Finite Group Databasebeta

← #13175167

Subgroups — #13175167

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×#89018C211yesmaximal, characteristic
2C2×C3⋊D3×Dic6C211yesmaximal, characteristic, complemented
3#11723916C211yesmaximal, characteristic, complemented
4C3×#64081C211yesmaximal, characteristic, complemented
5C32×#869C211yesmaximal, characteristic, complemented
6#11723495C214yesmaximal, complemented
7#11723493C214yesmaximal, complemented
8#11723455C211yesmaximal, characteristic, complemented
9#11723440C211yesmaximal, characteristic, complemented
10#10596195—31nomaximal
11#10593893—34nomaximal
12C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
13C2×#6308C2211yescharacteristic
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C32×Dic6C2211yescharacteristic
16C2×C32×Dic6C2211yescharacteristic
17C2×#6293C2212yes—
18C2×#6292C2212yes—
19C2×#6289C2212yes—
20C2×Dic3×C3⋊D3C2212yes—
21#89018—24no—
22C3⋊D3×Dic6—24no—
23C33×C42C2211yescharacteristic
24#88859C2214yes—
25C3×#3752C2212yes—
26C3×#3750C2212yes—
27C3×C4×C3⋊Dic3C2211yescharacteristic
28C32×C4⋊Dic3C2214yes—
29#88807C2214yes—
30#88804C2214yes—
31C2×#3832—31no—
32C2×Q8×C3⋊D3—31no—
33C2×#3799—34no—
34C2×D3×Dic6—34no—
35C32×C4⋊Q8—31no—
36#64156—31no—
37#64153—31no—
38#64105—34no—
39#64103—34no—
40#64084—31no—
41#64084—34no—
42#64084—64no—
43#64081D311yescharacteristic, complemented
44C3×#872—34no—
45C3×#869D314yescomplemented
46#63895—316no—
47#63890—316no—
48#63866—34no—
49#63864—34no—
50C22×C3⋊(C3⋊D3)C2312yes—
51C22×C3×C3⋊D3C2311yescharacteristic
52C2×C33×C4C2312yes—
53C2×C33×C4C2311yescharacteristic
54C2×C3⋊(C3⋊Dic3)C2312yes—
55C4⋊(C3⋊(C3⋊D3))—22no—
56#6308—22no—
57C2×C3×C3⋊Dic3C2312yes—
58C2×C3×C3⋊Dic3C2311yescharacteristic
59C3×C4×C3⋊D3D414yes—
60C2×C32×Dic3C2312yes—
61C2×C32×Dic3C2312yes—
62C32×Dic6—22no—
63C32×Dic6—22no—
64#6293—24no—
65#6292—24no—
66#6289—24no—
67Dic3×C3⋊D3—24no—
68#5724—94no—
69#5526—91no—
70C2×C32×Q8—31no—
71C2×C32×Q8—31no—
72#3832—64no—
73Q8×C3⋊D3—64no—
74C2×C4⋊(C3⋊D3)—32no—
75C2×C4⋊(C3⋊D3)—31no—
76C2×C4⋊(C3⋊D3)—34no—
77C2×C4⋊(C3⋊D3)—64no—
78C2×C4×C3⋊D3—32no—
79C2×C4×C3⋊D3—32no—
80C2×C4×C3⋊D3C2×D311yescharacteristic
81C2×#634—31no—
82C2×#634—34no—
83C2×#634—32no—
84C2×#634—64no—
85C2×C3×C4×D3—34no—
86C2×C3×Dic6C2×D314yes—
87C2×C3×Dic6C2×D314yes—
88C2×#627—38no—
89C2×#626—38no—
90C2×#624—38no—
91C2×D3×Dic3—38no—
