Finite Group Databasebeta

← #13155471

Subgroups — #13155471

Order 1728.

690 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 144 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#11723689C211yesmaximal, characteristic
2C2×C3⋊D3×C3⋊C8C211yesmaximal, characteristic
3C3×#63900C211yesmaximal, characteristic
4#11722056C212yesmaximal, complemented
5C3×#63575C212yesmaximal, complemented
6#10590687—31nomaximal
7D3×#807—32nomaximal
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×#6294—21no—
10C2×C3×#624—21no—
11C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
12C3×C4⋊(C3⋊Dic3)C2212yes—
13C32×C4⋊Dic3C2212yes—
14C2×C32×C3⋊C8C2211yescharacteristic
15#88815—21no—
16C3⋊D3×C3⋊C8C414yescomplemented
17C3×#3724—21no—
18C2×C8×C3⋊D3—31no—
19C3×D3×C4⋊C4—32no—
20#63900—31no—
21#63900D311yescharacteristic
22D3×C4⋊Dic3—32no—
23C2×D3×C3⋊C8—32no—
24C2×D3×C3⋊C8—61no—
25#63634—32no—
26C3×#818—32no—
27C3×#807—32no—
28#63576—32no—
29#63576—32no—
30#63576—62no—
31#63575D312yescomplemented
32C22×C3×C3⋊D3D411yescharacteristic
33C2×C33×C4C2311yescharacteristic
34C2×C3×C3⋊Dic3—22no—
35C2×C3×C3⋊Dic3D411yescharacteristic
36C3×C4×C3⋊D3C2×C412yes—
37C3×C4×C3⋊D3Q812yes—
38C2×C32×Dic3—22no—
39#6294—42no—
40C3×#624—42no—
41C3⋊(C3⋊(C3⋊C8))C2×C412yescomplemented
42C32×C3⋊C8C2×C412yescomplemented
43C2×#807—91no—
44D3×#107—92no—
45C2×C4×C3⋊D3C2×D311yescharacteristic
46C2×C3×C4×D3—32no—
47C2×C3×C4×D3—61no—
48C2×C3×C4×D3—62no—
49C2×#624—21no—
50C2×#624—61no—
51C2×D3×Dic3—62no—
52C2×C32×C8—31no—
53C32×C4⋊C4—32no—
54C4⋊(C3⋊Dic3)C2×D312yes—
55C2×C3⋊(C3⋊C8)—32no—
56C2×C3⋊(C3⋊C8)—31no—
57C2×C3⋊(C3⋊C8)—61no—
58C2×C3⋊(C3⋊C8)—62no—
59C2×C3⋊(C3⋊C8)—121no—
60C8×C3⋊D3—34no—
61C3×C4⋊Dic3—22no—
62C3×C4⋊Dic3—32no—
63C3×C4⋊Dic3—32no—
64C3×C4⋊Dic3—62no—
65C3×C4⋊Dic3C2×D312yes—
66C3×C4⋊Dic3C2×D312yes—
67C3×Dic3⋊C4—62no—
68C2×C3×C3⋊C8—21no—
69C2×C3×C3⋊C8C2×D312yes—
70#3724—21no—
71#3724—61no—
72Dic3⋊Dic3—62no—
73D3×C3⋊C8—38no—
74D3×C3⋊C8—64no—
75C22×C33C2×D411yescharacteristic
76C2×C3×C3⋊D3C4⋊C412yes—
77C2×C3×C3⋊D3D812yes—
78C2×C3×C3⋊D3Q1612yes—
79C33×C4C22×C411yescharacteristic
80C33×C4C2×Q811yescharacteristic
81C3×C3⋊Dic3—42no—
82C3×C3⋊Dic3C4⋊C412yes—
83C32×Dic3—42no—
84C2×C3×C4⋊C4—91no—
85C2×C4⋊Dic3—91no—
86C22×C3⋊C8—91no—
87C2×D3×C8—92no—
88C2×D3×C8—181no—
89D3×C4⋊C4—32no—
90D3×C4⋊C4—92no—
91C3×#107—92no—
92#818—32no—
93#807—32no—
94#807—92no—
95#807—92no—
96#807—92no—
97#807—182no—
98C22×C3⋊D3#6711yescharacteristic
99C22×C3×D3—32no—
100C22×C3×D3—61no—
101C2×C32×C4—21no—
102C2×C32×C4—22no—
103C2×C32×C4—41no—
104C2×C32×C4—62no—
105C2×C32×C4C22×D311yescharacteristic
106C2×C32×C4C22×D312yes—
107C2×C3⋊Dic3—22no—
108C2×C3⋊Dic3#6711yescharacteristic
109C4×C3⋊D3Dic612yes—
110C4×C3⋊D3C4×D312yes—
111C2×C3×Dic3—22no—
112C2×C3×Dic3—22no—
113C2×C3×Dic3—32no—
