Finite Group Databasebeta

← #13167069

Subgroups — #13167069

Order 1728.

398 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 51 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
1#11723673C211yesmaximal, characteristic
2C3×#63865C211yesmaximal, characteristic
3#11723103C211yesmaximal, characteristic
4#11722779C211yesmaximal, characteristic, complemented
5#11722778C211yesmaximal, characteristic, complemented
6#11722775C211yesmaximal, characteristic, complemented
7#11722775C211yesmaximal, characteristic, complemented
8#10593370—31nomaximal
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C3×#627C2211yescharacteristic
11C2×C3⋊#622C2211yescharacteristic
12C2×C3⋊#622C2211yescharacteristic
13C3×C4×C3⋊Dic3—21no—
14C32×Dic3⋊C4—21no—
15#88815C2211yescharacteristic
16#88815C2211yescharacteristic
17#88814—21no—
18C3×#3724C2211yescharacteristic
19C3×#3722—21no—
20C3×#896—61no—
21#63874—31no—
22#63865D311yescharacteristic
23#63855—61no—
24#63778—31no—
25#63743—31no—
26#63742—31no—
27#63740—31no—
28#63740—31no—
29C22×C3×C3⋊D3D411yescharacteristic
30C2×C33×C4D411yescharacteristic
31C2×C3×C3⋊Dic3—21no—
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3C2311yescharacteristic
34C3×C4×C3⋊D3—21no—
35C3×C4×C3⋊D3—21no—
36C2×C32×Dic3—21no—
37C2×C32×Dic3—21no—
38C3×#627—21no—
39C3×#627—21no—
40C3⋊#622—21no—
41C3⋊#622—21no—
42#5045—91nomaximal
43C2×C4×C3⋊D3C2×D311yescharacteristic
44C2×C3×C4×D3—61no—
45C2×C3×C4×D3—61no—
46C2×C3×Dic6—61no—
47C2×#627C2×D311yescharacteristic
48C2×#622—31no—
49C2×#622—31no—
50C32×C4⋊C4—61no—
51C4×C3⋊Dic3—21no—
52C3×(C2×D3)⋊C4—61no—
53C3×(C2×D3)⋊C4—61no—
54C3×Dic3⋊C4—21no—
55C3×Dic3⋊C4—21no—
56C3×Dic3⋊C4—41no—
57C3×Dic3⋊C4—61no—
58C3×C4×Dic3—61no—
59C3×C4×Dic3—61no—
60C3×C4×Dic3—121no—
61#3724—31no—
62#3724—31no—
63#3724C2×D311yescharacteristic
64Dic3⋊Dic3—61no—
65Dic3⋊Dic3—61no—
66#3722—21no—
67#3722—61no—
68(C2×D3)⋊Dic3—61no—
69(C2×D3)⋊Dic3—61no—
70C22×C33C2×D411yescharacteristic
71C2×C3×C3⋊D3—21no—
72C2×C3×C3⋊D3—21no—
73C2×C3×C3⋊D3—21no—
74C33×C4—21no—
75C3×C3⋊Dic3—41no—
76C3×C3⋊Dic3—41no—
77C3×C3⋊Dic3#4111yescharacteristic
78C3×C3⋊Dic3#4111yescharacteristic
79C32×Dic3—41no—
80C32×Dic3—41no—
81C3×C22⋊Q8—91no—
82#926—91no—
83#896—61no—
84#872—181no—
85#835—91no—
86#835—91no—
87#830—91no—
88#808—91no—
89#807—91no—
90C22×C3⋊D3#6711yescharacteristic
91C22×C3×D3—61no—
92C22×C3×D3—61no—
93C2×C32×C4—21no—
94C2×C32×C4—21no—
95C2×C32×C4—41no—
96C2×C32×C4—41no—
97C2×C32×C4—61no—
98C2×C32×C4—61no—
99C2×C32×C4#6711yescharacteristic
100C2×C3⋊Dic3—21no—
101C2×C3⋊Dic3—21no—
102C2×C3⋊Dic3C22×D311yescharacteristic
103C4×C3⋊D3—21no—
104C4×C3⋊D3—21no—
105C2×C3×Dic3—21no—
