Finite Group Databasebeta

← #13166990

Subgroups — #13166990

Order 1728.

640 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C2×C3×#3798C211yesmaximal, characteristic
2C2×#88988C211yesmaximal, characteristic, complemented
3#11723673C211yesmaximal, characteristic, complemented
4#11722775C212yesmaximal, complemented
5#11722773C212yesmaximal, complemented
6#10593349—31nomaximal
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C32×C4×D3—21no—
9C2×C3×#627C2211yescharacteristic
10C2×C3×#626—21no—
11C2×C3×#625C2211yescharacteristic
12C3×#3798—22no—
13C3×#3798C414yescomplemented
14C2×C3⋊#622C2212yes—
15#88988—22no—
16C3×C4×C3⋊Dic3—21no—
17#88815C2212yes—
18#88814—21no—
19C2×C3×#235—61no—
20C2×#3798D311yescharacteristic
21C2×#3787—31no—
22#63874—31no—
23#63855—61no—
24#63740—32no—
25#63739—32no—
26C22×C3×C3⋊D3D411yescharacteristic
27C22×C32×D3—21no—
28C2×C33×C4D411yescharacteristic
29C2×C3×C3⋊Dic3—22no—
30C2×C3×C3⋊Dic3C2311yescharacteristic
31C3×C4×C3⋊D3C2×C412yes—
32C3×C4×C3⋊D3D412yes—
33C2×C32×Dic3—21no—
34C32×C4×D3—24no—
35C3×#627—21no—
36C3×#627C2×C412yes—
37C3×#626—24no—
38C3×#625—21no—
39C3×#625C2×C412yes—
40C3⋊#622—22no—
41#5233—91nomaximal
42C22×C32×C4—61no—
43C2×C4×C3⋊D3C2×D311yescharacteristic
44C2×C3×#67—61no—
45C2×C3×#67—61no—
46C3×#235—64no—
47C3×#235—64no—
48C2×C3×D12—61no—
49C2×C3×C4×D3—21no—
50C2×C3×C4×D3—21no—
51C2×C3×C4×D3—41no—
52C2×C3×C4×D3—61no—
53C2×C3×C4×D3—61no—
54C2×C3×Dic6—61no—
55C2×#627C2×D311yescharacteristic
56C2×#626—21no—
57C2×#625C2×D311yescharacteristic
58#3798—22no—
59#3798Dic314yescomplemented
60C2×#622—32no—
61#3787—62no—
62C4×C3⋊Dic3—21no—
63C3×C4×Dic3—61no—
64C3×C4×Dic3—61no—
65C3×C4×Dic3—121no—
66#3724—32no—
67Dic3⋊Dic3—62no—
68#3722—61no—
69(C2×D3)⋊Dic3—62no—
70C22×C33C2×D411yescharacteristic
71C2×C3×C3⋊D3—22no—
72C2×C3×C3⋊D3C22⋊C412yes—
73C2×C32×D3—22no—
74C2×C32×D3—42no—
75C33×C4C22⋊C412yes—
76C3×C3⋊Dic3—42no—
77C3×C3⋊Dic3C22×C411yescharacteristic
78C3×C3⋊Dic3C2×D411yescharacteristic
79C32×Dic3—22no—
80C2×C3×#41—91no—
81C2×#235—61no—
82#926—91no—
83C22×C3⋊C8—91no—
84#872—181no—
85#835—92no—
86#832—92no—
87C23×C32—61no—
88C22×C3⋊D3#6711yescharacteristic
89C22×C3×D3—21no—
90C22×C3×D3—21no—
91C22×C3×D3—41no—
92C22×C3×D3—61no—
93C22×C3×D3—61no—
94C2×C32×C4—21no—
95C2×C32×C4—21no—
96C2×C32×C4—41no—
97C2×C32×C4—41no—
98C2×C32×C4—64no—
99C2×C32×C4—61no—
100C2×C32×C4#6711yescharacteristic
101C2×C3⋊Dic3—22no—
102C2×C3⋊Dic3C22×D311yescharacteristic
103C4×C3⋊D3C2×Dic312yes—
104C4×C3⋊D3#6712yes—
105C3×#67—62no—
106C3×#67—64no—
107C3×#67—62no—
108C2×C3×Dic3—21no—
109C2×C3×Dic3—21no—
110C2×C3×Dic3—41no—
