Finite Group Databasebeta

← #13175359

Subgroups — #13175359

Order 1728.

1624 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#11723916C211yesmaximal, characteristic, complemented
2C3×#64084C211yesmaximal, characteristic, complemented
3C32×#870C211yesmaximal, characteristic, complemented
4#11723504C212yesmaximal, complemented
5#11723497C212yesmaximal, complemented
6#11723493C212yesmaximal, complemented
7#11723489C212yesmaximal, complemented
8#11723486C212yesmaximal, complemented
9#11723482C212yesmaximal, complemented
10#10596171—31nomaximal
11#10593953—34nomaximal
12C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
13C2×#6308C2211yescharacteristic
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C3×#634C2211yescharacteristic
16C2×#6292C2212yes—
17C2×#6292C2212yes—
18C2×#6289C2212yes—
19C2×Dic3×C3⋊D3C2212yes—
20C33×C42C2211yescharacteristic
21#88859C2212yes—
22#88859C2212yes—
23C3×#3752C2212yes—
24C3×#3752C2212yes—
25C32×C4⋊Dic3C2211yescharacteristic
26C32×C4⋊Dic3C2211yescharacteristic
27C32×Dic3⋊C4C2212yes—
28C32×Dic3⋊C4C2212yes—
29#88807C2212yes—
30#88806C2212yes—
31#88804C2212yes—
32#88803C2212yes—
33C32×#125—31no—
34#64105—32no—
35#64105—32no—
36#64104—32no—
37#64104—32no—
38#64103—32no—
39#64103—32no—
40#64084—31no—
41#64084—34no—
42#64084—64no—
43#64084D311yescharacteristic, complemented
44C3×#875—34no—
45C3×#870D314yescomplemented
46#63910—38no—
47#63896—38no—
48#63890—38no—
49#63888—38no—
50#63886—38no—
51#63854—38no—
52C22×C3⋊(C3⋊D3)C2312yes—
53C22×C3×C3⋊D3C2312yes—
54C2×C33×C4C2311yescharacteristic
55C2×C33×C4C2311yescharacteristic
56C2×C33×C4C2311yescharacteristic
57C2×C3⋊(C3⋊Dic3)C2312yes—
58C4⋊(C3⋊(C3⋊D3))—41no—
59#6308—41no—
60C2×C3×C3⋊Dic3C2312yes—
61C3×C4⋊(C3⋊D3)—41no—
62C3×#634—41no—
63C2×C32×Dic3C2312yes—
64C2×C32×Dic3C2312yes—
65#6292—42no—
66#6292—42no—
67#6289—42no—
68Dic3×C3⋊D3—42no—
69#5700—94no—
70#5684—91no—
71C2×C4⋊(C3⋊D3)—32no—
72C2×C4⋊(C3⋊D3)—32no—
73C2×C4⋊(C3⋊D3)—31no—
74C2×C4⋊(C3⋊D3)—34no—
75C2×C4⋊(C3⋊D3)—64no—
76C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
77C2×C4×C3⋊D3—32no—
78C2×C4×C3⋊D3—32no—
79C2×#634—31no—
80C2×#634—34no—
81C2×#634—64no—
82C2×#634C2×D311yescharacteristic
83C2×C3×D12—34no—
84C2×C3×Dic6—34no—
85C2×#626—38no—
86C2×#626—38no—
87C2×#624—38no—
88C2×D3×Dic3—38no—
89C32×C4⋊C4—31no—
90C32×C4⋊C4—31no—
91C32×C4⋊C4—32no—
92C32×C4⋊C4—32no—
93C32×C42—24no—
94C32×C42C2×D314yes—
95C32×C42C2×D311yescharacteristic
96#3752—32no—
97#3752—32no—
98#3752—32no—
99#3752—38no—
100#3752—38no—
101#3752—32no—
