Finite Group Databasebeta

← #13178713

Subgroups — #13178713

Order 1728.

1490 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q8×C3⋊D3C211yesmaximal, characteristic
2C2×#89040C211yesmaximal, characteristic
3C3×#64153C212yesmaximal, complemented
4#11723682C214yesmaximal
5C4×#6294C211yesmaximal, characteristic
6#11723678C212yesmaximal, complemented
7#11723666C214yesmaximal, complemented
8#10595365—31nomaximal
9D3×#944—32nomaximal
10C2×C33×Q8C2211yescharacteristic
11C3×Q8×C3⋊D3—24no—
12C2×C3×C4×C3⋊D3C2212yes—
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C3×#634C2212yes—
15C2×C3×#634C2212yes—
16C2×C3×#634C2211yescharacteristic
17C2×#6296C2212yes—
18C2×#6294C2212yes—
19C2×#6294C2212yes—
20#89040—24no—
21C3×#3750C2218yes—
22C3×C4×C3⋊Dic3C2212yes—
23#88815C2218yes—
24#88813C2212yes—
25C2×Q8×C3⋊D3D311yescharacteristic
26C2×C3×D3×Q8—32no—
27C2×C3×D3×Q8—61no—
28C2×#3797—31no—
29C2×D3×Dic6—32no—
30#64153D312yescomplemented
31C3×#944—32no—
32C3×#944—32no—
33C3×#944—62no—
34#63883—34no—
35C4×#624—31no—
36D3×Dic3⋊C4—38no—
37C4×D3×Dic3—32no—
38#63873—32no—
39#63866—32no—
40#63866—32no—
41#63840—34no—
42#63839—38no—
43C22×C3×C3⋊D3C2311yescharacteristic
44C33×Q8—22no—
45C2×C33×C4C2312yes—
46C2×C33×C4C2311yescharacteristic
47C2×C3×C3⋊Dic3C2314yes—
48C2×C3×C3⋊Dic3C2312yes—
49C2×C3×C3⋊Dic3C2311yescharacteristic
50C2×C3×C3⋊Dic3C2314yes—
51C3×C4×C3⋊D3—24no—
52C3×C4×C3⋊D3D414yes—
53C3×#634—22no—
54C3×#634—24no—
55C3×#634—24no—
56#6296—24no—
57#6294—24no—
58#6294Q818yescomplemented
59C2×#944—91no—
60D3×C4⋊Q8—92no—
61C2×C32×Q8—21no—
62C2×C32×Q8—22no—
63C2×C32×Q8—41no—
64C2×C32×Q8C2×D311yescharacteristic
65C2×C32×Q8C2×D312yes—
66Q8×C3⋊D3—24no—
67C2×C4×C3⋊D3C2×D312yes—
68C2×C4×C3⋊D3C2×D311yescharacteristic
69C2×#634C2×D312yes—
70C2×#634C2×D312yes—
71C2×#634C2×D311yescharacteristic
72C3×D3×Q8—68no—
73C3×D3×Q8—68no—
74C2×C3×C4×D3—32no—
75C2×C3×C4×D3—34no—
76C2×C3×C4×D3—61no—
77C2×C3×C4×D3—62no—
78C2×C3×Dic6—32no—
79C2×C3×Dic6—32no—
80C2×C3×Dic6—34no—
81C2×C3×Dic6—32no—
82C2×C3×Dic6—62no—
83C2×C3×Dic6—61no—
84C2×C3×Dic6—62no—
85C2×#627—34no—
86C2×#627—32no—
87C2×#624—32no—
88C2×#624—32no—
89C2×D3×Dic3—34no—
90C2×D3×Dic3—34no—
91#3797—64no—
92D3×Dic6—68no—
93#3750C2×D318yes—
94C4×C3⋊Dic3C2×D312yes—
95C3×Dic3⋊C4—38no—
96C3×Dic3⋊C4—38no—
97C3×Dic3⋊C4—68no—
98C3×C4×Dic3—32no—
99C3×C4×Dic3—32no—
100C3×C4×Dic3—62no—
