Finite Group Databasebeta

← #13178727

Subgroups — #13178727

Order 1728.

1202 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 1202 conjugacy classes of subgroups — too many to draw. Only its 175 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64156C212yesmaximal, complemented
2C3×#64153C211yesmaximal, characteristic, complemented
3#11723687C214yesmaximal, complemented
4#11723685C212yesmaximal, complemented
5#11723683C214yesmaximal, complemented
6#11723670C212yesmaximal, complemented
7#10595390—31nomaximal
8#10595385—32nomaximal
9C2×C33×Q8C2211yescharacteristic
10C2×C3×C4⋊(C3⋊D3)C2212yes—
11C2×C3×#634C2211yescharacteristic
12C2×#6295C2212yes—
13C2×#6295C2212yes—
14C2×#6294C2214yes—
15C3×#3752C2214yes—
16C3×#3752C2214yes—
17C3×#3750C2214yes—
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C3×C4×C3⋊Dic3C2212yes—
20#88815C2212yes—
21#88814C2214yes—
22#88813C2212yes—
23#64156D312yescomplemented
24#64153D311yescharacteristic, complemented
25C3×#947—32no—
26C3×#947—32no—
27C3×#947—62no—
28C3×#944—32no—
29C3×#944—61no—
30#63913—32no—
31#63910—34no—
32#63907—34no—
33#63902—32no—
34#63886—34no—
35#63885—34no—
36#63880—34no—
37#63878—34no—
38#63877—34no—
39#63836—34no—
40C22×C3×C3⋊D3C2314yes—
41C33×Q8—41no—
42C2×C33×C4C2312yes—
43C2×C33×C4C2311yescharacteristic
44C2×C3×C3⋊Dic3C2314yes—
45C2×C3×C3⋊Dic3C2312yes—
46C2×C3×C3⋊Dic3C2312yes—
47C3×C4⋊(C3⋊D3)—42no—
48C3×#634—41no—
49#6295—24no—
50#6295—42no—
51#6294—28no—
52#5734—91no—
53#5670—92no—
54C2×C32×Q8—21no—
55C2×C32×Q8—22no—
56C2×C32×Q8—41no—
57C2×C32×Q8C2×D312yes—
58C2×C32×Q8C2×D311yescharacteristic
59C2×C4⋊(C3⋊D3)C2×D312yes—
60C2×#634C2×D311yescharacteristic
61C2×C3×D12—32no—
62C2×C3×D12—32no—
63C2×C3×D12—62no—
64C2×C3×Dic6—32no—
65C2×C3×Dic6—61no—
66C2×#626—34no—
67C2×#626—34no—
68C2×#625—32no—
69C2×#625—32no—
70C2×#624—34no—
71C2×D3×Dic3—34no—
72C2×D3×Dic3—34no—
73#3752C2×D314yes—
74#3752C2×D314yes—
75#3750C2×D314yes—
76C4×C3⋊Dic3C2×D312yes—
77C4×C3⋊Dic3C2×D311yescharacteristic
78C3×(C2×D3)⋊C4—34no—
79C3×(C2×D3)⋊C4—34no—
80C3×(C2×D3)⋊C4—34no—
81C3×(C2×D3)⋊C4—34no—
82C3×(C2×D3)⋊C4—64no—
83C3×(C2×D3)⋊C4—64no—
84C3×Dic3⋊C4—34no—
85C3×Dic3⋊C4—34no—
86C3×Dic3⋊C4—64no—
87C3×C4×Dic3—32no—
88C3×C4×Dic3—32no—
89C3×C4×Dic3—32no—
90C3×C4×Dic3—61no—
91C3×C4×Dic3—62no—
92#3724—32no—
93Dic3⋊Dic3—34no—
94#3722—34no—
95(C2×D3)⋊Dic3—34no—
96(C2×D3)⋊Dic3—34no—
97Dic32—34no—
98Dic32—32no—
99C22×C33C2411yescharacteristic
100C2×C3×C3⋊D3—24no—
101C2×C3×C3⋊D3—24no—
102C2×C3×C3⋊D3—24no—
103C33×C4—21no—
104C33×C4—22no—
105C3×C3⋊Dic3—24no—
106C3×C3⋊Dic3—22no—
107C3×C3⋊Dic3#4114yes—
108C3×C4⋊Q8—91no—
109C3×#124—92no—
110#947—32no—
111#947—32no—
112#947—62no—
113#947—92no—
114#944—32no—
115#944—61no—
116#939—94no—
117#895—94no—
118#892—94no—
119#885—94no—
120#884—94no—
121#67⋊C4—94no—
122#877—94no—
123C4×D12—92no—
124#872—94no—
125C22×C3⋊D3C22×D314yes—
126C22×C3×D3—34no—
127C22×C3×D3—34no—
128C22×C3×D3—64no—
129C32×Q8—41no—
130C32×Q8—42no—
