Finite Group Databasebeta

← #13175180

Subgroups — #13175180

Order 1728.

2164 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89022C211yesmaximal, characteristic, complemented
2C2×#89021C211yesmaximal, characteristic, complemented
3#11723916C211yesmaximal, characteristic, complemented
4C3×#64084C211yesmaximal, characteristic, complemented
5C32×C42×D3C211yesmaximal, characteristic, complemented
6#11723517C211yesmaximal, characteristic, complemented
7#11723514C214yesmaximal, complemented
8#11723471C214yesmaximal, complemented
9#11723443C211yesmaximal, characteristic, complemented
10#10595948—31nomaximal
11#10593969—34nomaximal
12C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
13C2×#6308C2211yescharacteristic
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C3×#634C2211yescharacteristic
16C2×C32×C4×D3C2212yes—
17C2×C32×C4×D3C2211yescharacteristic
18C2×#6292C2214yes—
19C2×#6291C2212yes—
20C2×#6290C2212yes—
21C2×#6289C2212yes—
22C2×Dic3×C3⋊D3C2212yes—
23#89022—24no—
24#89021—24no—
25C33×C42C2211yescharacteristic
26#88859C2214yes—
27C3×#3752C2214yes—
28C32×C4×Dic3C2211yescharacteristic
29C32×C4×Dic3C2212yes—
30Dic3⋊(C3⋊Dic3)C2214yes—
31C2×#3828—31no—
32C2×#3828—31no—
33C2×#3799—34no—
34C2×#3795—34no—
35C2×C32×C42—31no—
36#64139—34no—
37#64084—31no—
38#64084—34no—
39#64084—64no—
40#64084D311yescharacteristic, complemented
41#64083—31no—
42C4×C4⋊(C3⋊D3)—34no—
43#64078—31no—
44C3×#875—34no—
45C3×C42×D3D314yescomplemented
46#63911—34no—
47#63906—316no—
48#63880—316no—
49#63866—34no—
50C22×C3⋊(C3⋊D3)C2312yes—
51C22×C3×C3⋊D3C2312yes—
52C22×C32×D3C2311yescharacteristic
53C2×C33×C4C2312yes—
54C2×C33×C4C2311yescharacteristic
55C2×C3⋊(C3⋊Dic3)C2312yes—
56C4⋊(C3⋊(C3⋊D3))—22no—
57#6308—22no—
58C2×C3×C3⋊Dic3C2312yes—
59C3×C4⋊(C3⋊D3)—22no—
60C3×#634—22no—
61C2×C32×Dic3C2311yescharacteristic
62C2×C32×Dic3C2312yes—
63C32×C4×D3—24no—
64C32×C4×D3D414yescomplemented
65#6292—28no—
66#6291—24no—
67#6290—24no—
68#6289—24no—
69Dic3×C3⋊D3—24no—
70#5668—91no—
71#5477—94no—
72C22×C32×C4—31no—
73C22×C32×C4—32no—
74C2×#638—32no—
75C2×#638—32no—
76#3828—64no—
77#3828—64no—
78C2×C4⋊(C3⋊D3)—34no—
79C2×C4⋊(C3⋊D3)—31no—
80C2×C4⋊(C3⋊D3)—34no—
81C2×C4⋊(C3⋊D3)—64no—
82C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
83C2×C4×C3⋊D3—32no—
84C2×C4×C3⋊D3—32no—
85C2×#634—31no—
86C2×#634—34no—
87C2×#634—64no—
88C2×#634C2×D311yescharacteristic
89C2×C3×D12—34no—
90C2×C3×C4×D3C2×D318yes—
91C2×C3×C4×D3C2×D314yes—
92C2×C3×Dic6—34no—
93C2×#626—38no—
94C2×#626—316no—
95C2×#625—38no—
96C2×#624—38no—
97C2×D3×Dic3—38no—