92#3799—616no—
93D3×Dic6—616no—
94C32×C4⋊C4—34no—
95C32×C42—24no—
96C32×C42C2×D314yes—
97C32×C42C2×D311yescharacteristic
98#3752—34no—
99#3752—316no—
100#3752—34no—
101#3752—616no—
102#3752C2×D312yes—
103#3750—34no—
104#3750C2×D312yes—
105C4×C3⋊Dic3C2×D311yescharacteristic
106C3×(C2×D3)⋊C4—38no—
107C3×C4⋊Dic3C2×D3116yes—
108C3×Dic3⋊C4—38no—
109C3×C4×Dic3—34no—
110Dic3⋊Dic3—316no—
111#3722—316no—
112C22×C33C2411yescharacteristic
113C2×C3⋊(C3⋊D3)—22no—
114C2×C3⋊(C3⋊D3)—24no—
115C2×C3×C3⋊D3—22no—
116C2×C3×C3⋊D3C2×D412yes—
117C33×C4—22no—
118C33×C4C2×D412yes—
119C3⋊(C3⋊Dic3)—22no—
120C3×C3⋊Dic3—22no—
121C3×C3⋊Dic3C2×D412yes—
122C32×Dic3—22no—
123C32×Dic3—22no—
124C2×#239—94no—
125C2×D3×Q8—94no—
126C2×#235—91no—
127C22×Dic6—91no—
128C3×C4⋊Q8—34no—
129C3×#117—91no—
130#947—94no—
131#944—94no—
132#897—94no—
133#896—916no—
134#894—916no—
135#886—94no—
136#875—91no—
137#875—94no—
138#875—91no—
139#875—184no—
140#872—34no—
141#869—91no—
142#869C3⋊D311yescharacteristic, complemented
143C22×C3⋊D3—32no—
144C22×C3⋊D3—38no—
145C22×C3⋊D3—68no—
146C22×C3⋊D3C22×D311yescharacteristic
147C22×C3×D3—34no—
148C32×Q8—62no—
149C32×Q8—62no—
150C2×C32×C4—24no—
151C2×C32×C4—28no—
152C2×C32×C4—32no—
153C2×C32×C4—32no—
154C2×C32×C4C22×D318yes—
155C2×C32×C4C22×D314yes—
156C2×C32×C4C22×D312yes—
157C2×C32×C4C22×D311yescharacteristic
158C2×C3⋊Dic3—32no—
159C2×C3⋊Dic3—38no—
160C2×C3⋊Dic3—68no—
161C2×C3⋊Dic3C22×D312yes—
162C2×C3⋊Dic3C22×D311yescharacteristic
163C4⋊(C3⋊D3)—64no—
164C4⋊(C3⋊D3)—62no—
165C4⋊(C3⋊D3)—68no—
166C4⋊(C3⋊D3)—128no—
167C4×C3⋊D3—64no—
168C4×C3⋊D3—64no—
169C4×C3⋊D3D1214yes—
170#634—62no—
171#634—64no—
172#634—68no—
173#634—128no—
174C2×C3×Dic3—34no—
175C2×C3×Dic3—38no—
176C2×C3×Dic3C22×D318yes—
177C2×C3×Dic3C22×D318yes—
178C3×C4×D3—316no—
179C3×Dic6—28no—
180C3×Dic6—28no—
181#627—616no—
182#626—616no—
183#624—616no—
184D3×Dic3—616no—
185#536—271noSylow
186C2×C33C22×D411yescharacteristic
187C2×C332−1+412yes—
188C3⋊(C3⋊D3)—42no—
189C3×C3⋊D3—22no—
190C2×C3×Q8—34no—
191C2×C3×Q8—34no—
192C22×C3×C4—91no—
193C2×#67—92no—
194C22×Dic3—92no—
195#239—1816no—
196D3×Q8—1816no—
197#235—184no—
198C2×D12—98no—
199C2×D12—94no—
200C2×D12—91no—
201C2×D12—184no—
202C2×C4×D3—34no—
203C2×C4×D3—98no—
204C2×C4×D3—98no—
205C2×C4×D3—92no—
206C2×Dic6—98no—
207C2×Dic6—94no—
208C2×Dic6—91no—
209C2×Dic6—92no—
210C2×Dic6—184no—
211C2×Dic6—184no—