114C2×C3×Dic3—42no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—62no—
117C2×C3×Dic3—62no—
118C2×C3×Dic3—122no—
119C3×C4×D3—34no—
120C3×C4×D3—34no—
121C3×C4×D3—62no—
122C3×C4×D3—62no—
123C3×C4×D3—124no—
124#624—42no—
125#624—122no—
126D3×Dic3—124no—
127C32×C8—32no—
128C3⋊(C3⋊C8)—34no—
129C3⋊(C3⋊C8)—32no—
130C3⋊(C3⋊C8)—62no—
131C3⋊(C3⋊C8)—64no—
132C3⋊(C3⋊C8)—122no—
133C3×C3⋊C8—22no—
134C3×C3⋊C8C4×D314yescomplemented
135C2×#107—271noSylow
136C2×C33C2×C4⋊C411yescharacteristic
137C2×C33C2×D811yescharacteristic
138C2×C33C2×Q1611yescharacteristic
139C3×C3⋊D3#10714yescomplemented
140C22×C3×C4—91no—
141C22×C3×C4—181no—
142C22×Dic3—181no—
143C2×C4×D3—32no—
144C2×C4×D3—61no—
145C2×C4×D3—62no—
146C2×C4×D3—182no—
147C2×C3×C8—32no—
148C2×C3×C8—61no—
149C3×C4⋊C4—32no—
150C3×C4⋊C4—32no—
151C3×C4⋊C4—62no—
152C3×C4⋊C4—92no—
153C3×C4⋊C4—181no—
154C4⋊Dic3—32no—
155C4⋊Dic3—32no—
156C4⋊Dic3—62no—
157C4⋊Dic3—181no—
158C4⋊Dic3D3212yes—
159Dic3⋊C4—62no—
160Dic3⋊C4—182no—
161C2×C3⋊C8—92no—
162C2×C3⋊C8—91no—
163C2×C3⋊C8—94no—
164C2×C3⋊C8—182no—
165C2×C3⋊C8—181no—
166C2×C3⋊C8—361no—
167C2×C3⋊C8D3211yescharacteristic
168D3×C8—98no—
169D3×C8—184no—
170C22×C32—21no—
171C22×C32—22no—
172C22×C32—41no—
173C22×C32D3×D412yes—
174C22×C32C2×#6711yescharacteristic
175C2×C3⋊D3Dic3⋊C412yes—
176C2×C3⋊D3#21312yes—
177C2×C3⋊D3#21612yes—
178C2×C3×D3—34no—
179C2×C3×D3—34no—
180C2×C3×D3—34no—
181C2×C3×D3—62no—
182C2×C3×D3—62no—
183C2×C3×D3—62no—
184C32×C4—22no—
185C32×C4—21no—
186C32×C4—22no—
187C32×C4—21no—
188C32×C4—41no—
189C32×C4—41no—
190C32×C4—122no—
191C32×C4C2×Dic611yescharacteristic
192C32×C4C2×C4×D311yescharacteristic
193C32×C4C2×C4×D312yes—
194C32×C4D3×Q812yes—
195C3⋊Dic3—42no—
196C3⋊Dic3Dic3⋊C412yes—
197C3×Dic3—34no—
198C3×Dic3—42no—
199C3×Dic3—42no—
200C3×Dic3—62no—
201C3×Dic3—82no—
202C3×Dic3—122no—
203C3×Dic3—122no—
204C3×Dic3—242no—
205C22×C8—271no—
206C2×C4⋊C4—91no—
207C2×C4⋊C4—271no—
208#107—92no—
209#107—272no—
210C33C2×#10711yescharacteristic, complemented, Sylow
211C23×C3—91no—
212C22×D3—32no—
213C22×D3—61no—
214C2×C3×C4—22no—
215C2×C3×C4—21no—
216C2×C3×C4—41no—
217C2×C3×C4—62no—
218C2×C3×C4—62no—
219C2×C3×C4—91no—
220C2×C3×C4—92no—
221C2×C3×C4—92no—
222C2×C3×C4—122no—
223C2×C3×C4—182no—
224C2×C3×C4—362no—
225C2×C3×C4C2×D3212yes—
226C2×C3×C4C2×D3211yescharacteristic
227C2×Dic3—22no—
228C2×Dic3—32no—
229C2×Dic3—61no—
230C2×Dic3—62no—
231C2×Dic3—62no—
232C2×Dic3—122no—
233C2×Dic3—362no—
234C4×D3—34no—
235C4×D3—34no—
236C4×D3—62no—
237C4×D3—62no—
238C4×D3—124no—
239C4×D3—364no—
240C3×C8—34no—
241C3×C8—62no—
242C3⋊C8—94no—
243C3⋊C8—92no—
244C3⋊C8—182no—
245C3⋊C8—184no—
246C3⋊C8—362no—
247C3⋊C8#62412yescomplemented
248C2×C32—21no—
249C2×C32—21no—
250C2×C32—22no—
251C2×C32—21no—
252C2×C32—22no—
253C2×C32—22no—
254C2×C32—41no—
255C2×C32—41no—
256C2×C32—41no—