111C2×C3×Dic3—61no—
112C2×C3×Dic3—61no—
113C2×C3×Dic3—62no—
114C2×C3×Dic3—62no—
115C2×C3×Dic3—122no—
116C3×D12—64no—
117C3×C4×D3—24no—
118C3×C4×D3—24no—
119C3×C4×D3—44no—
120C3×C4×D3—62no—
121C3×C4×D3—62no—
122C3×C4×D3—62no—
123C3×C4×D3—62no—
124C3×Dic6—62no—
125C3×Dic6—62no—
126#627—21no—
127#627C2×Dic312yes—
128#626—24no—
129#625—21no—
130#625C2×Dic312yes—
131#622—62no—
132#416—271noSylow
133C2×C33C2×C22⋊C411yescharacteristic
134C2×C33#13512yes—
135C3×C3⋊D3—22no—
136C32×D3—42no—
137C3×#41—94no—
138C3×#41—182no—
139C2×C3×Q8—91no—
140C2×C3×D4—91no—
141C2×C3×D4—181no—
142C22×C3×C4—61no—
143C22×C3×C4—61no—
144C22×C3×C4—91no—
145C22×C3×C4—121no—
146C2×#67—61no—
147C2×#67—61no—
148#235—64no—
149#235—64no—
150C2×D12—61no—
151C2×C4×D3—21no—
152C2×C4×D3—61no—
153C2×C4×D3—61no—
154C2×Dic6—61no—
155C3×C42—181no—
156#217—181no—
157(C2×D3)⋊C4—182no—
158C4⋊Dic3—92no—
159Dic3⋊C4—182no—
160C4×Dic3—61no—
161C4×Dic3—61no—
162C4×Dic3—121no—
163C2×C3⋊C8—92no—
164C2×C3⋊C8—182no—
165C22×C32—21no—
166C22×C32—21no—
167C22×C32—41no—
168C22×C32—41no—
169C22×C32—62no—
170C22×C32—122no—
171C22×C32C2×#6711yescharacteristic
172C2×C3⋊D3—22no—
173C2×C3⋊D3#21712yes—
174C2×C3×D3—22no—
175C2×C3×D3—22no—
176C2×C3×D3—42no—
177C2×C3×D3—42no—
178C2×C3×D3—44no—
179C2×C3×D3—42no—
180C2×C3×D3—62no—
181C2×C3×D3—62no—
182C2×C3×D3—122no—
183C2×C3×D3—122no—
184C32×C4—22no—
185C32×C4—22no—
186C32×C4—42no—
187C32×C4—42no—
188C32×C4—62no—
189C32×C4#21712yes—
190C3⋊Dic3—42no—
191C3⋊Dic3C22×Dic311yescharacteristic
192C3⋊Dic3C2×#6711yescharacteristic
193C3×Dic3—22no—
194C3×Dic3—22no—
195C3×Dic3—42no—
196C3×Dic3—61no—
197C3×Dic3—61no—
198C3×Dic3—61no—
199C3×Dic3—61no—
200C3×Dic3—122no—
201C3×Dic3—124no—
202C3×Dic3—122no—
203C2×#41—91no—
204C22×C8—271no—
205#117—271no—
206#103—272no—
207#102—272no—
208C33#41611yescharacteristic, complemented, Sylow
209C23×C3—61no—
210C23×C3—61no—
211C23×C3—91no—
212C23×C3—121no—
213C22×D3—21no—
214C22×D3—61no—
215C22×D3—61no—
216C3×Q8—92no—
217C3×Q8—181no—
218C3×D4—92no—
219C3×D4—184no—
220C3×D4—181no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—64no—
226C2×C3×C4—61no—
227C2×C3×C4—61no—
228C2×C3×C4—64no—
229C2×C3×C4—91no—
230C2×C3×C4—92no—
231C2×C3×C4—92no—
232C2×C3×C4—124no—
233C2×C3×C4—121no—
234C2×C3×C4—182no—
235C2×C3×C4GO+(4,𝔽2)11yescharacteristic
236#67—62no—
237#67—64no—
238#67—62no—
239C2×Dic3—21no—
240C2×Dic3—61no—
241C2×Dic3—61no—
242C2×Dic3—62no—
243C2×Dic3—62no—
244C2×Dic3—122no—
245D12—64no—
246C4×D3—24no—
247C4×D3—62no—
248C4×D3—62no—
249C4×D3—62no—
250C4×D3—62no—
251Dic6—62no—
252Dic6—62no—
253C3⋊C8—182no—
254C2×C32—22no—