106C2×C3×Dic3—21no—
107C2×C3×Dic3—21no—
108C2×C3×Dic3—21no—
109C2×C3×Dic3—41no—
110C2×C3×Dic3—41no—
111C2×C3×Dic3—61no—
112C2×C3×Dic3—61no—
113C2×C3×Dic3—61no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—121no—
118C2×C3×Dic3—121no—
119C3×C4×D3—62no—
120C3×C4×D3—62no—
121C3×C4×D3—121no—
122C3×C4×D3—121no—
123C3×Dic6—121no—
124C3×Dic6—121no—
125#627—21no—
126#627—21no—
127#622—61no—
128#622—61no—
129#485—271noSylow
130C2×C33#12311yescharacteristic
131C2×C33#13511yescharacteristic
132C2×C33#13711yescharacteristic
133C3×C3⋊D3—41no—
134C2×C3×Q8—91no—
135C22×C3×C4—91no—
136C2×C4×D3—61no—
137C2×C4×D3—61no—
138C2×Dic6—61no—
139C3×C4⋊C4—61no—
140C3×C4⋊C4—61no—
141C3×C4⋊C4—91no—
142C3×C4⋊C4—121no—
143C3×C22⋊C4—181no—
144C3×C42—181no—
145#217—181no—
146(C2×D3)⋊C4—61no—
147(C2×D3)⋊C4—61no—
148(C2×D3)⋊C4—181no—
149(C2×D3)⋊C4—181no—
150C4⋊Dic3—91no—
151C4⋊Dic3—91no—
152Dic3⋊C4—21no—
153Dic3⋊C4—61no—
154Dic3⋊C4—181no—
155Dic3⋊C4—181no—
156C4×Dic3—61no—
157C4×Dic3—61no—
158C4×Dic3—121no—
159C2×C3⋊C8—91no—
160C2×C3⋊C8—91no—
161C22×C32—21no—
162C22×C32—21no—
163C22×C32—41no—
164C22×C32—41no—
165C22×C32C2×#6711yescharacteristic
166C2×C3⋊D3—21no—
167C2×C3⋊D3—21no—
168C2×C3⋊D3—21no—
169C2×C3×D3—62no—
170C2×C3×D3—62no—
171C2×C3×D3—121no—
172C2×C3×D3—121no—
173C2×C3×D3—121no—
174C2×C3×D3—121no—
175C32×C4—22no—
176C32×C4—21no—
177C32×C4—41no—
178C32×C4—42no—
179C32×C4—81no—
180C32×C4—121no—
181C32×C4—121no—
182C3⋊Dic3—41no—
183C3⋊Dic3—41no—
184C3⋊Dic3#23711yescharacteristic
185C3⋊Dic3#23711yescharacteristic
186C3×Dic3—41no—
187C3×Dic3—41no—
188C3×Dic3—42no—
189C3×Dic3—41no—
190C3×Dic3—41no—
191C3×Dic3—61no—
192C3×Dic3—61no—
193C3×Dic3—61no—
194C3×Dic3—61no—
195C3×Dic3—81no—
196C3×Dic3—121no—
197C3×Dic3—121no—
198C3×Dic3—122no—
199C3×Dic3—122no—
200C3×Dic3—121no—
201C3×Dic3—121no—
202C22⋊Q8—91no—
203#117—271no—
204#107—271no—
205#106—271no—
206#103—271no—
207#103—271no—
208C22⋊C8—271no—
209C33#48511yescharacteristic, complemented, Sylow
210C23×C3—91no—
211C22×D3—61no—
212C22×D3—61no—
213C3×Q8—181no—
214C3×Q8—181no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—61no—
220C2×C3×C4—61no—
221C2×C3×C4—61no—
222C2×C3×C4—61no—
223C2×C3×C4—91no—
224C2×C3×C4—121no—
225C2×C3×C4—121no—
226C2×C3×C4—181no—
227C2×C3×C4—181no—
228C2×C3×C4—181no—
229C2×C3×C4—181no—
230C2×C3×C4GO+(4,𝔽2)11yescharacteristic
231C2×Dic3—21no—
232C2×Dic3—21no—
233C2×Dic3—61no—
234C2×Dic3—61no—
235C2×Dic3—61no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—61no—
239C2×Dic3—121no—
240C2×Dic3—121no—
241C4×D3—62no—
242C4×D3—62no—
243C4×D3—121no—
244C4×D3—121no—
245Dic6—121no—
246Dic6—121no—
247C3⋊C8—181no—
248C3⋊C8—181no—
249C2×C32—21no—