102#3752—68no—
103#3752—68no—
104#3752C2×D312yes—
105#3752C2×D312yes—
106#3750—32no—
107#3750—32no—
108C3×(C2×D3)⋊C4—38no—
109C3×(C2×D3)⋊C4—38no—
110C3×C4⋊Dic3C2×D314yes—
111C3×C4⋊Dic3C2×D314yes—
112C3×Dic3⋊C4C2×D318yes—
113C3×Dic3⋊C4C2×D318yes—
114#3724—38no—
115Dic3⋊Dic3—38no—
116#3722—38no—
117(C2×D3)⋊Dic3—38no—
118C22×C33C2411yescharacteristic
119C2×C3⋊(C3⋊D3)—22no—
120C2×C3⋊(C3⋊D3)—22no—
121C2×C3⋊(C3⋊D3)—22no—
122C2×C3×C3⋊D3—22no—
123C2×C3×C3⋊D3—22no—
124C2×C3×C3⋊D3—22no—
125C33×C4—21no—
126C33×C4—21no—
127C33×C4—21no—
128C3⋊(C3⋊Dic3)—22no—
129C3×C3⋊Dic3—22no—
130C32×Dic3—22no—
131C32×Dic3—22no—
132C3×#125—34no—
133C3×#124—91no—
134#946—92no—
135#939—92no—
136#896—98no—
137#896—98no—
138#895—98no—
139#895—98no—
140#894—98no—
141#894—98no—
142#886—92no—
143#884—92no—
144#883—92no—
145#878—92no—
146#875—34no—
147#875—94no—
148#875—91no—
149#875—184no—
150#870C3⋊D311yescharacteristic, complemented
151C22×C3⋊D3—32no—
152C22×C3⋊D3—38no—
153C22×C3⋊D3—68no—
154C22×C3⋊D3C22×D312yes—
155C22×C3×D3—38no—
156C2×C32×C4—24no—
157C2×C32×C4—24no—
158C2×C32×C4—24no—
159C2×C32×C4—32no—
160C2×C32×C4—32no—
161C2×C32×C4C22×D314yes—
162C2×C32×C4C22×D314yes—
163C2×C32×C4C22×D314yes—
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C22×D311yescharacteristic
167C2×C3⋊Dic3—32no—
168C2×C3⋊Dic3—38no—
169C2×C3⋊Dic3—68no—
170C2×C3⋊Dic3C22×D312yes—
171C4⋊(C3⋊D3)—41no—
172C4⋊(C3⋊D3)—121no—
173C4⋊(C3⋊D3)—124no—
174C4⋊(C3⋊D3)—128no—
175C4⋊(C3⋊D3)—122no—
176C4⋊(C3⋊D3)—122no—
177C4×C3⋊D3—122no—
178C4×C3⋊D3—122no—
179#634—41no—
180#634—121no—
181#634—124no—
182#634—128no—
183C2×C3×Dic3—38no—
184C2×C3×Dic3C22×D318yes—
185C2×C3×Dic3C22×D318yes—
186C3×D12—124no—
187C3×Dic6—124no—
188#626—128no—
189#626—128no—
190#624—128no—
191D3×Dic3—128no—
192#562—271noSylow
193C2×C33H(5,𝔽2)11yescharacteristic
194C2×C33H(5,𝔽2)11yescharacteristic
195C2×C332−1+411yescharacteristic
196C3⋊(C3⋊D3)—42no—
197C3×C3⋊D3—42no—
198C2×C3×Q8—91no—
199C2×C3×D4—91no—
200C2×#67—92no—
201C2×#67—92no—
202C22×Dic3—92no—
203C2×D12—34no—
204C2×D12—98no—
205C2×D12—98no—
206C2×D12—94no—
207C2×D12—91no—
208C2×D12—184no—
209C2×C4×D3—98no—
210C2×C4×D3—98no—
211C2×C4×D3—92no—
212C2×Dic6—34no—
213C2×Dic6—94no—
214C2×Dic6—91no—
215C2×Dic6—184no—
216C3×C4⋊C4—34no—
217C3×C4⋊C4—34no—
218C3×C4⋊C4—38no—
219C3×C4⋊C4—38no—
220C3×C22⋊C4—92no—
221C3×C22⋊C4—92no—
222C3×C42—24no—
223C3×C42D3214yes—
224C3×C42C2×C3⋊D311yescharacteristic
225#217—92no—
226(C2×D3)⋊C4—38no—
227(C2×D3)⋊C4—38no—
228(C2×D3)⋊C4—98no—
229(C2×D3)⋊C4—98no—
230(C2×D3)⋊C4—98no—