131C32×Q8—42no—
132C32×Q8—44no—
133C32×Q8—44no—
134C2×C32×C4—21no—
135C2×C32×C4—22no—
136C2×C32×C4—22no—
137C2×C32×C4—22no—
138C2×C32×C4—22no—
139C2×C32×C4—41no—
140C2×C32×C4—42no—
141C2×C32×C4C22×D312yes—
142C2×C32×C4C22×D312yes—
143C2×C32×C4C22×D312yes—
144C2×C32×C4C22×D312yes—
145C2×C32×C4C22×D311yescharacteristic
146C2×C3⋊Dic3C22×D312yes—
147C2×C3⋊Dic3C22×D314yes—
148C2×C3⋊Dic3C22×D312yes—
149C4⋊(C3⋊D3)—42no—
150#634—41no—
151C2×C3×Dic3—34no—
152C2×C3×Dic3—34no—
153C2×C3×Dic3—34no—
154C2×C3×Dic3—34no—
155C2×C3×Dic3—62no—
156C2×C3×Dic3—64no—
157C2×C3×Dic3—62no—
158C3×D12—122no—
159C3×D12—122no—
160C3×D12—124no—
161C3×Dic6—122no—
162C3×Dic6—122no—
163#626—68no—
164#626—124no—
165#625—64no—
166#625—122no—
167#624—68no—
168D3×Dic3—68no—
169D3×Dic3—68no—
170#555—271noSylow
171C2×C33C2×#4112yes—
172C2×C33H(5,𝔽2)11yescharacteristic
173C3×C3⋊D3—44no—
174C2×C3×Q8—21no—
175C2×C3×Q8—22no—
176C2×C3×Q8—41no—
177C2×C3×Q8—91no—
178C2×C3×Q8D3212yes—
179C2×C3×Q8D3211yescharacteristic
180C2×C3×D4—92no—
181C2×#67—94no—
182C2×#67—94no—
183C22×Dic3—94no—
184C2×D12—32no—
185C2×D12—32no—
186C2×D12—62no—
187C2×D12—92no—
188C2×D12—92no—
189C2×C4×D3—94no—
190C2×C4×D3—94no—
191C2×Dic6—32no—
192C2×Dic6—61no—
193C3×C4⋊C4—94no—
194C3×C22⋊C4—94no—
195C3×C22⋊C4—94no—
196C3×C42—92no—
197C3×C42—91no—
198#217—94no—
199(C2×D3)⋊C4—34no—
200(C2×D3)⋊C4—34no—
201(C2×D3)⋊C4—34no—
202(C2×D3)⋊C4—34no—
203(C2×D3)⋊C4—64no—
204(C2×D3)⋊C4—64no—
205(C2×D3)⋊C4—94no—
206(C2×D3)⋊C4—94no—
207C4⋊Dic3—92no—
208Dic3⋊C4—34no—
209Dic3⋊C4—34no—
210Dic3⋊C4—64no—
211Dic3⋊C4—94no—
212C4×Dic3—32no—
213C4×Dic3—32no—
214C4×Dic3—32no—
215C4×Dic3—61no—
216C4×Dic3—62no—
217C4×Dic3—94no—
218C4×Dic3—92no—
219C22×C32—21no—
220C22×C32—22no—
221C22×C32—41no—
222C22×C32C23×D311yescharacteristic
223C22×C32C23×D312yes—
224C2×C3⋊D3—24no—
225C2×C3⋊D3—24no—
226C2×C3⋊D3—24no—
227C2×C3×D3—64no—
228C2×C3×D3—64no—
229C2×C3×D3—64no—
230C2×C3×D3—64no—
231C2×C3×D3—64no—
232C2×C3×D3—64no—
233C2×C3×D3—124no—
234C2×C3×D3—124no—
235C2×C3×D3—124no—
236C32×C4—21no—
237C32×C4—22no—
238C32×C4—22no—
239C32×C4—24no—
240C32×C4—22no—
241C32×C4—22no—
242C32×C4—22no—
243C32×C4—42no—
244C32×C4—42no—
245C32×C4—42no—
246C32×C4—42no—
247C32×C4—44no—
248C3⋊Dic3—22no—
249C3⋊Dic3—24no—
250C3⋊Dic3#23914yes—
251C3×Dic3—38no—
252C3×Dic3—64no—
253C3×Dic3—64no—
254C3×Dic3—64no—
255C3×Dic3—64no—
256C3×Dic3—122no—
257C3×Dic3—124no—
258C4⋊Q8—91no—
259#124—92no—
260#124—272no—
261#121—274no—
262C4×D4—272no—
263#117—274no—
264C33#55511yescharacteristic, complemented, Sylow
265C23×C3—94no—
266C22×D3—34no—
267C22×D3—34no—
268C22×D3—64no—
269C3×Q8—42no—
270C3×Q8—41no—
271C3×Q8—42no—
272C3×Q8—44no—
273C3×Q8—44no—
274C3×Q8—361no—
275C3×D4—362no—
276C2×C3×C4—21no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—41no—
282C2×C3×C4—42no—
283C2×C3×C4—92no—
284C2×C3×C4—94no—
285C2×C3×C4—92no—
286C2×C3×C4C2×D3212yes—
287C2×C3×C4C2×D3212yes—
288C2×C3×C4C2×D3212yes—
289C2×C3×C4C2×D3212yes—
290C2×C3×C4C2×D3211yescharacteristic