98#3799—616no—
99#3795—616no—
100C32×C42—24no—
101C32×C42—31no—
102C32×C42—32no—
103C32×C42C2×D314yes—
104C32×C42C2×D311yescharacteristic
105#3752—34no—
106#3752—316no—
107#3752—616no—
108#3752C2×D314yes—
109C4⋊(C3⋊Dic3)—34no—
110C3×(C2×D3)⋊C4—316no—
111C3×C4×Dic3C2×D314yes—
112C3×C4×Dic3C2×D318yes—
113Dic3⋊Dic3—316no—
114C22×C33C2411yescharacteristic
115C2×C3⋊(C3⋊D3)—22no—
116C2×C3⋊(C3⋊D3)—24no—
117C2×C3×C3⋊D3—22no—
118C2×C3×C3⋊D3—24no—
119C2×C32×D3—22no—
120C2×C32×D3C2×D412yes—
121C33×C4—22no—
122C33×C4C2×D412yes—
123C3⋊(C3⋊Dic3)—22no—
124C3×C3⋊Dic3—22no—
125C32×Dic3C2×D412yes—
126C32×Dic3#4114yescomplemented
127C2×#239—91no—
128C2×#237—91no—
129C2×#235—94no—
130C2×#235—94no—
131C3×#124—91no—
132C2×C3×C42—34no—
133#944—91no—
134#940—91no—
135#930—916no—
136#884—94no—
137#67⋊C4—94no—
138#875—34no—
139#875—94no—
140#875—91no—
141#875—184no—
142#874—94no—
143C4×D12—916no—
144C42×D3C3⋊D311yescharacteristic, complemented
145#869—94no—
146C23×C32—31no—
147C22×C3⋊D3—32no—
148C22×C3⋊D3—38no—
149C22×C3⋊D3—68no—
150C22×C3⋊D3C22×D312yes—
151C22×C3×D3—38no—
152C22×C3×D3C22×D314yes—
153C2×C32×C4—24no—
154C2×C32×C4—28no—
155C2×C32×C4—32no—
156C2×C32×C4—31no—
157C2×C32×C4—34no—
158C2×C32×C4—64no—
159C2×C32×C4C22×D318yes—
160C2×C32×C4C22×D314yes—
161C2×C32×C4C22×D312yes—
162C2×C32×C4C22×D311yescharacteristic
163#638—64no—
164#638—64no—
165C2×C3⋊Dic3—32no—
166C2×C3⋊Dic3—38no—
167C2×C3⋊Dic3—68no—
168C2×C3⋊Dic3C22×D312yes—
169C4⋊(C3⋊D3)—22no—
170C4⋊(C3⋊D3)—68no—
171C4⋊(C3⋊D3)—62no—
172C4⋊(C3⋊D3)—68no—
173C4⋊(C3⋊D3)—128no—
174C4×C3⋊D3—64no—
175C4×C3⋊D3—64no—
176#634—22no—
177#634—62no—
178#634—68no—
179#634—128no—
180C2×C3×Dic3—38no—
181C2×C3×Dic3C22×D314yes—
182C2×C3×Dic3C22×D318yes—
183C3×D12—68no—
184C3×C4×D3—216no—
185C3×C4×D3D12116yescomplemented
186C3×Dic6—68no—
187#626—616no—
188#626—632no—
189#625—616no—
190#624—616no—
191D3×Dic3—616no—
192#532—271noSylow
193C2×C33C22×D411yescharacteristic
194C2×C33C2×#4112yes—
195C3⋊(C3⋊D3)—42no—
196C3×C3⋊D3—42no—
197C32×D3—22no—
198C2×C3×Q8—91no—
199C2×C3×D4—91no—
200C22×C3×C4—34no—
201C22×C3×C4—38no—
202C2×#67—98no—
203C2×#67—98no—
204C2×#67—92no—
205C2×#67—94no—
206C22×Dic3—92no—
207#239—184no—
208#237—184no—
209#235—1816no—
210#235—1816no—
211C2×D12—34no—
212C2×D12—916no—
213C2×D12—92no—
214C2×D12—94no—
215C2×D12—91no—
216C2×D12—184no—
217C2×C4×D3—98no—
218C2×C4×D3—98no—
219C2×C4×D3—92no—
220C2×C4×D3C2×C3⋊D312yes—
221C2×C4×D3C2×C3⋊D311yescharacteristic
222C2×Dic6—34no—
223C2×Dic6—94no—
224C2×Dic6—91no—
225C2×Dic6—184no—
226C3×C22⋊C4—94no—
227C3×C42—24no—
