Finite Group Databasebeta

← #13177405

Subgroups — #13177405

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×C32×#237C211yesmaximal, characteristic, complemented
2C2×#89021C211yesmaximal, characteristic, complemented
3#11723977C211yesmaximal, characteristic, complemented
4C3×#64149C211yesmaximal, characteristic, complemented
5Dic3×#638C214yesmaximal, complemented
6#11723615C214yesmaximal, complemented
7C4×D3×C3⋊Dic3C211yesmaximal, characteristic, complemented
8#11723455C211yesmaximal, characteristic, complemented
9#11723436C211yesmaximal, characteristic, complemented
10#10596303—31nomaximal
11#10595152—34nomaximal
12C2×C33×D4C2211yescharacteristic
13C2×#6308C2211yescharacteristic
14C2×C3×#638C2214yes—
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C32×#67C2212yes—
17C22×C32×Dic3C2212yes—
18C32×#237—24no—
19C2×C32×C4×D3C2211yescharacteristic
20C2×C32×Dic6C2211yescharacteristic
21C2×#6290C2212yes—
22C2×Dic3×C3⋊D3C2212yes—
23C2×D3×C3⋊Dic3C2212yes—
24#89021—24no—
25#88864C2214yes—
26C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
27C3×C4×C3⋊Dic3C2211yescharacteristic
28#88807C2214yes—
29#88803C2214yes—
30Dic3×C3⋊Dic3C2212yes—
31C2×C32×#41—31no—
32C2×#3828—31no—
33C2×C3×#237D314yescomplemented
34C2×#3795—34no—
35#64156—31no—
36#64153—31no—
37#64149D311yescharacteristic, complemented
38#64148—34no—
39#64145—31no—
40#64145—34no—
41#64145—64no—
42C4×#638—34no—
43C2×C4×C3⋊Dic3—31no—
44C3×#940—34no—
45Dic3×#67—316no—
46#63970—316no—
47C4×D3×Dic3—34no—
48#63866—34no—
49#63863—34no—
50C23×C33C2312yes—
51C22×C3×C3⋊D3C2312yes—
52C22×C32×D3C2311yescharacteristic
53C33×D4—22no—
54C2×C33×C4C2311yescharacteristic
55C2×C3⋊(C3⋊Dic3)C2312yes—
56C2×C3⋊(C3⋊Dic3)C2312yes—
57#6308—22no—
58C3×#638—28no—
59C2×C3×C3⋊Dic3C2312yes—
60C3×C4⋊(C3⋊D3)—22no—
61C32×#67—24no—
62C2×C32×Dic3—24no—
63C2×C32×Dic3C2312yes—
64C2×C32×Dic3C2311yescharacteristic
65C32×C4×D3D414yescomplemented
66C32×Dic6—22no—
67#6290—24no—
68Dic3×C3⋊D3—24no—
69D3×C3⋊Dic3—24no—
70#5832—94no—
71#5716—91no—
72C32×#41—64no—
73C2×C32×Q8—31no—
74C2×C32×D4—24no—
75C2×C32×D4—32no—
76C2×C32×D4C2×D314yes—
77C2×C32×D4C2×D311yescharacteristic
78C22×C32×C4—31no—
79C22×C32×C4—32no—
80C2×#638—32no—
81C2×#638C2×D314yes—
82C22×C3⋊Dic3—32no—
83#3828—64no—
84C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
85C2×C4×C3⋊D3—32no—
86C2×#634—31no—
87C2×#634—34no—
88C2×#634—64no—
89C2×C3×#67—316no—
90C2×C3×#67C2×D318yes—
91C22×C3×Dic3C2×D318yes—
92C3×#237—216no—
93C2×C3×D12—34no—
94C2×C3×C4×D3C2×D314yes—