101#3724—38no—
102Dic3⋊Dic3—38no—
103Dic3⋊Dic3—38no—
104Dic32—34no—
105Dic32—32no—
106C22×C33C2411yescharacteristic
107C2×C3×C3⋊D3C2×D412yes—
108C2×C3×C3⋊D3C2×Q814yes—
109C33×C4—22no—
110C33×C4C2×D412yes—
111C3×C3⋊Dic3—22no—
112C3×C3⋊Dic3—24no—
113C3×C3⋊Dic3C2×D412yes—
114C3×C3⋊Dic3C2×Q818yescomplemented
115C22×C3×Q8—91no—
116C2×D3×Q8—32no—
117C2×D3×Q8—61no—
118C2×D3×Q8—92no—
119C22×Dic6—91no—
120C3×C4⋊Q8—92no—
121#944—32no—
122#944—32no—
123#944—62no—
124#944—92no—
125#944—92no—
126#944—94no—
127C2×Dic3⋊C4—94no—
128C2×C4×Dic3—91no—
129D3×C4⋊C4—98no—
130#888—98no—
131C42×D3—92no—
132#869—92no—
133C22×C3⋊D3C22×D311yescharacteristic
134C22×C3×D3—32no—
135C22×C3×D3—61no—
136C32×Q8—22no—
137C32×Q8—24no—
138C32×Q8—24no—
139C32×Q8—44no—
140C32×Q8—44no—
141C2×C32×C4—22no—
142C2×C32×C4—22no—
143C2×C32×C4—21no—
144C2×C32×C4—24no—
145C2×C32×C4—41no—
146C2×C32×C4—42no—
147C2×C32×C4C22×D312yes—
148C2×C32×C4C22×D314yes—
149C2×C32×C4C22×D312yes—
150C2×C32×C4C22×D311yescharacteristic
151C2×C3⋊Dic3C22×D312yes—
152C2×C3⋊Dic3C22×D314yes—
153C2×C3⋊Dic3C22×D314yes—
154C2×C3⋊Dic3C22×D311yescharacteristic
155C4×C3⋊D3—24no—
156C4×C3⋊D3#6714yes—
157#634—22no—
158#634—24no—
159#634—24no—
160C2×C3×Dic3—32no—
161C2×C3×Dic3—34no—
162C2×C3×Dic3—34no—
163C2×C3×Dic3—34no—
164C2×C3×Dic3—34no—
165C2×C3×Dic3—34no—
166C2×C3×Dic3—64no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3—62no—
169C2×C3×Dic3—64no—
170C3×C4×D3—38no—
171C3×C4×D3—64no—
172C3×C4×D3—68no—
173C3×C4×D3—68no—
174C3×Dic6—68no—
175C3×Dic6—68no—
176C3×Dic6—64no—
177C3×Dic6—64no—
178C3×Dic6—64no—
179C3×Dic6—64no—
180C3×Dic6—124no—
181#627—68no—
182#627—64no—
183#624—38no—
184#624—64no—
185D3×Dic3—316no—
186D3×Dic3—68no—
187C2×C4⋊Q8—271noSylow
188C2×C33C22×D411yescharacteristic
189C2×C33C22×Q812yes—
190C3×C3⋊D3C4⋊Q814yescomplemented
191C2×C3×Q8—21no—
192C2×C3×Q8—22no—
193C2×C3×Q8—41no—
194C2×C3×Q8—92no—
195C2×C3×Q8—91no—
196C2×C3×Q8—92no—
197C2×C3×Q8—184no—
198C2×C3×Q8D3211yescharacteristic
199C2×C3×Q8D3212yes—
200C22×C3×C4—91no—
201C22×C3×C4—92no—
202C22×Dic3—92no—
203C22×Dic3—92no—
204D3×Q8—68no—
205D3×Q8—68no—
206D3×Q8—188no—
207C2×C4×D3—32no—
208C2×C4×D3—34no—
209C2×C4×D3—61no—
210C2×C4×D3—62no—
211C2×C4×D3—94no—
212C2×C4×D3—94no—
213C2×Dic6—32no—
214C2×Dic6—32no—
215C2×Dic6—34no—
216C2×Dic6—32no—
217C2×Dic6—62no—
218C2×Dic6—61no—
219C2×Dic6—62no—
220C2×Dic6—94no—