228C3×C42—34no—
229C3×C42—38no—
230C3×C42D3214yes—
231C3×C42C2×C3⋊D311yescharacteristic
232(C2×D3)⋊C4—316no—
233(C2×D3)⋊C4—916no—
234(C2×D3)⋊C4—94no—
235(C2×D3)⋊C4—1816no—
236C4⋊Dic3—916no—
237Dic3⋊C4—94no—
238C4×Dic3C2×C3⋊D312yes—
239C4×Dic3C2×C3⋊D311yescharacteristic
240C22×C32—24no—
241C22×C32—32no—
242C22×C32—62no—
243C22×C32C23×D314yes—
244C22×C32C23×D311yescharacteristic
245C2×C3⋊D3—22no—
246C2×C3⋊D3—24no—
247C2×C3⋊D3—62no—
248C2×C3⋊D3—64no—
249C2×C3⋊D3—68no—
250C2×C3⋊D3—616no—
251C2×C3⋊D3—128no—
252C2×C3⋊D3—1216no—
253C2×C3×D3—28no—
254C2×C3×D3—68no—
255C2×C3×D3—616no—
256C2×C3×D3C2×D1218yes—
257C32×C4—22no—
258C32×C4—28no—
259C32×C4—28no—
260C32×C4—32no—
261C32×C4—34no—
262C32×C4—48no—
263C32×C4C2×D1218yes—
264C32×C4D3×D412yes—
265C3⋊Dic3—22no—
266C3⋊Dic3—62no—
267C3⋊Dic3—68no—
268C3⋊Dic3—128no—
269C3×Dic3—68no—
270C3×Dic3C2×D1218yes—
271C3×Dic3#235116yescomplemented
272C2×#41—271no—
273C2×#41—271no—
274C4⋊Q8—271no—
275#127—271no—
276#124—91no—
277#124—271no—
278#121—274no—
279C4×D4—274no—
280C2×C42—31no—
281C33#53211yescharacteristic, complemented, Sylow
282C23×C3—34no—
283C23×C3—92no—
284C22×D3—38no—
285C22×D3—98no—
286C22×D3—92no—
287C22×D3—188no—
288C22×D3C22×C3⋊D311yescharacteristic
289C3×Q8—182no—
290C3×D4—182no—
291C2×C3×C4—24no—
292C2×C3×C4—28no—
293C2×C3×C4—38no—
294C2×C3×C4—34no—
295C2×C3×C4—316no—
296C2×C3×C4—616no—
297C2×C3×C4—92no—
298C2×C3×C4C2×D3218yes—
299C2×C3×C4C2×D3214yes—
300C2×C3×C4C22×C3⋊D312yes—
301C2×C3×C4C22×C3⋊D311yescharacteristic
302#67—1816no—
303#67—1816no—
304#67—184no—
305#67—188no—
306C2×Dic3—38no—
307C2×Dic3—98no—
308C2×Dic3—92no—
309C2×Dic3—188no—
310C2×Dic3—184no—
311C2×Dic3C22×C3⋊D311yescharacteristic
312C2×Dic3C22×C3⋊D312yes—
313D12—68no—
314D12—1832no—
315D12—184no—
316D12—188no—
317D12—182no—
318D12—368no—
319C4×D3—24no—
320C4×D3—1816no—
321C4×D3—1816no—
322C4×D3—184no—
323C4×D3C4⋊(C3⋊D3)14yescomplemented
324Dic6—68no—
325Dic6—188no—
326Dic6—182no—
327Dic6—368no—
328C2×C32—24no—
329C2×C32—28no—
330C2×C32—62no—
331C2×C32C22×D1214yes—
332C2×C32C2×#23518yes—
333C2×C32C2×D3×D411yescharacteristic
334C2×C32D3×#4112yes—
335C3⋊D3—42no—
336C3⋊D3—122no—
337C3⋊D3—128no—
338C3⋊D3—248no—
339C3×D3—28no—
340C3×D3—128no—
341#41—544no—
342#41—544no—
343C2×Q8—91no—
344C2×Q8—271no—
345C2×D4—91no—
346C2×D4—274no—
347C2×D4—272no—
348C2×D4—272no—
349C2×D4—271no—
350C22×C4—31no—
351C22×C4—32no—
352C22×C4—272no—
353C22×C4—272no—
354C4⋊C4—274no—
355C22⋊C4—94no—
356C22⋊C4—274no—
357C42—32no—
358C42—31no—
359C42D3×C3⋊D311yescharacteristic
360C22×C3—24no—
361C22×C3—38no—
