Finite Group Databasebeta

← #13178266

Subgroups — #13178266

Order 1728.

1476 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 1476 conjugacy classes of subgroups — too many to draw. Only its 199 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
1C22×#6296C211yesmaximal, characteristic, complemented
2C2×#89042C211yesmaximal, characteristic, complemented
3C3×#64090C213yesmaximal, complemented
4#11723717C213yesmaximal, complemented
5C4×#6296C211yesmaximal, characteristic, complemented
6#11723672C213yesmaximal, complemented
7#11723671C213yesmaximal, complemented
8#10594189—33nomaximal
9C2×C3×#638C2213yes—
10C22×C3×C3⋊Dic3C2213yes—
11C2×C3×C4×C3⋊D3C2213yes—
12C2×#6296—24no—
13C2×#6296C2213yes—
14C2×#6296C2211yescharacteristic
15C2×#6295C2213yes—
16#89042—24no—
17C33×C22⋊C4C2211yescharacteristic
18C3×#3752C2213yes—
19C3×#3750C2213yes—
20C3×C4×C3⋊Dic3C2213yes—
21#88815C2213yes—
22#88814C2216yes—
23C22×#627—33no—
24C2×#3798—33no—
25#64090D313yescomplemented
26C3×#67⋊C4—36no—
27C3×#67⋊C4—63no—
28#63980—33no—
29#63978—36no—
30C4×#627—33no—
31#63915—33no—
32#63865—33no—
33#63852—36no—
34#63851—36no—
35C23×C33C2311yescharacteristic
36C22×C3×C3⋊D3C2313yes—
37C2×C33×C4C2311yescharacteristic
38C2×C33×C4C2311yescharacteristic
39C3×#638—26no—
40C2×C3×C3⋊Dic3—26no—
41C2×C3×C3⋊Dic3C2313yes—
42C2×C3×C3⋊Dic3C2313yes—
43C2×C3×C3⋊Dic3C2313yes—
44C3×C4×C3⋊D3—26no—
45#6296—26no—
46#6296D414yescomplemented
47#6295—26no—
48#5552—93no—
49C2×#638C2×D313yes—
50C22×C3⋊Dic3C2×D313yes—
51C2×C4×C3⋊D3C2×D313yes—
52C2×C3×#67—36no—
53C2×C3×#67—63no—
54C22×C3×Dic3—36no—
55C22×C3×Dic3—63no—
56C2×C3×C4×D3—36no—
57C2×C3×C4×D3—63no—
58C2×#627—33no—
59C2×#627—36no—
60C2×#627—33no—
61C2×#627—612no—
62C2×#626—36no—
63C2×#625—33no—
64#3798—612no—
65C32×C22⋊C4—23no—
66C32×C22⋊C4—41no—
67C32×C22⋊C4C2×D313yes—
68#3752C2×D313yes—
69#3750C2×D313yes—
70C4×C3⋊Dic3C2×D313yes—
71C3×(C2×D3)⋊C4—36no—
72C3×(C2×D3)⋊C4—63no—
73C3×Dic3⋊C4—36no—
74C3×Dic3⋊C4—63no—
75C3×C4×Dic3—36no—
76C3×C4×Dic3—63no—
77#3724—33no—
78Dic3⋊Dic3—36no—
79#3722—36no—
80(C2×D3)⋊Dic3—36no—
81(C2×D3)⋊Dic3—36no—
82C22×C33—21no—
83C22×C33—21no—
84C22×C33#4112yescomplemented
85C22×C33C2411yescharacteristic
86C2×C3×C3⋊D3—23no—
87C2×C3×C3⋊D3—23no—
88C2×C3×C3⋊D3—23no—
89C33×C4—21no—
90C33×C4—21no—
91C3×C3⋊Dic3—23no—
92C3×C3⋊Dic3—23no—
93C3×C3⋊Dic3C2×D416yes—
94C2×#235—93no—
95C22×Dic6—93no—
96C3×C4×D4—93no—
97#939—96no—
98#930—93no—
99#924—93no—
100#896—96no—
101#885—96no—
102#67⋊C4—36no—
103#67⋊C4—63no—
104#875—93no—
105C4×Dic6—93no—
106C23×C32—23no—
107C23×C32—41no—
108C23×C32C22×D313yes—
109C22×C3⋊D3C22×D313yes—
110C22×C3×D3—36no—
111C22×C3×D3—63no—
112C2×C32×C4—23no—
113C2×C32×C4—23no—
114C2×C32×C4—41no—
