C3⋊Dic3—61no—
292C3⋊Dic3—64no—
293C3⋊Dic3—124no—
294C3×Dic3—24no—
295C3×Dic3—24no—
296C3×Dic3—28no—
297C3×Dic3—64no—
298C3×Dic3—64no—
299C3×Dic3—68no—
300C4⋊Q8—271no—
301#125—31no—
302#125—91no—
303C22⋊Q8—272no—
304C22⋊Q8—272no—
305C22⋊Q8—272no—
306C4×Q8—271no—
307C4×D4—271no—
308C4×D4—271no—
309C4×D4—271no—
310C2×C4⋊C4—272no—
311C33#55411yescharacteristic, complemented, Sylow
312C22×D3—98no—
313C22×D3—92no—
314C22×D3—188no—
315C2×C3×C4—24no—
316C2×C3×C4—28no—
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×C4C2×D3218yes—
324C2×C3×C4C2×D3214yes—
325C2×C3×C4C22×C3⋊D312yes—
326C2×C3×C4C22×C3⋊D311yescharacteristic
327C2×Dic3—34no—
328C2×Dic3—34no—
329C2×Dic3—38no—
330C2×Dic3—94no—
331C2×Dic3—91no—
332C2×Dic3—94no—
333C2×Dic3—91no—
334C2×Dic3—184no—
335C2×Dic3—184no—
336C2×Dic3C22×C3⋊D311yescharacteristic
337C2×Dic3C22×C3⋊D311yescharacteristic
338C2×Dic3C22×C3⋊D312yes—
339D12—916no—
340D12—94no—
341D12—1816no—
342C4×D3—184no—
343C4×D3—1816no—
344C4×D3—3616no—
345C4×D3—362no—
346C4×D3—362no—
347C4×D3—368no—
348C4×D3—368no—
349Dic6—182no—
350Dic6—188no—
351Dic6—362no—
352Dic6—368no—
353C2×C32—24no—
354C2×C32—24no—
355C2×C32—24no—
356C2×C32C2×D3×Q814yes—
357C2×C32C2×D3×Q811yescharacteristic
358C2×C32D3×#4114yes—
359C2×C32D3×#4111yescharacteristic
360C2×C32#100914yes—
361C2×C32#100911yescharacteristic
362C3⋊D3—64no—
363C3⋊D3—616no—
364C3⋊D3—1216no—
365C2×Q8—271no—
366C2×Q8—272no—
367C2×D4—271no—
368C22×C4—272no—
369C22×C4—272no—
370C22×C4—272no—
371C4⋊C4—32no—
372C4⋊C4—31no—
373C4⋊C4—32no—
374C4⋊C4—92no—
375C4⋊C4—91no—
376C4⋊C4—92no—
377C4⋊C4—271no—
378C4⋊C4—272no—
379C4⋊C4—271no—
380C4⋊C4—271no—
381C4⋊C4D3×C3⋊D311yescharacteristic
382C22⋊C4—272no—
383C22⋊C4—272no—
384C22⋊C4—272no—
385C42—31no—
386C42—91no—
387C42—271no—
388C22×C3—24no—
389C22×C3C22×D3214yes—
390C22×C3C23×C3⋊D311yescharacteristic
391C2×D3—916no—
392C2×D3—94no—
393C2×D3—1816no—
394C2×D3—188no—
395C2×D3—188no—
396C2×D3—182no—
397C2×D3—182no—
398C2×D3—368no—
399C2×D3—368no—
400C3×C4—28no—
401C3×C4—22no—
402C3×C4—28no—
403C3×C4—48no—
404C3×C4—64no—
405C3×C4—68no—
406C3×C4—64no—
407C3×C4—181no—
408C3×C4—182no—
409C3×C4—181no—
410C3×C4#379718yes—
411C3×C4Q8×C3⋊D312yes—
412Dic3—21no—
413Dic3—22no—
414Dic3—21no—
415Dic3—64no—
416Dic3—68no—
417Dic3—64no—
418Dic3—98no—
419Dic3—92no—
420Dic3—188no—
421Dic3—184no—
422Dic3—181no—
423Dic3—364no—
424C32—24no—
425C32#569014yescomplemented
426C32#569011yescharacteristic, complemented
427C23—272no—
428Q8—542no—
429Q8—1082no—
430D4—274no—
431C2×C4—31no—
432C2×C4—31no—
433C2×C4—32no—
434C2×C4—91no—
435C2×C4—91no—
436C2×C4—92no—
437C2×C4—271no—
438C2×C4—271no—
439C2×C4—544no—
440C2×C4—1082no—
441C2×C4—1082no—
442C2×C4C2×D3×C3⋊D312yes—
443C2×C4C2×D3×C3⋊D311yescharacteristic
444C2×C3—24no—
445C2×C3—24no—
446C2×C3—24no—
447C2×C3C2×#379714yes—
448C2×C3#6431214yes—
449C2×C3#6432014yes—
450C2×C3C2×Q8×C3⋊D311yescharacteristic
451C2×C3#41×C3⋊D311yescharacteristic
452C2×C3#6436611yescharacteristic
453D3—184no—
454D3—1816no—
455D3—3616no—
456C22—274no—
457C22—542no—
458C22—542no—
459C22C22×D3×C3⋊D311yescharacteristic
460C4—22no—
461C4—61no—
462C4—61no—
463C4—62no—
464C4—181no—
465C4—181no—
466C4—182no—
467C4—272no—
468C4—541no—
469C4#8901912yes—
470C3—24no—
471C3#1059448414yescomplemented
472C3#1059616111yescharacteristic, complemented
473C2—544no—
474C2C2×#8901911yescharacteristic
475C2#1172458611yescharacteristic
476C2#1172459711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#554
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×#890C2semidirect
4. #11723536C2semidirect
5. C4×#6293C2semidirect
6. #11723503C2semidirect
7. #11723495C2semidirect
9. #11723485C2semidirect
11. #11723442C2semidirect
46. #64099D3semidirect
51. C3×#890D3semidirect
65. C4⋊(C3⋊(C3⋊D3))Q8semidirect
160. #890C3⋊D3semidirect
311. C33#554semidirect
425. C32#5690semidirect
426. C32#5690semidirect
471. C3#10594484semidirect
472. C3#10596161semidirect

Maximal subgroups

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

subgroupindex
#117239342
C3×#640992
C32×#8902
#117235362
C4×#62932
#117235032
#117234952
#117234942
#117234852
#117234842
#117234422
#105961613
#105944843