255C2×C32—22no—
256C2×C32—21no—
257C2×C32—21no—
258C2×C32—41no—
259C2×C32—41no—
260C2×C32—42no—
261C2×C32—42no—
262C2×C32—122no—
263C2×C32#95012yes—
264C2×C32C2×#21711yescharacteristic
265C3⋊D3—22no—
266C3×D3—42no—
267C3×D3—44no—
268C3×D3—42no—
269C3×D3—122no—
270C3×D3—122no—
271#41—94no—
272#41—182no—
273C2×Q8—91no—
274C2×D4—91no—
275C2×D4—181no—
276C22×C4—61no—
277C22×C4—91no—
278C2×C8—272no—
279C2×C8—542no—
280C4⋊C4—272no—
281C22⋊C4—541no—
282C42—181no—
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3—62no—
288C22×C3—62no—
289C22×C3—92no—
290C22×C3—122no—
291C22×C3—122no—
292C22×C3—124no—
293C22×C3—122no—
294C22×C3—182no—
295C22×C3C2×GO+(4,𝔽2)11yescharacteristic
296C2×D3—22no—
297C2×D3—42no—
298C2×D3—62no—
299C2×D3—62no—
300C2×D3—122no—
301C2×D3—122no—
302C3×C4—22no—
303C3×C4—22no—
304C3×C4—42no—
305C3×C4—42no—
306C3×C4—62no—
307C3×C4—62no—
308C3×C4—91no—
309C3×C4—91no—
310C3×C4—122no—
311C3×C4—362no—
312C3×C4D32⋊C412yes—
313Dic3—22no—
314Dic3—61no—
315Dic3—61no—
316Dic3—61no—
317Dic3—61no—
318Dic3—122no—
319Dic3—124no—
320Dic3—122no—
321C32—21no—
322C32—21no—
323C32—41no—
324C32—41no—
325C32#523311yescharacteristic, complemented
326C23—61no—
327C23—91no—
328Q8—92no—
329Q8—181no—
330D4—92no—
331D4—184no—
332D4—181no—
333C2×C4—61no—
334C2×C4—64no—
335C2×C4—91no—
336C2×C4—92no—
337C2×C4—92no—
338C2×C4—182no—
339C2×C4D32⋊D311yescharacteristic
340C8—542no—
341C2×C3—22no—
342C2×C3—22no—
343C2×C3—21no—
344C2×C3—21no—
345C2×C3—42no—
346C2×C3—42no—
347C2×C3—41no—
348C2×C3—41no—
349C2×C3—122no—
350C2×C3—124no—
351C2×C3—122no—
352C2×C3—182no—
353C2×C3#6422312yes—
354C2×C3C2×D32⋊C411yescharacteristic
355D3—42no—
356D3—122no—
357D3—122no—
358C22—62no—
359C22—92no—
360C22—122no—
361C22—182no—
362C22C2×D32⋊D311yescharacteristic
363C4—62no—
364C4—91no—
365C4—91no—
366C4—362no—
367C4D32⋊Dic312yes—
368C3—21no—
369C3—21no—
370C3—41no—
371C3—41no—
372C3#1059334911yescharacteristic, complemented
373C2—122no—
374C2—182no—
375C2#1172430312yes—
376C2C2×D32⋊Dic311yescharacteristic
subgroupG/H
center, hypercenterC2×C4D32⋊D3
commutatorC3×C3⋊Dic3C22×C4
FrattiniC22C2×D32⋊D3
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×C3×#3798C2
cosocleC2×C3×C3⋊Dic3C23
hypocenterC33#416
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88988C2semidirect
3. #11723673C2semidirect
4. #11722775C2semidirect
5. #11722773C2semidirect
13. C3×#3798C4semidirect
59. #3798Dic3semidirect
208. C33#416semidirect
325. C32#5233semidirect
372. C3#10593349semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37982
C2×#889882
#117236732
#117227752
#117227732
#105933493
#52339