212C2×Dic6C2×C3⋊D311yescharacteristic
213C2×Dic6C2×C3⋊D311yescharacteristic
214C3×C4⋊C4—316no—
215C3×C4⋊C4—92no—
216C3×C22⋊C4—92no—
217C3×C42—24no—
218C3×C42—91no—
219C3×C42D3214yes—
220C3×C42C2×C3⋊D311yescharacteristic
221(C2×D3)⋊C4—38no—
222(C2×D3)⋊C4—916no—
223(C2×D3)⋊C4—916no—
224(C2×D3)⋊C4—94no—
225(C2×D3)⋊C4—94no—
226(C2×D3)⋊C4—1816no—
227C4⋊Dic3—94no—
228C4⋊Dic3C2×C3⋊D314yes—
229Dic3⋊C4—38no—
230Dic3⋊C4—916no—
231C4×Dic3—34no—
232C22×C32—24no—
233C22×C32C23×D314yes—
234C22×C32C23×D311yescharacteristic
235C2×C3⋊D3—22no—
236C2×C3⋊D3—62no—
237C2×C3⋊D3—64no—
238C2×C3⋊D3—68no—
239C2×C3⋊D3—616no—
240C2×C3⋊D3—128no—
241C2×C3⋊D3—1216no—
242C2×C3⋊D3C2×D1212yes—
243C2×C3×D3—38no—
244C2×C3×D3—68no—
245C32×C4—22no—
246C32×C4—28no—
247C32×C4—28no—
248C32×C4—48no—
249C32×C4—62no—
250C32×C4—62no—
251C32×C4C2×D1212yes—
252C32×C4D3×D418yes—
253C3⋊Dic3—22no—
254C3⋊Dic3—62no—
255C3⋊Dic3—68no—
256C3⋊Dic3—128no—
257C3⋊Dic3C2×D1212yes—
258C3×Dic3—28no—
259C3×Dic3—28no—
260C3×Dic3—38no—
261C3×Dic3—68no—
262C2×#41—271no—
263C22×Q8—271no—
264C4⋊Q8—31no—
265C4⋊Q8—271no—
266#124—271no—
267#124—271no—
268#123—274no—
269C22⋊Q8—274no—
270#117—91no—
271C33#53611yescharacteristic, complemented, Sylow
272C23×C3—91no—
273C22×D3—34no—
274C22×D3—98no—
275C22×D3—92no—
276C22×D3—188no—
277C3×Q8—68no—
278C3×Q8—68no—
279C2×C3×C4—24no—
280C2×C3×C4—28no—
281C2×C3×C4—38no—
282C2×C3×C4—38no—
283C2×C3×C4—91no—
284C2×C3×C4—94no—
285C2×C3×C4—92no—
286C2×C3×C4C2×D3218yes—
287C2×C3×C4C2×D3214yes—
288C2×C3×C4C22×C3⋊D312yes—
289C2×C3×C4C22×C3⋊D311yescharacteristic
290#67—184no—
291C2×Dic3—34no—
292C2×Dic3—38no—
293C2×Dic3—98no—
294C2×Dic3—92no—
295C2×Dic3—188no—
296C2×Dic3—184no—
297C2×Dic3C22×C3⋊D312yes—
298C2×Dic3C22×C3⋊D312yes—
299D12—1816no—
300D12—188no—
301D12—182no—
302D12—368no—
303C4×D3—316no—
304C4×D3—184no—
305C4×D3—1816no—
306C4×D3—1816no—
307Dic6—22no—
308Dic6—22no—
309Dic6—184no—
310Dic6—1816no—
311Dic6—188no—
312Dic6—182no—
313Dic6—368no—
314C2×C32—28no—
315C2×C32—24no—
316C2×C32C22×D1211yescharacteristic
317C2×C32C2×D3×D414yes—
318C2×C32#100718yes—
319C2×C32#101012yes—
320C3⋊D3—22no—
321C3⋊D3—122no—
322C3⋊D3—128no—
323C3⋊D3—248no—
324C3×D3—68no—
325#41—544no—
326C2×Q8—31no—
327C2×Q8—31no—
328C2×Q8—272no—
329C2×Q8—271no—
330C2×Q8—544no—
331C2×D4—272no—
332C2×D4—271no—
333C22×C4—91no—
334C22×C4—272no—
335C22×C4—272no—
336C4⋊C4—34no—
