Finite Group Databasebeta

← #13177414

Subgroups — #13177414

Order 1728.

1924 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#237C211yesmaximal, characteristic
2C2×D3×#634C211yesmaximal, characteristic, complemented
3#11723977C211yesmaximal, characteristic, complemented
4C3×#64145C211yesmaximal, characteristic, complemented
5#11723640C214yesmaximal, complemented
6#11723614C214yesmaximal, complemented
7#11723455C211yesmaximal, characteristic, complemented
8#11723451C211yesmaximal, characteristic, complemented
9#11723423C211yesmaximal, characteristic, complemented
10#10596296—31nomaximal
11#10595147—34nomaximal
12C2×C33×D4C2211yescharacteristic
13C2×#6308C2211yescharacteristic
14C2×C3×#634C2211yescharacteristic
15C2×C32×#67C2212yes—
16C22×C32×Dic3C2212yes—
17C32×#237—24no—
18C2×C32×C4×D3C2211yescharacteristic
19C2×C32×Dic6C2211yescharacteristic
20C2×#6293C2212yes—
21C2×D3×C3⋊Dic3C2212yes—
22D3×#634—24no—
23#88864C2214yes—
24C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
25C3×#3757C2214yes—
26C3×C4×C3⋊Dic3C2211yescharacteristic
27#88807C2214yes—
28#88806C2214yes—
29Dic3⋊(C3⋊Dic3)C2212yes—
30(C2×D3)⋊(C3⋊Dic3)C2212yes—
31C2×C32×#41—31no—
32C22×#634—31no—
33C2×C3×#237D314yes—
34C2×D3×Dic6—34no—
35#64153—31no—
36#64153—31no—
37#64145—31no—
38#64145—34no—
39#64145—64no—
40#64145D311yescharacteristic, complemented
41#64144—34no—
42#64135—31no—
43#64133—34no—
44C3×#936—34no—
45#63969—316no—
46#63964—316no—
47#63873—34no—
48#63866—34no—
49#63855—34no—
50C23×C33C2312yes—
51C22×C32×D3C2311yescharacteristic
52C33×D4—22no—
53C2×C33×C4C2311yescharacteristic
54C2×C3⋊(C3⋊Dic3)C2312yes—
55C2×C3⋊(C3⋊Dic3)C2312yes—
56#6308—22no—
57C2×C3×C3⋊Dic3C2312yes—
58C2×C3×C3⋊Dic3C2312yes—
59C3×#634—22no—
60C32×#67—24no—
61C2×C32×Dic3—24no—
62C2×C32×Dic3C2312yes—
63C2×C32×Dic3C2311yescharacteristic
64C32×C4×D3D414yes—
65C32×Dic6—22no—
66#6293—24no—
67D3×C3⋊Dic3—24no—
68#5825—94no—
69#5675—91no—
70C32×#41—64no—
71C2×C32×Q8—31no—
72C2×C32×D4—24no—
73C2×C32×D4—32no—
74C2×C32×D4C2×D314yes—
75C2×C32×D4C2×D311yescharacteristic
76C22×C32×C4—31no—
77C22×C32×C4—32no—
78C22×C3⋊Dic3—32no—
79C2×#634—31no—
80C2×#634—34no—
81C2×#634—32no—
82C2×#634—64no—
83C2×#634—64no—
84C2×#634C2×D311yescharacteristic
85C2×C3×#67C2×D318yes—
86C22×C3×Dic3C2×D318yes—
87C3×#237—216no—
88C2×C3×C4×D3C2×D314yes—
89C2×C3×Dic6—34no—
90C2×C3×Dic6C2×D314yes—
91C2×#627—38no—
92C2×D3×Dic3—38no—
93D3×Dic6—616no—
94#3757—34no—
95#3757—316no—
96#3757—32no—
97#3757—616no—
98#3757C2×D314yes—
99C4⋊(C3⋊Dic3)—32no—
100#3750—34no—
101#3750—34no—
102C4×C3⋊Dic3—31no—
103C4×C3⋊Dic3—34no—
104C4×C3⋊Dic3—64no—
105C4×C3⋊Dic3C2×D311yescharacteristic
106C3×#217—316no—
107C3×C4×Dic3—34no—
108#3724—316no—
109Dic3⋊Dic3—38no—
110Dic3⋊Dic3—316no—
111(C2×D3)⋊Dic3—38no—
112C22×C33—22no—
113C22×C33—24no—
114C22×C33C2411yescharacteristic
115C2×C32×D3—22no—
116C2×C32×D3C2×D412yes—
117C33×C4C2×D412yes—
118C3⋊(C3⋊Dic3)—22no—
119C3⋊(C3⋊Dic3)—22no—
120C3×C3⋊Dic3—22no—
121C3×C3⋊Dic3—22no—
122C32×Dic3—22no—
123C32×Dic3C2×D412yes—
124C2×C3×#41—34no—
125C2×D3×Q8—91no—
126C2×#237C3⋊D311yescharacteristic
127C22×Dic6—94no—