115C2×C32×C4—41no—
116C2×C32×C4C22×D313yes—
117C2×C32×C4C22×D313yes—
118#638—26no—
119C2×C3⋊Dic3—26no—
120C2×C3⋊Dic3C22×D313yes—
121C2×C3⋊Dic3C22×D313yes—
122C2×C3⋊Dic3C22×D313yes—
123C4×C3⋊D3—26no—
124C3×#67—612no—
125C3×#67—612no—
126C2×C3×Dic3—36no—
127C2×C3×Dic3—36no—
128C2×C3×Dic3—36no—
129C2×C3×Dic3—63no—
130C2×C3×Dic3—63no—
131C2×C3×Dic3—63no—
132C2×C3×Dic3—612no—
133C2×C3×Dic3—126no—
134C3×C4×D3—612no—
135C3×C4×D3—126no—
136#627—312no—
137#627—612no—
138#627—66no—
139#626—612no—
140#625—66no—
141#548—271noSylow
142C2×C33—22no—
143C2×C33C22×D411yescharacteristic
144C2×C33C2×#4111yescharacteristic
145C2×C332−1+411yescharacteristic
146C3×C3⋊D3—43no—
147C2×C3×D4—93no—
148C22×C3×C4—93no—
149C22×C3×C4—93no—
150C2×#67—36no—
151C2×#67—63no—
152C2×#67—96no—
153C22×Dic3—36no—
154C22×Dic3—63no—
155#235—1812no—
156C2×D12—93no—
157C2×C4×D3—36no—
158C2×C4×D3—63no—
159C2×Dic6—93no—
160C2×Dic6—93no—
161C2×Dic6—96no—
162C2×Dic6—1812no—
163C3×C4⋊C4—93no—
164C3×C22⋊C4—23no—
165C3×C22⋊C4—41no—
166C3×C22⋊C4—93no—
167C3×C22⋊C4D3213yes—
168C3×C42—93no—
169#217—96no—
170(C2×D3)⋊C4—36no—
171(C2×D3)⋊C4—63no—
172(C2×D3)⋊C4—96no—
173(C2×D3)⋊C4—96no—
174C4⋊Dic3—93no—
175Dic3⋊C4—36no—
176Dic3⋊C4—63no—
177Dic3⋊C4—96no—
178C4×Dic3—36no—
179C4×Dic3—63no—
180C22×C32—23no—
181C22×C32—23no—
182C22×C32—23no—
183C22×C32—26no—
184C22×C32—41no—
185C22×C32—42no—
186C22×C32—43no—
187C22×C32—43no—
188C22×C32—81no—
189C22×C32—81no—
190C22×C32#23516yescomplemented
191C22×C32C23×D313yes—
192C2×C3⋊D3—23no—
193C2×C3⋊D3—23no—
194C2×C3⋊D3—23no—
195C2×C3×D3—66no—
196C2×C3×D3—66no—
197C2×C3×D3—66no—
198C2×C3×D3—66no—
199C2×C3×D3—123no—
200C2×C3×D3—123no—
201C32×C4—23no—
202C32×C4—23no—
203C32×C4—43no—
204C32×C4—43no—
205C32×C4—81no—
206C32×C4—81no—
207C3⋊Dic3—23no—
208C3⋊Dic3—23no—
209C3⋊Dic3D3×D416yes—
210C3×Dic3—312no—
211C3×Dic3—66no—
212C3×Dic3—66no—
213C3×Dic3—66no—
214C3×Dic3—66no—
215C3×Dic3—123no—
216C2×#41—271no—
217C22×Q8—271no—
218#124—273no—
219C22⋊Q8—273no—
220#121—273no—
221C4×Q8—271no—
222C4×D4—93no—
223C33#54811yescharacteristic, complemented, Sylow
224C23×C3—23no—
225C23×C3—41no—
226C23×C3—93no—
227C23×C3C2×D3213yes—
228C22×D3—36no—
229C22×D3—63no—
230C3×D4—186no—
231C2×C3×C4—23no—
232C2×C3×C4—23no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—93no—
236C2×C3×C4—93no—
237C2×C3×C4—93no—
238C2×C3×C4—186no—
239C2×C3×C4—186no—
240C2×C3×C4C2×D3213yes—
241C2×C3×C4C2×D3213yes—
242#67—612no—
243#67—612no—
244#67—1812no—
245C2×Dic3—36no—
246C2×Dic3—36no—
247C2×Dic3—36no—
248C2×Dic3—63no—
249C2×Dic3—63no—
250C2×Dic3—63no—
251C2×Dic3—612no—
252C2×Dic3—126no—
253D12—186no—
254C4×D3—612no—
255C4×D3—126no—
256Dic6—912no—
257Dic6—1812no—
258Dic6—186no—
259C2×C32—26no—
260C2×C32—23no—