291#67—188no—
292#67—364no—
293C2×Dic3—34no—
294C2×Dic3—34no—
295C2×Dic3—34no—
296C2×Dic3—34no—
297C2×Dic3—62no—
298C2×Dic3—64no—
299C2×Dic3—62no—
300C2×Dic3—188no—
301D12—122no—
302D12—122no—
303D12—124no—
304D12—184no—
305D12—362no—
306C4×D3—188no—
307C4×D3—188no—
308Dic6—122no—
309Dic6—122no—
310C2×C32—24no—
311C2×C32—22no—
312C2×C32—21no—
313C2×C32—22no—
314C2×C32—42no—
315C2×C32—41no—
316C2×C32#100412yes—
317C2×C32#100411yescharacteristic
318C2×C32C2×#23912yes—
319C2×C32D3×#4114yes—
320C3⋊D3—44no—
321C3×D3—124no—
322C3×D3—124no—
323C3×D3—244no—
324C2×Q8—91no—
325C2×Q8#111311yescharacteristic
326C2×D4—92no—
327C2×D4—272no—
328C2×D4—272no—
329C22×C4—274no—
330C4⋊C4—94no—
331C4⋊C4—272no—
332C22⋊C4—94no—
333C22⋊C4—94no—
334C22⋊C4—274no—
335C42—92no—
336C42—91no—
337C42—272no—
338C22×C3—21no—
339C22×C3—22no—
340C22×C3—41no—
341C22×C3—184no—
342C22×C3—184no—
343C22×C3—184no—
344C22×C3C22×D3212yes—
345C22×C3C22×D3211yescharacteristic
346C2×D3—64no—
347C2×D3—64no—
348C2×D3—64no—
349C2×D3—64no—
350C2×D3—64no—
351C2×D3—64no—
352C2×D3—124no—
353C2×D3—124no—
354C2×D3—124no—
355C3×C4—22no—
356C3×C4—24no—
357C3×C4—22no—
358C3×C4—22no—
359C3×C4—22no—
360C3×C4—22no—
361C3×C4—21no—
362C3×C4—42no—
363C3×C4—42no—
364C3×C4—42no—
365C3×C4—44no—
366C3×C4—42no—
367C3×C4—94no—
368C3×C4—184no—
369C3×C4—182no—
370Dic3—38no—
371Dic3—64no—
372Dic3—64no—
373Dic3—64no—
374Dic3—64no—
375Dic3—122no—
376Dic3—124no—
377C32—22no—
378C32—21no—
379C32—41no—
380C32#567012yescomplemented
381C32#573411yescharacteristic, complemented
382C23—94no—
383Q8—41no—
384Q8—361no—
385D4—362no—
386D4—544no—
387D4—1082no—
388C2×C4—92no—
389C2×C4—94no—
390C2×C4—92no—
391C2×C4—548no—
392C2×C4C2×#111311yescharacteristic
393C2×C4C2×#111312yes—
394C2×C3—22no—
395C2×C3—24no—
396C2×C3—21no—
397C2×C3—22no—
398C2×C3—42no—
399C2×C3—41no—
400C2×C3—364no—
401C2×C3D3×#23914yes—
402C2×C3#6431912yes—
403C2×C3#6433012yes—
404C2×C3#6433011yescharacteristic
405D3—124no—
406D3—124no—
407D3—244no—
408C22—184no—
409C22—184no—
410C22—184no—
411C22C22×#111311yescharacteristic
412C4—22no—
413C4—21no—
414C4—94no—
415C4—184no—
416C4—182no—
417C3—22no—
418C3—21no—
419C3—41no—
420C3#1059538512yescomplemented
421C3#1059539011yescharacteristic, complemented
422C2—364no—
423C2#1172462612yes—
424C2#1172463311yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×Q8C22
hypercenterC2×Q8#1113
hypocenterC33#555
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C3×#64156C2semidirect
2. C3×#64153C2semidirect
3. #11723687C2semidirect
4. #11723685C2semidirect
5. #11723683C2semidirect
6. #11723670C2semidirect
23. #64156D3semidirect
24. #64153D3semidirect
264. C33#555semidirect
380. C32#5670semidirect
381. C32#5734semidirect
420. C3#10595385semidirect
421. C3#10595390semidirect

Maximal subgroups

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

subgroupindex
C3×#641562
C3×#641532
#117236872
#117236852
#117236832
#117236702
#105953903
#105953853