95C2×C3×Dic6C2×D314yes—
96C2×#625—38no—
97C2×D3×Dic3—38no—
98C2×D3×Dic3—38no—
99#3795—616no—
100#3757—34no—
101#3757—316no—
102#3757—616no—
103#3752—34no—
104#3750—34no—
105C4×C3⋊Dic3—31no—
106C4×C3⋊Dic3—34no—
107C4×C3⋊Dic3—32no—
108C4×C3⋊Dic3—64no—
109C4×C3⋊Dic3C2×D311yescharacteristic
110C3×C4×Dic3—34no—
111Dic3⋊Dic3—316no—
112(C2×D3)⋊Dic3—316no—
113Dic32—38no—
114C22×C33—22no—
115C22×C33—24no—
116C22×C33C2411yescharacteristic
117C2×C3×C3⋊D3—22no—
118C2×C3×C3⋊D3—24no—
119C2×C32×D3—22no—
120C2×C32×D3C2×D412yes—
121C33×C4C2×D412yes—
122C3⋊(C3⋊Dic3)—22no—
123C3⋊(C3⋊Dic3)—22no—
124C3×C3⋊Dic3#4114yescomplemented
125C32×Dic3—22no—
126C32×Dic3C2×D412yes—
127C2×C3×#41—34no—
128C2×#237—91no—
129C2×#237C3⋊D311yescharacteristic, complemented
130C2×#235—94no—
131C3×#127—91no—
132#947—94no—
133#944—94no—
134#940—34no—
135#939—916no—
136#938—94no—
137#936—94no—
138#936—91no—
139#936—91no—
140#936—184no—
141D4×Dic3—94no—
142C4×#67—916no—
143C2×C4×Dic3—94no—
144C42×D3—91no—
145#869—91no—
146C23×C32—28no—
147C23×C32—31no—
148C23×C32C22×D318yes—
149C23×C32C22×D312yes—
150C22×C3⋊D3C22×D312yes—
151C22×C3×D3—38no—
152C22×C3×D3C22×D314yes—
153C32×Q8—62no—
154C32×D4—22no—
155C32×D4—28no—
156C32×D4—48no—
157C32×D4—64no—
158C2×C32×C4—24no—
159C2×C32×C4—31no—
160C2×C32×C4—34no—
161C2×C32×C4—32no—
162C2×C32×C4—64no—
163C2×C32×C4C22×D314yes—
164C2×C32×C4C22×D311yescharacteristic
165#638—28no—
166#638—64no—
167C2×C3⋊Dic3—32no—
168C2×C3⋊Dic3—32no—
169C2×C3⋊Dic3—38no—
170C2×C3⋊Dic3—38no—
171C2×C3⋊Dic3—68no—
172C2×C3⋊Dic3—68no—
173C2×C3⋊Dic3—64no—
174C2×C3⋊Dic3C22×D312yes—
175C4⋊(C3⋊D3)—22no—
176C4×C3⋊D3—64no—
177#634—62no—
178#634—68no—
179#634—128no—
180C3×#67—216no—
181C3×#67—632no—
182C2×C3×Dic3—216no—
183C2×C3×Dic3—38no—
184C2×C3×Dic3C22×D318yes—
185C2×C3×Dic3C22×D314yes—
186C3×D12—68no—
187C3×C4×D3#67116yescomplemented
188C3×Dic6—28no—
189#625—616no—
190D3×Dic3—616no—
191D3×Dic3—616no—
192#532—271noSylow
193C2×C33—42no—
194C2×C33C22×D411yescharacteristic
195C2×C33C2×#4112yes—
196C3×C3⋊D3—42no—
197C32×D3—22no—
198C3×#41—616no—
199C2×C3×Q8—34no—
200C2×C3×D4—24no—
201C2×C3×D4—38no—
202C2×C3×D4—94no—
203C2×C3×D4—91no—
204C2×C3×D4D3214yes—
205C2×C3×D4C2×C3⋊D311yescharacteristic
206C22×C3×C4—34no—
207C22×C3×C4—38no—
208C2×#67—316no—
209C2×#67—98no—
210C2×#67—92no—
211C2×#67C2×C3⋊D312yes—
212C22×Dic3—98no—
213C22×Dic3—92no—
214C22×Dic3C2×C3⋊D312yes—
215#237—24no—
216#237—184no—
217#235—1816no—
218C2×D12—34no—
219C2×C4×D3—98no—
220C2×C4×D3—92no—