221C2×Dic6—92no—
222C2×Dic6—184no—
223C3×C4⋊C4—98no—
224C3×C42—92no—
225C4⋊Dic3—98no—
226Dic3⋊C4—38no—
227Dic3⋊C4—38no—
228Dic3⋊C4—68no—
229Dic3⋊C4—98no—
230Dic3⋊C4—98no—
231C4×Dic3—32no—
232C4×Dic3—32no—
233C4×Dic3—62no—
234C4×Dic3—94no—
235C4×Dic3—92no—
236C22×C32—22no—
237C22×C32—21no—
238C22×C32—41no—
239C22×C32C23×D312yes—
240C22×C32C23×D311yescharacteristic
241C2×C3⋊D3D3×Q814yes—
242C2×C3⋊D3C2×#6712yes—
243C2×C3×D3—34no—
244C2×C3×D3—38no—
245C2×C3×D3—62no—
246C2×C3×D3—64no—
247C32×C4—22no—
248C32×C4—24no—
249C32×C4—24no—
250C32×C4—24no—
251C32×C4—22no—
252C32×C4—42no—
253C32×C4—44no—
254C32×C4—44no—
255C32×C4D3×D414yes—
256C32×C4C2×#6712yes—
257C3⋊Dic3—22no—
258C3⋊Dic3—24no—
259C3⋊Dic3D3×Q818yescomplemented
260C3⋊Dic3C2×#6712yes—
261C3×Dic3—34no—
262C3×Dic3—38no—
263C3×Dic3—38no—
264C3×Dic3—62no—
265C3×Dic3—68no—
266C3×Dic3—64no—
267C3×Dic3—64no—
268C3×Dic3—64no—
269C3×Dic3—64no—
270C3×Dic3—124no—
271C22×Q8—91no—
272C22×Q8—271no—
273C4⋊Q8—92no—
274C4⋊Q8—272no—
275C4⋊Q8—274no—
276C2×C4⋊C4—274no—
277C2×C42—271no—
278C33C2×C4⋊Q811yescharacteristic, complemented, Sylow
279C23×C3—91no—
280C22×D3—32no—
281C22×D3—61no—
282C3×Q8—24no—
283C3×Q8—24no—
284C3×Q8—22no—
285C3×Q8—44no—
286C3×Q8—44no—
287C3×Q8—184no—
288C3×Q8—184no—
289C3×Q8—182no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—21no—
293C2×C3×C4—24no—
294C2×C3×C4—42no—
295C2×C3×C4—41no—
296C2×C3×C4—91no—
297C2×C3×C4—94no—
298C2×C3×C4—94no—
299C2×C3×C4—92no—
300C2×C3×C4—94no—
301C2×C3×C4—184no—
302C2×C3×C4C2×D3212yes—
303C2×C3×C4C2×D3214yes—
304C2×C3×C4C2×D3212yes—
305C2×C3×C4C2×D3211yescharacteristic
306C2×Dic3—34no—
307C2×Dic3—34no—
308C2×Dic3—32no—
309C2×Dic3—34no—
310C2×Dic3—34no—
311C2×Dic3—34no—
312C2×Dic3—64no—
313C2×Dic3—61no—
314C2×Dic3—62no—
315C2×Dic3—64no—
316C2×Dic3—98no—
317C2×Dic3—184no—
318C4×D3—38no—
319C4×D3—64no—
320C4×D3—68no—
321C4×D3—68no—
322C4×D3—916no—
323C4×D3—188no—
324Dic6—68no—
325Dic6—68no—
326Dic6—64no—
327Dic6—64no—
328Dic6—64no—
329Dic6—64no—
330Dic6—124no—
331Dic6—188no—
332Dic6—184no—
333C2×C32—24no—
334C2×C32—22no—
335C2×C32—22no—
336C2×C32—21no—
337C2×C32—41no—
338C2×C32—42no—
339C2×C32C2×D3×D412yes—
340C2×C32C2×D3×Q814yes—
341C2×C32C2×D3×Q812yes—
342C2×C32C22×#6711yescharacteristic
343C3⋊D3#94414yescomplemented
344C3×D3—38no—
345C3×D3—64no—
346C2×Q8—92no—
347C2×Q8—92no—
348C2×Q8—91no—
349C2×Q8—184no—
350C2×Q8—272no—