231(C2×D3)⋊C4—98no—
232(C2×D3)⋊C4—92no—
233(C2×D3)⋊C4—92no—
234(C2×D3)⋊C4—92no—
235(C2×D3)⋊C4—188no—
236(C2×D3)⋊C4—188no—
237C4⋊Dic3—92no—
238C4⋊Dic3C2×C3⋊D311yescharacteristic
239C4⋊Dic3C2×C3⋊D311yescharacteristic
240Dic3⋊C4—98no—
241Dic3⋊C4—98no—
242Dic3⋊C4—92no—
243Dic3⋊C4C2×C3⋊D312yes—
244Dic3⋊C4C2×C3⋊D312yes—
245C22×C32—24no—
246C22×C32C23×D314yes—
247C22×C32C23×D311yescharacteristic
248C2×C3⋊D3—22no—
249C2×C3⋊D3—22no—
250C2×C3⋊D3—22no—
251C2×C3⋊D3—62no—
252C2×C3⋊D3—62no—
253C2×C3⋊D3—62no—
254C2×C3⋊D3—68no—
255C2×C3⋊D3—68no—
256C2×C3⋊D3—68no—
257C2×C3⋊D3—128no—
258C2×C3⋊D3—128no—
259C2×C3⋊D3—128no—
260C2×C3×D3—68no—
261C2×C3×D3—68no—
262C2×C3×D3—68no—
263C32×C4—28no—
264C32×C4—21no—
265C32×C4—24no—
266C32×C4—21no—
267C32×C4—21no—
268C32×C4—24no—
269C32×C4—24no—
270C32×C4—44no—
271C32×C4—44no—
272C32×C4—62no—
273C32×C4—62no—
274C3⋊Dic3—22no—
275C3⋊Dic3—62no—
276C3⋊Dic3—68no—
277C3⋊Dic3—128no—
278C3×Dic3—28no—
279C3×Dic3—28no—
280C3×Dic3—68no—
281#125—31no—
282#124—91no—
283#124—271no—
284#123—272no—
285#123—272no—
286C22⋊Q8—272no—
287C22⋊Q8—272no—
288#121—272no—
289#121—272no—
290C33#56211yescharacteristic, complemented, Sylow
291C23×C3—92no—
292C22×D3—38no—
293C22×D3—98no—
294C22×D3—92no—
295C22×D3—188no—
296C3×Q8—361no—
297C3×D4—361no—
298C2×C3×C4—24no—
299C2×C3×C4—24no—
300C2×C3×C4—24no—
301C2×C3×C4—38no—
302C2×C3×C4—38no—
303C2×C3×C4—92no—
304C2×C3×C4C2×D3214yes—
305C2×C3×C4C2×D3214yes—
306C2×C3×C4C2×D3214yes—
307C2×C3×C4C22×C3⋊D311yescharacteristic
308C2×C3×C4C22×C3⋊D311yescharacteristic
309C2×C3×C4C22×C3⋊D311yescharacteristic
310#67—362no—
311#67—362no—
312C2×Dic3—38no—
313C2×Dic3—98no—
314C2×Dic3—92no—
315C2×Dic3—188no—
316C2×Dic3—362no—
317C2×Dic3C22×C3⋊D312yes—
318C2×Dic3C22×C3⋊D312yes—
319D12—124no—
320D12—364no—
321D12—361no—
322D12—368no—
323D12—368no—
324D12—368no—
325C4×D3—362no—
326C4×D3—368no—
327C4×D3—368no—
328Dic6—124no—
329Dic6—364no—
330Dic6—361no—
331Dic6—368no—
332C2×C32—24no—
333C2×C32—24no—
334C2×C32—24no—
335C2×C32#100411yescharacteristic
336C2×C32#100714yes—
337C2×C32#100914yes—
338C2×C32#100914yes—
339C2×C32#100911yescharacteristic
340C2×C32#101011yescharacteristic
341C3⋊D3—42no—
342C3⋊D3—122no—
343C3⋊D3—128no—
344C3⋊D3—248no—
345C3×D3—128no—
346C2×Q8—91no—
347C2×Q8—271no—
348C2×D4—91no—
349C2×D4—272no—
350C2×D4—272no—
351C2×D4—271no—
352C22×C4—272no—
353C22×C4—272no—
354C4⋊C4—32no—
355C4⋊C4—32no—
356C4⋊C4—31no—
357C4⋊C4—31no—
358C4⋊C4—272no—
359C4⋊C4—272no—
360C22⋊C4—92no—
361C22⋊C4—92no—
362C22⋊C4—272no—
363C22⋊C4—272no—
364C22⋊C4—272no—