221C2×C4×D3C2×C3⋊D311yescharacteristic
222C2×Dic6—94no—
223C2×Dic6—91no—
224C2×Dic6—184no—
225C2×Dic6C2×C3⋊D311yescharacteristic
226C3×C42—91no—
227#217—916no—
228#217—94no—
229#217—94no—
230#217—1816no—
231(C2×D3)⋊C4—916no—
232C4⋊Dic3—94no—
233Dic3⋊C4—916no—
234C4×Dic3—34no—
235C4×Dic3—98no—
236C4×Dic3—94no—
237C4×Dic3—91no—
238C4×Dic3—92no—
239C4×Dic3—184no—
240C22×C32—22no—
241C22×C32—24no—
242C22×C32—24no—
243C22×C32—216no—
244C22×C32—28no—
245C22×C32—32no—
246C22×C32—48no—
247C22×C32—416no—
248C22×C32—62no—
249C22×C32C23×D314yes—
250C22×C32C23×D311yescharacteristic
251C2×C3⋊D3—22no—
252C2×C3⋊D3—24no—
253C2×C3×D3—28no—
254C2×C3×D3—616no—
255C2×C3×D3—68no—
256C2×C3×D3C2×#6718yes—
257C32×C4—28no—
258C32×C4—32no—
259C32×C4—62no—
260C32×C4D3×D412yes—
261C32×C4C2×#6718yes—
262C3⋊Dic3—62no—
263C3⋊Dic3—62no—
264C3⋊Dic3—68no—
265C3⋊Dic3—68no—
266C3⋊Dic3—128no—
267C3⋊Dic3—128no—
268C3⋊Dic3#23714yescomplemented
269C3×Dic3—28no—
270C3×Dic3—316no—
271C3×Dic3C2×#6718yes—
272C2×#41—31no—
273C2×#41—271no—
274C4⋊Q8—271no—
275#127—91no—
276#124—271no—
277#124—271no—
278#121—274no—
279C4×D4—274no—
280C2×C42—271no—
281C33#53211yescharacteristic, complemented, Sylow
282C23×C3—28no—
283C23×C3—34no—
284C23×C3—92no—
285C23×C3C2×D3218yes—
286C23×C3C22×C3⋊D312yes—
287C22×D3—38no—
288C22×D3C22×C3⋊D311yescharacteristic
289C3×Q8—68no—
290C3×D4—28no—
291C3×D4—22no—
292C3×D4—48no—
293C3×D4—616no—
294C3×D4—188no—
295C3×D4—182no—
296C2×C3×C4—24no—
297C2×C3×C4—34no—
298C2×C3×C4—316no—
299C2×C3×C4—38no—
300C2×C3×C4—616no—
301C2×C3×C4—92no—
302C2×C3×C4C2×D3214yes—
303C2×C3×C4C22×C3⋊D311yescharacteristic
304#67—24no—
305#67—632no—
306#67—1816no—
307#67—184no—
308C2×Dic3—24no—
309C2×Dic3—38no—
310C2×Dic3—98no—
311C2×Dic3—98no—
312C2×Dic3—92no—
313C2×Dic3—92no—
314C2×Dic3—188no—
315C2×Dic3—188no—
316C2×Dic3—1816no—
317C2×Dic3—184no—
318C2×Dic3C22×C3⋊D312yes—
319C2×Dic3C22×C3⋊D311yescharacteristic
320D12—68no—
321C4×D3—1816no—
322C4×D3—184no—
323C4×D3#63814yescomplemented
324Dic6—22no—
325Dic6—188no—
326Dic6—182no—
327Dic6—368no—
328C2×C32—28no—
329C2×C32—24no—
330C2×C32—42no—
331C2×C32—48no—
332C2×C32—62no—
333C2×C32—88no—
334C2×C32C2×D3×D411yescharacteristic
335C2×C32C2×#23712yes—
336C2×C32D3×#4118yes—
337C2×C32C22×#6714yes—
338C3⋊D3—42no—
339C3×D3—28no—
340C3×D3—128no—
341#41—64no—
342#41—544no—
343C2×Q8—31no—
344C2×Q8—271no—
345C2×D4—32no—
346C2×D4—94no—
347C2×D4—91no—
348C2×D4—272no—
349C2×D4D3×C3⋊D311yescharacteristic
350C22×C4—31no—
351C22×C4—32no—
352C22×C4—272no—
353C22×C4—272no—