351C2×Q8—544no—
352C2×Q8#111311yescharacteristic
353C22×C4—91no—
354C22×C4—92no—
355C22×C4—272no—
356C22×C4—272no—
357C4⋊C4—98no—
358C4⋊C4—278no—
359C42—92no—
360C42—272no—
361C22×C3—22no—
362C22×C3—21no—
363C22×C3—41no—
364C22×C3—94no—
365C22×C3—92no—
366C22×C3C22×D3212yes—
367C22×C3C22×D3211yescharacteristic
368C2×D3—38no—
369C2×D3—34no—
370C2×D3—62no—
371C2×D3—64no—
372C3×C4—24no—
373C3×C4—24no—
374C3×C4—22no—
375C3×C4—22no—
376C3×C4—24no—
377C3×C4—44no—
378C3×C4—44no—
379C3×C4—42no—
380C3×C4—98no—
381C3×C4—92no—
382C3×C4—184no—
383C3×C4—182no—
384C3×C4D3×#6714yes—
385C3×C4#381112yes—
386Dic3—38no—
387Dic3—38no—
388Dic3—34no—
389Dic3—62no—
390Dic3—68no—
391Dic3—64no—
392Dic3—64no—
393Dic3—64no—
394Dic3—64no—
395Dic3—124no—
396C32—22no—
397C32—21no—
398C32—41no—
399C32D3×C4⋊Q812yescomplemented
400C32C2×#94411yescharacteristic, complemented
401C23—91no—
402Q8—22no—
403Q8—184no—
404Q8—182no—
405Q8—184no—
406Q8—544no—
407C2×C4—94no—
408C2×C4—91no—
409C2×C4—94no—
410C2×C4—94no—
411C2×C4—92no—
412C2×C4—184no—
413C2×C4—278no—
414C2×C4—544no—
415C2×C4C2×#111312yes—
416C2×C4C2×#111311yescharacteristic
417C2×C3—24no—
418C2×C3—22no—
419C2×C3—21no—
420C2×C3—22no—
421C2×C3—42no—
422C2×C3—41no—
423C2×C3—94no—
424C2×C3D32×Q814yes—
425C2×C3D32×Q812yes—
426C2×C3C2×D3×#6712yes—
427C2×C3C2×#381111yescharacteristic
428D3—38no—
429D3—64no—
430C22—94no—
431C22—92no—
432C22C22×#111311yescharacteristic
433C4—22no—
434C4—98no—
435C4—92no—
436C4—182no—
437C4—184no—
438C4#8904912yes—
439C3—21no—
440C3—22no—
441C3—41no—
442C3D3×#94412yescomplemented
443C3#1059536511yescharacteristic, complemented
444C2—94no—
445C2Q8×#111312yes—
446C2C2×#8904911yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×Q8C22
prime radicalC22×C3×C3⋊D3C23
hypercenterC2×Q8#1113
hypocenterC33C2×C4⋊Q8
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
3. C3×#64153C2semidirect
6. #11723678C2semidirect
7. #11723666C2semidirect
30. #64153D3semidirect
58. #6294Q8semidirect
114. C3×C3⋊Dic3C2×Q8semidirect
190. C3×C3⋊D3C4⋊Q8semidirect
259. C3⋊Dic3D3×Q8semidirect
278. C33C2×C4⋊Q8semidirect
343. C3⋊D3#944semidirect
399. C32D3×C4⋊Q8semidirect
400. C32C2×#944semidirect
442. C3D3×#944semidirect
443. C3#10595365semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×C3⋊D32
C2×#890402
C3×#641532
#117236822
C4×#62942
#117236782
#117236662
#105953653
D3×#9443