365C22⋊C4—272no—
366C42D3×C3⋊D311yescharacteristic
367C22×C3—24no—
368C22×C3—182no—
369C22×C3—182no—
370C22×C3—182no—
371C22×C3C22×D3214yes—
372C22×C3C23×C3⋊D311yescharacteristic
373C2×D3—68no—
374C2×D3—68no—
375C2×D3—68no—
376C2×D3—188no—
377C2×D3—188no—
378C2×D3—188no—
379C2×D3—182no—
380C2×D3—182no—
381C2×D3—182no—
382C2×D3—368no—
383C2×D3—368no—
384C2×D3—368no—
385C3×C4—28no—
386C3×C4—24no—
387C3×C4—24no—
388C3×C4—24no—
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—44no—
393C3×C4—44no—
394C3×C4—68no—
395C3×C4—68no—
396C3×C4—182no—
397Dic3—22no—
398Dic3—22no—
399Dic3—68no—
400Dic3—188no—
401Dic3—182no—
402Dic3—368no—
403C32—24no—
404C32#568411yescharacteristic, complemented
405C32#570014yescomplemented
406C23—92no—
407C23—272no—
408Q8—361no—
409Q8—1081no—
410D4—361no—
411D4—1081no—
412D4—1082no—
413D4—1082no—
414C2×C4—32no—
415C2×C4—32no—
416C2×C4—92no—
417C2×C4—272no—
418C2×C4—1082no—
419C2×C4—1082no—
420C2×C4C2×D3×C3⋊D311yescharacteristic
421C2×C4C2×D3×C3⋊D311yescharacteristic
422C2×C4C2×D3×C3⋊D311yescharacteristic
423C2×C3—24no—
424C2×C3—24no—
425C2×C3—24no—
426C2×C3—362no—
427C2×C3#6429714yes—
428C2×C3#6430714yes—
429C2×C3#6430814yes—
430C2×C3#6436411yescharacteristic
431C2×C3#6436611yescharacteristic
432C2×C3#6436611yescharacteristic
433D3—128no—
434D3—362no—
435D3—368no—
436D3—728no—
437C22—182no—
438C22—182no—
439C22—182no—
440C22—542no—
441C22—542no—
442C22—542no—
443C22C22×D3×C3⋊D311yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—62no—
448C4—62no—
449C4—182no—
450C4—542no—
451C3—24no—
452C3#1059395314yescomplemented
453C3#1059617111yescharacteristic, complemented
454C2—362no—
455C2—1082no—
456C2#1172456611yescharacteristic
457C2#1172458011yescharacteristic
458C2#1172458111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
hypercenterC42D3×C3⋊D3
hypocenterC33#562
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11723916C2semidirect
2. C3×#64084C2semidirect
3. C32×#870C2semidirect
4. #11723504C2semidirect
5. #11723497C2semidirect
6. #11723493C2semidirect
7. #11723489C2semidirect
8. #11723486C2semidirect
9. #11723482C2semidirect
43. #64084D3semidirect
45. C3×#870D3semidirect
150. #870C3⋊D3semidirect
290. C33#562semidirect
404. C32#5684semidirect
405. C32#5700semidirect
452. C3#10593953semidirect
453. C3#10596171semidirect

Maximal subgroups

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

subgroupindex
#117239162
C3×#640842
C32×#8702
#117235042
#117234972
#117234932
#117234892
#117234862
#117234822
#105961713
#105939533