128C3×#124—91no—
129#944—94no—
130#944—94no—
131#944—91no—
132#936—34no—
133#936—94no—
134#936—91no—
135#936—184no—
136#935—916no—
137#926—94no—
138#924—916no—
139#883—94no—
140#878—94no—
141#872—91no—
142#869—91no—
143C23×C32—28no—
144C23×C32—31no—
145C23×C32C22×D318yes—
146C23×C32C22×D312yes—
147C22×C3×D3C22×D314yes—
148C32×Q8—62no—
149C32×D4—22no—
150C32×D4—28no—
151C32×D4—48no—
152C32×D4—64no—
153C2×C32×C4—24no—
154C2×C32×C4—31no—
155C2×C32×C4—34no—
156C2×C32×C4—32no—
157C2×C32×C4—64no—
158C2×C32×C4C22×D314yes—
159C2×C32×C4C22×D311yescharacteristic
160C2×C3⋊Dic3—32no—
161C2×C3⋊Dic3—32no—
162C2×C3⋊Dic3—38no—
163C2×C3⋊Dic3—38no—
164C2×C3⋊Dic3—68no—
165C2×C3⋊Dic3—68no—
166C2×C3⋊Dic3—64no—
167C2×C3⋊Dic3C22×D312yes—
168C2×C3⋊Dic3C22×D312yes—
169#634—22no—
170#634—64no—
171#634—62no—
172#634—68no—
173#634—128no—
174C3×#67—216no—
175C2×C3×Dic3—216no—
176C2×C3×Dic3—38no—
177C2×C3×Dic3—38no—
178C2×C3×Dic3C22×D318yes—
179C2×C3×Dic3C22×D314yes—
180C3×C4×D3#67116yes—
181C3×Dic6—28no—
182C3×Dic6—68no—
183#627—616no—
184D3×Dic3—616no—
185#536—271noSylow
186C2×C33—42no—
187C2×C33C22×D411yescharacteristic
188C2×C332−1+412yes—
189C32×D3—22no—
190C3×#41—616no—
191C2×C3×Q8—34no—
192C2×C3×Q8—91no—
193C2×C3×D4—24no—
194C2×C3×D4—38no—
195C2×C3×D4D3214yes—
196C2×C3×D4C2×C3⋊D311yescharacteristic
197C22×C3×C4—34no—
198C22×C3×C4—38no—
199C2×#67C2×C3⋊D312yes—
200C22×Dic3—98no—
201C22×Dic3C2×C3⋊D312yes—
202D3×Q8—184no—
203#237—24no—
204C2×C4×D3—92no—
205C2×C4×D3C2×C3⋊D311yescharacteristic
206C2×Dic6—34no—
207C2×Dic6—98no—
208C2×Dic6—94no—
209C2×Dic6—91no—
210C2×Dic6—92no—
211C2×Dic6—184no—
212C2×Dic6—1816no—
213C2×Dic6C2×C3⋊D311yescharacteristic
214C3×C22⋊C4—94no—
215C3×C42—91no—
216#217—316no—
217#217—98no—
218#217—916no—
219#217—94no—
220#217—1816no—
221(C2×D3)⋊C4—92no—
222C4⋊Dic3—98no—
223C4⋊Dic3—94no—
224Dic3⋊C4—916no—
225Dic3⋊C4—916no—
226Dic3⋊C4—92no—
227Dic3⋊C4—94no—
228C4×Dic3—34no—
229C4×Dic3—94no—
230C4×Dic3—91no—
231C4×Dic3—184no—
232C22×C32—22no—
233C22×C32—24no—
234C22×C32—24no—
235C22×C32—216no—
236C22×C32—28no—
237C22×C32—32no—
238C22×C32—48no—
239C22×C32—416no—
240C22×C32—62no—
241C22×C32C23×D314yes—
242C22×C32C23×D311yescharacteristic
243C2×C3×D3—28no—
244C2×C3×D3C2×#6718yes—
245C32×C4—28no—
246C32×C4—32no—
247C32×C4—62no—
248C32×C4D3×D412yes—
249C32×C4C2×#6718yes—
250C3⋊Dic3—22no—
251C3⋊Dic3—22no—
252C3⋊Dic3—62no—
253C3⋊Dic3—62no—
254C3⋊Dic3—68no—
255C3⋊Dic3—68no—
256C3⋊Dic3—128no—
257C3⋊Dic3—128no—
258C3×Dic3—28no—
259C3×Dic3—68no—
260C3×Dic3—68no—
261C3×Dic3C2×#6718yes—
262C2×#41—31no—
263C22×Q8—271no—
264C4⋊Q8—271no—
265C4⋊Q8—271no—
266#124—91no—
267#124—271no—
268#123—274no—
269C22⋊Q8—274no—
270#117—271no—
271C33#53611yescharacteristic, complemented, Sylow
272C23×C3—28no—
273C23×C3—34no—
274C23×C3C2×D3218yes—
275C23×C3C22×C3⋊D312yes—
276C22×D3C22×C3⋊D311yescharacteristic
277C3×Q8—68no—
278C3×Q8—182no—
279C3×D4—28no—
280C3×D4—22no—
281C3×D4—48no—
282C3×D4—616no—
283C2×C3×C4—24no—
284C2×C3×C4—34no—
285C2×C3×C4—316no—
286C2×C3×C4—38no—
287C2×C3×C4—616no—
288C2×C3×C4—92no—
289C2×C3×C4—92no—
290C2×C3×C4C2×D3214yes—