261C2×C32—23no—
262C2×C32—23no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32—41no—
266C2×C32—46no—
267C2×C32—82no—
268C2×C32C2×#23513yes—
269C2×C32C2×D3×D413yes—
270C2×C32#101013yes—
271C3⋊D3—43no—
272C3×D3—126no—
273C3×D3—126no—
274#41—544no—
275C2×Q8—271no—
276C2×Q8—273no—
277C2×Q8—544no—
278C2×D4—93no—
279C2×D4—273no—
280C22×C4—93no—
281C22×C4—93no—
282C4⋊C4—93no—
283C4⋊C4—273no—
284C22⋊C4—93no—
285C22⋊C4—276no—
286C22⋊C4#111311yescharacteristic
287C42—93no—
288C22×C3—23no—
289C22×C3—23no—
290C22×C3—23no—
291C22×C3—26no—
292C22×C3—43no—
293C22×C3—43no—
294C22×C3—41no—
295C22×C3—42no—
296C22×C3—81no—
297C22×C3—81no—
298C22×C3—183no—
299C22×C3—183no—
300C22×C3—183no—
301C22×C3#379816yescomplemented
302C22×C3C22×D3213yes—
303C2×D3—66no—
304C2×D3—66no—
305C2×D3—66no—
306C2×D3—66no—
307C2×D3—123no—
308C2×D3—123no—
309C3×C4—23no—
310C3×C4—23no—
311C3×C4—43no—
312C3×C4—43no—
313C3×C4—81no—
314C3×C4—81no—
315C3×C4—96no—
316C3×C4—183no—
317C3×C4—183no—
318Dic3—312no—
319Dic3—66no—
320Dic3—66no—
321Dic3—66no—
322Dic3—66no—
323Dic3—123no—
324C32—23no—
325C32—41no—
326C32#555213yescomplemented
327C23—93no—
328C23C2×#111311yescharacteristic
329Q8—274no—
330Q8—546no—
331D4—186no—
332D4—546no—
333C2×C4—93no—
334C2×C4—93no—
335C2×C4—93no—
336C2×C4—186no—
337C2×C4—186no—
338C2×C4C2×#111311yescharacteristic
339C2×C4C2×#111311yescharacteristic
340C2×C3—26no—
341C2×C3—23no—
342C2×C3—23no—
343C2×C3—23no—
344C2×C3—46no—
345C2×C3—41no—
346C2×C3—41no—
347C2×C3—41no—
348C2×C3—82no—
349C2×C3—363no—
350C2×C3C2×#379813yes—
351C2×C3#6430913yes—
352C2×C3D32×D413yes—
353D3—126no—
354D3—126no—
355C22—21no—
356C22—21no—
357C22—183no—
358C22—183no—
359C22—183no—
360C22#8904212yescomplemented
361C22C22×#111311yescharacteristic
362C4—21no—
363C4—21no—
364C4—96no—
365C4—183no—
366C4—183no—
367C3—23no—
368C3—41no—
369C3#1059418913yescomplemented
370C2—22no—
371C2—363no—
372C2C2×#8904211yescharacteristic
373C2#1172462011yescharacteristic
374C2D4×#111311yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊C4C22
hypercenterC22⋊C4#1113
hypocenterC33#548
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#6296C2semidirect
2. C2×#89042C2semidirect
3. C3×#64090C2semidirect
4. #11723717C2semidirect
5. C4×#6296C2semidirect
6. #11723672C2semidirect
7. #11723671C2semidirect
25. #64090D3semidirect
46. #6296D4semidirect
84. C22×C33#41semidirect
190. C22×C32#235semidirect
223. C33#548semidirect
301. C22×C3#3798semidirect
326. C32#5552semidirect
360. C22#89042semidirect
369. C3#10594189semidirect

Maximal subgroups

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

subgroupindex
C22×#62962
C2×#890422
C3×#640902
#117237172
C4×#62962
#117236722
#117236712
#105941893