250C2×C32—21no—
251C2×C32—21no—
252C2×C32—21no—
253C2×C32—21no—
254C2×C32—21no—
255C2×C32—41no—
256C2×C32—41no—
257C2×C32—41no—
258C2×C32—41no—
259C2×C32—41no—
260C2×C32—41no—
261C2×C32#93511yescharacteristic
262C2×C32#94211yescharacteristic
263C2×C32#95011yescharacteristic
264C3⋊D3—41no—
265C3×D3—122no—
266C3×D3—122no—
267C2×Q8—91no—
268C22×C4—91no—
269C2×C8—271no—
270C2×C8—271no—
271C4⋊C4—61no—
272C4⋊C4—91no—
273C4⋊C4—271no—
274C4⋊C4—271no—
275C22⋊C4—181no—
276C22⋊C4—541no—
277C42—181no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—181no—
283C22×C3—181no—
284C22×C3—181no—
285C22×C3C2×GO+(4,𝔽2)11yescharacteristic
286C2×D3—62no—
287C2×D3—62no—
288C2×D3—121no—
289C2×D3—121no—
290C2×D3—121no—
291C2×D3—121no—
292C3×C4—22no—
293C3×C4—21no—
294C3×C4—41no—
295C3×C4—42no—
296C3×C4—81no—
297C3×C4—91no—
298C3×C4—91no—
299C3×C4—121no—
300C3×C4—121no—
301C3×C4—122no—
302C3×C4—121no—
303C3×C4—121no—
304C3×C4—241no—
305C3×C4—361no—
306C3×C4—361no—
307Dic3—41no—
308Dic3—41no—
309Dic3—61no—
310Dic3—61no—
311Dic3—61no—
312Dic3—61no—
313Dic3—121no—
314Dic3—121no—
315Dic3—122no—
316Dic3—122no—
317Dic3—121no—
318Dic3—121no—
319C32—21no—
320C32—21no—
321C32—41no—
322C32—41no—
323C32#504511yescharacteristic, complemented
324C23—91no—
325Q8—181no—
326Q8—181no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—91no—
330C2×C4—181no—
331C2×C4—181no—
332C2×C4—181no—
333C2×C4—181no—
334C2×C4D32⋊D311yescharacteristic
335C8—541no—
336C8—541no—
337C2×C3—21no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3—41no—
344C2×C3—41no—
345C2×C3—41no—
346C2×C3—41no—
347C2×C3—41no—
348C2×C3—41no—
349C2×C3—361no—
350C2×C3#6422311yescharacteristic
351C2×C3#6422611yescharacteristic
352C2×C3#6423311yescharacteristic
353D3—122no—
354D3—122no—
355C22—181no—
356C22—181no—
357C22—181no—
358C22C2×D32⋊D311yescharacteristic
359C4—21no—
360C4—91no—
361C4—91no—
362C4—121no—
363C4—121no—
364C4—361no—
365C4—361no—
366C3—21no—
367C3—21no—
368C3—41no—
369C3—41no—
370C3#1059337011yescharacteristic, complemented
371C2—361no—
372C2#1172430311yescharacteristic
373C2#1172430711yescharacteristic
374C2#1172431911yescharacteristic
subgroupG/H
center, FrattiniC22C2×D32⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC2×C4D32⋊D3
hypocenterC33#485
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722779C2semidirect
5. #11722778C2semidirect
6. #11722775C2semidirect
7. #11722775C2semidirect
209. C33#485semidirect
323. C32#5045semidirect
370. C3#10593370semidirect

Maximal subgroups

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

subgroupindex
#117236732
C3×#638652
#117231032
#117227792
#117227782
#117227752
#105933703
#50459