291C2×C3×C4C22×C3⋊D311yescharacteristic
292#67—24no—
293C2×Dic3—24no—
294C2×Dic3—38no—
295C2×Dic3—38no—
296C2×Dic3—98no—
297C2×Dic3—98no—
298C2×Dic3—92no—
299C2×Dic3—92no—
300C2×Dic3—188no—
301C2×Dic3—188no—
302C2×Dic3—1816no—
303C2×Dic3C22×C3⋊D312yes—
304C2×Dic3C22×C3⋊D311yescharacteristic
305C4×D3—184no—
306C4×D3#63814yes—
307Dic6—22no—
308Dic6—68no—
309Dic6—1816no—
310Dic6—188no—
311Dic6—182no—
312Dic6—184no—
313Dic6—368no—
314C2×C32—28no—
315C2×C32—24no—
316C2×C32—42no—
317C2×C32—48no—
318C2×C32—62no—
319C2×C32—88no—
320C2×C32C2×D3×D411yescharacteristic
321C2×C32#101018yes—
322C2×C32#101012yes—
323C2×C32C22×#6714yes—
324C3×D3—28no—
325#41—64no—
326C2×Q8—31no—
327C2×Q8—91no—
328C2×Q8—272no—
329C2×Q8—271no—
330C2×Q8—544no—
331C2×D4—32no—
332C2×D4D3×C3⋊D311yescharacteristic
333C22×C4—31no—
334C22×C4—32no—
335C22×C4—272no—
336C4⋊C4—272no—
337C4⋊C4—274no—
338C4⋊C4—274no—
339C22⋊C4—94no—
340C22⋊C4—272no—
341C22⋊C4—274no—
342C42—91no—
343C42—271no—
344C22×C3—216no—
345C22×C3—28no—
346C22×C3—24no—
347C22×C3—22no—
348C22×C3—24no—
349C22×C3—38no—
350C22×C3—48no—
351C22×C3—416no—
352C22×C3—68no—
353C22×C3C22×D3214yes—
354C22×C3C23×C3⋊D311yescharacteristic
355C2×D3—22no—
356C2×D3C2×#63812yes—
357C3×C4—28no—
358C3×C4—38no—
359C3×C4—68no—
360C3×C4—182no—
361C3×C4—182no—
362C3×C4D3×#6718yes—
363C3×C4C2×#63812yes—
364Dic3—22no—
365Dic3—68no—
366Dic3—68no—
367Dic3—188no—
368Dic3—188no—
369Dic3—182no—
370Dic3—182no—
371Dic3—368no—
372Dic3—368no—
373Dic3C2×#63812yes—
374C32—24no—
375C32#567511yescharacteristic, complemented
376C32#582514yescomplemented
377C23—31no—
378C23C2×D3×C3⋊D312yes—
379Q8—62no—
380Q8—182no—
381Q8—544no—
382Q8—542no—
383D4—22no—
384D4—64no—
385C2×C4—31no—
386C2×C4—34no—
387C2×C4—32no—
388C2×C4—64no—
389C2×C4—92no—
390C2×C4—92no—
391C2×C4—272no—
392C2×C4—272no—
393C2×C4—544no—
394C2×C4C2×D3×C3⋊D311yescharacteristic
395C2×C3—28no—
396C2×C3—24no—
397C2×C3—48no—
398C2×C3—42no—
399C2×C3—68no—
400C2×C3—88no—
401C2×C3#6430918yes—
402C2×C3C2×D3×#6714yes—
403C2×C3#6436712yes—
404C2×C3C22×#63811yescharacteristic
405D3—22no—
406C22—24no—
407C22—22no—
408C22—32no—
409C22—62no—
410C22C22×D3×C3⋊D311yescharacteristic
411C4—32no—
412C4—62no—
413C4—182no—
414C4—182no—
415C4—542no—
416C4—542no—
417C4D3×#63812yes—
418C3—24no—
419C3#1059514714yescomplemented
420C3#1059629611yescharacteristic, complemented
421C2—42no—
422C2—62no—
423C2#1172458212yes—
424C2C2×D3×#63811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
prime radicalC2×C32×#237C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#536
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×D3×#634C2semidirect
3. #11723977C2semidirect
4. C3×#64145C2semidirect
5. #11723640C2semidirect
6. #11723614C2semidirect
7. #11723455C2semidirect
8. #11723451C2semidirect
9. #11723423C2semidirect
40. #64145D3semidirect
271. C33#536semidirect
375. C32#5675semidirect
376. C32#5825semidirect
419. C3#10595147semidirect
420. C3#10596296semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2372
C2×D3×#6342
#117239772
C3×#641452
#117236402
#117236142
#117234552
#117234512
#117234232
#105962963
#105951473