354C4⋊C4—274no—
355C22⋊C4—274no—
356C22⋊C4—274no—
357C42—91no—
358C42—272no—
359C42—271no—
360C22×C3—216no—
361C22×C3—28no—
362C22×C3—24no—
363C22×C3—22no—
364C22×C3—24no—
365C22×C3—38no—
366C22×C3—48no—
367C22×C3—416no—
368C22×C3—68no—
369C22×C3—184no—
370C22×C3—182no—
371C22×C3C22×D3214yes—
372C22×C3C23×C3⋊D311yescharacteristic
373C2×D3—22no—
374C2×D3—616no—
375C2×D3—68no—
376C2×D3C2×#63812yes—
377C3×C4—28no—
378C3×C4—38no—
379C3×C4—68no—
380C3×C4—94no—
381C3×C4D3×#6718yes—
382C3×C4C2×#63812yes—
383Dic3—22no—
384Dic3—316no—
385Dic3—188no—
386Dic3—188no—
387Dic3—182no—
388Dic3—182no—
389Dic3—368no—
390Dic3—368no—
391Dic3C2×#63812yes—
392C32—24no—
393C32#571611yescharacteristic, complemented
394C32#583214yescomplemented
395C23—31no—
396C23—92no—
397C23C2×D3×C3⋊D312yes—
398Q8—62no—
399Q8—542no—
400D4—22no—
401D4—64no—
402D4—188no—
403D4—182no—
404D4—544no—
405C2×C4—31no—
406C2×C4—34no—
407C2×C4—32no—
408C2×C4—64no—
409C2×C4—92no—
410C2×C4—272no—
411C2×C4—272no—
412C2×C4—544no—
413C2×C4—544no—
414C2×C4C2×D3×C3⋊D311yescharacteristic
415C2×C3—28no—
416C2×C3—24no—
417C2×C3—48no—
418C2×C3—42no—
419C2×C3—68no—
420C2×C3—88no—
421C2×C3—362no—
422C2×C3D3×#23718yes—
423C2×C3C2×D3×#6714yes—
424C2×C3#41×C3⋊D312yes—
425C2×C3C22×#63811yescharacteristic
426D3—22no—
427D3—128no—
428C22—24no—
429C22—22no—
430C22—32no—
431C22—62no—
432C22—184no—
433C22—182no—
434C22C22×D3×C3⋊D311yescharacteristic
435C4—32no—
436C4—62no—
437C4—94no—
438C4—542no—
439C4—542no—
440C4D3×#63812yes—
441C3—24no—
442C3#1059515214yescomplemented
443C3#1059630311yescharacteristic, complemented
444C2—42no—
445C2—62no—
446C2—362no—
447C2C3⋊D3×#23712yes—
448C2C2×D3×#63811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#532
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C32×#237C2semidirect
2. C2×#89021C2semidirect
3. #11723977C2semidirect
4. C3×#64149C2semidirect
5. Dic3×#638C2semidirect
6. #11723615C2semidirect
7. C4×D3×C3⋊Dic3C2semidirect
8. #11723455C2semidirect
9. #11723436C2semidirect
33. C2×C3×#237D3semidirect
37. #64149D3semidirect
65. C32×C4×D3D4semidirect
124. C3×C3⋊Dic3#41semidirect
129. C2×#237C3⋊D3semidirect
187. C3×C4×D3#67semidirect
268. C3⋊Dic3#237semidirect
281. C33#532semidirect
323. C4×D3#638semidirect
393. C32#5716semidirect
394. C32#5832semidirect
442. C3#10595152semidirect
443. C3#10596303semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2372
C2×#890212
#117239772
C3×#641492
Dic3×#6382
#117236152
C4×D3×C3⋊Dic32
#117234552
#117234362
#105963033
#105951523