257C2×C32D3×C4⋊C412yes—
258C2×C32D3×D812yes—
259C2×C32D3×Q1612yes—
260C2×C32C2×Dic3⋊C411yescharacteristic
261C2×C32C2×#21311yescharacteristic
262C2×C32C2×#21611yescharacteristic
263C3⋊D3#80714yescomplemented
264C3×D3—38no—
265C3×D3—64no—
266C22×C4—91no—
267C22×C4—181no—
268C22×C4—541no—
269C2×C8—31no—
270C2×C8—274no—
271C2×C8—271no—
272C4⋊C4—32no—
273C4⋊C4—92no—
274C4⋊C4—181no—
275C4⋊C4—541no—
276C22×C3—21no—
277C22×C3—22no—
278C22×C3—41no—
279C22×C3—92no—
280C22×C3—92no—
281C22×C3—92no—
282C22×C3#380211yescharacteristic
283C22×C3D3×#6712yes—
284C2×D3—34no—
285C2×D3—34no—
286C2×D3—34no—
287C2×D3—62no—
288C2×D3—62no—
289C2×D3—62no—
290C3×C4—21no—
291C3×C4—22no—
292C3×C4—21no—
293C3×C4—22no—
294C3×C4—41no—
295C3×C4—41no—
296C3×C4—92no—
297C3×C4—122no—
298C3×C4—122no—
299C3×C4—242no—
300C3×C4—362no—
301C3×C4D3×Dic612yes—
302C3×C4#379711yescharacteristic
303C3×C4C4×D3212yes—
304C3×C4C2×#62411yescharacteristic
305Dic3—34no—
306Dic3—42no—
307Dic3—62no—
308Dic3—122no—
309Dic3—122no—
310Dic3—242no—
311C32—22no—
312C32—21no—
313C32—41no—
314C32D3×#10712yescomplemented
315C32C2×#80711yescharacteristic, complemented
316C23—91no—
317C2×C4—62no—
318C2×C4—91no—
319C2×C4—92no—
320C2×C4—92no—
321C2×C4—182no—
322C2×C4—362no—
323C2×C4—1082no—
324C2×C4D3311yescharacteristic
325C8—32no—
326C8—272no—
327C2×C3—22no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3—22no—
331C2×C3—21no—
332C2×C3—22no—
333C2×C3—41no—
334C2×C3—41no—
335C2×C3—41no—
336C2×C3—94no—
337C2×C3#6379711yescharacteristic
338C2×C3#6380211yescharacteristic
339C2×C3D3×Dic3⋊C412yes—
340C2×C3#6390011yescharacteristic
341C2×C3D3×#21312yes—
342C2×C3D3×#21612yes—
343D3—38no—
344D3—64no—
345C22—92no—
346C22—92no—
347C22—92no—
348C22#8895311yescharacteristic
349C4—92no—
350C4—122no—
351C4—362no—
352C4D3×#62411yescharacteristic
353C4#8895711yescharacteristic
354C3—21no—
355C3—22no—
356C3—41no—
357C3D3×#80712yescomplemented
358C3#1059068711yescharacteristic, complemented
359C2—94no—
360C2#1172283111yescharacteristic
361C2#1172284711yescharacteristic
362C2#1172291111yescharacteristic
subgroupG/H
centerC22#88953
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypocenterC33C2×#107
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722056C2semidirect
5. C3×#63575C2semidirect
16. C3⋊D3×C3⋊C8C4semidirect
31. #63575D3semidirect
41. C3⋊(C3⋊(C3⋊C8))C2×C4semidirect
42. C32×C3⋊C8C2×C4semidirect
134. C3×C3⋊C8C4×D3semidirect
139. C3×C3⋊D3#107semidirect
210. C33C2×#107semidirect
247. C3⋊C8#624semidirect
263. C3⋊D3#807semidirect
314. C32D3×#107semidirect
315. C32C2×#807semidirect
357. C3D3×#807semidirect
358. C3#10590687semidirect

Maximal subgroups

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

subgroupindex
#117236892
C2×C3⋊D3×C3⋊C82
C3×#639002
#117220562
C3×#635752
#105906873
D3×#8073