337C4⋊C4—92no—
338C4⋊C4—274no—
339C22⋊C4—92no—
340C22⋊C4—274no—
341C22⋊C4—274no—
342C42—91no—
343C42D3×C3⋊D311yescharacteristic
344C22×C3—24no—
345C22×C3—92no—
346C22×C3—182no—
347C22×C3C22×D3214yes—
348C22×C3C23×C3⋊D311yescharacteristic
349C2×D3—38no—
350C2×D3—68no—
351C2×D3—1816no—
352C2×D3—188no—
353C2×D3—182no—
354C2×D3—184no—
355C2×D3—368no—
356C2×D3—3616no—
357C3×C4—28no—
358C3×C4—22no—
359C3×C4—28no—
360C3×C4—48no—
361C3×C4—68no—
362C3×C4—68no—
363C3×C4—92no—
364C3×C4—182no—
365C3×C4D3×D1218yes—
366C3×C4D4×C3⋊D312yes—
367Dic3—22no—
368Dic3—22no—
369Dic3—38no—
370Dic3—68no—
371Dic3—188no—
372Dic3—182no—
373Dic3—368no—
374C32—24no—
375C32#552611yescharacteristic, complemented
376C32#572414yescomplemented
377C23—91no—
378C23—272no—
379Q8—62no—
380Q8—62no—
381Q8—544no—
382Q8—542no—
383D4—544no—
384D4—542no—
385C2×C4—32no—
386C2×C4—32no—
387C2×C4—91no—
388C2×C4—94no—
389C2×C4—92no—
390C2×C4—272no—
391C2×C4—544no—
392C2×C4—544no—
393C2×C4C2×D3×C3⋊D312yes—
394C2×C4C2×D3×C3⋊D311yescharacteristic
395C2×C3—28no—
396C2×C3—24no—
397C2×C3—182no—
398C2×C3#6429718yes—
399C2×C3C2×D3×D1214yes—
400C2×C3C2×D4×C3⋊D311yescharacteristic
401C2×C3#6436412yes—
402D3—68no—
403D3—362no—
404D3—368no—
405D3—728no—
406C22—92no—
407C22—182no—
408C22—544no—
409C22—542no—
410C22C22×D3×C3⋊D311yescharacteristic
411C4—22no—
412C4—62no—
413C4—62no—
414C4—92no—
415C4—182no—
416C4—542no—
417C4C3⋊D3×D1212yes—
418C3—24no—
419C3#1059389314yescomplemented
420C3#1059619511yescharacteristic, complemented
421C2—182no—
422C2—1082no—
423C2#1172456612yes—
424C2C2×C3⋊D3×D1211yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
prime radicalC2×#89018C2
hypercenterC42D3×C3⋊D3
hypocenterC33#536
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊D3×Dic6C2semidirect
3. #11723916C2semidirect
4. C3×#64081C2semidirect
5. C32×#869C2semidirect
6. #11723495C2semidirect
7. #11723493C2semidirect
8. #11723455C2semidirect
9. #11723440C2semidirect
43. #64081D3semidirect
45. C3×#869D3semidirect
142. #869C3⋊D3semidirect
271. C33#536semidirect
375. C32#5526semidirect
376. C32#5724semidirect
419. C3#10593893semidirect
420. C3#10596195semidirect

Maximal subgroups

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

subgroupindex
C2×#890182
C2×C3⋊D3×Dic62
#117239162
C3×#640812
C32×#8692
#117234952
#117234932
#117234552
#117234402
#105961953
#105938933