362C22×C3—68no—
363C22×C3—182no—
364C22×C3—184no—
365C22×C3C22×D3214yes—
366C22×C3C23×C3⋊D311yescharacteristic
367C2×D3—22no—
368C2×D3—68no—
369C2×D3—616no—
370C2×D3—188no—
371C2×D3—1816no—
372C2×D3—182no—
373C2×D3—184no—
374C2×D3—368no—
375C2×D3—3616no—
376C2×D3C2×C4⋊(C3⋊D3)12yes—
377C3×C4—28no—
378C3×C4—22no—
379C3×C4—28no—
380C3×C4—38no—
381C3×C4—316no—
382C3×C4—48no—
383C3×C4—182no—
384C3×C4D3×D1218yes—
385C3×C4C2×C4⋊(C3⋊D3)12yes—
386Dic3—68no—
387Dic3—188no—
388Dic3—182no—
389Dic3—368no—
390Dic3C2×C4⋊(C3⋊D3)12yes—
391Dic3#382814yescomplemented
392C32—24no—
393C32#547714yescomplemented
394C32#566811yescharacteristic, complemented
395C23—31no—
396C23—92no—
397C23—272no—
398Q8—182no—
399Q8—542no—
400D4—182no—
401D4—548no—
402D4—544no—
403D4—544no—
404D4—542no—
405C2×C4—32no—
406C2×C4—31no—
407C2×C4—34no—
408C2×C4—64no—
409C2×C4—92no—
410C2×C4—272no—
411C2×C4—544no—
412C2×C4—544no—
413C2×C4C2×D3×C3⋊D312yes—
414C2×C4C2×D3×C3⋊D311yescharacteristic
415C2×C3—24no—
416C2×C3—28no—
417C2×C3—68no—
418C2×C3—362no—
419C2×C3C2×D3×D1214yes—
420C2×C3D3×#23518yes—
421C2×C3C22×C4⋊(C3⋊D3)11yescharacteristic
422C2×C3C2×#382812yes—
423D3—22no—
424D3—128no—
425D3—362no—
426D3—368no—
427D3—728no—
428C22—32no—
429C22—62no—
430C22—182no—
431C22—184no—
432C22—542no—
433C22—544no—
434C22C22×D3×C3⋊D311yescharacteristic
435C4—22no—
436C4—32no—
437C4—34no—
438C4—182no—
439C4—542no—
440C4D3×C4⋊(C3⋊D3)12yes—
441C3—24no—
442C3#1059396914yescomplemented
443C3#1059594811yescharacteristic, complemented
444C2—62no—
445C2—362no—
446C2—1082no—
447C2C2×D3×C4⋊(C3⋊D3)11yescharacteristic
448C2D3×#382812yes—
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
hypercenterC42D3×C3⋊D3
hypocenterC33#532
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#89022C2semidirect
2. C2×#89021C2semidirect
3. #11723916C2semidirect
4. C3×#64084C2semidirect
5. C32×C42×D3C2semidirect
6. #11723517C2semidirect
7. #11723514C2semidirect
8. #11723471C2semidirect
9. #11723443C2semidirect
40. #64084D3semidirect
45. C3×C42×D3D3semidirect
64. C32×C4×D3D4semidirect
126. C32×Dic3#41semidirect
144. C42×D3C3⋊D3semidirect
185. C3×C4×D3D12semidirect
271. C3×Dic3#235semidirect
281. C33#532semidirect
323. C4×D3C4⋊(C3⋊D3)semidirect
391. Dic3#3828semidirect
393. C32#5477semidirect
394. C32#5668semidirect
442. C3#10593969semidirect
443. C3#10595948semidirect

Maximal subgroups

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

subgroupindex
C2×#890222
C2×#890212
#117239162
C3×#640842
C32×C42×D32
#117235172
#117235142
#117234712
#117234432
#105959483
#105939693