Finite Group Databasebeta

← #13167422

Subgroups — #13167422

Order 1728.

582 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 75 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×#89040C211yesmaximal, characteristic
2C3×#64286C211yesmaximal, characteristic
3#11723656C211yesmaximal, characteristic
4#11722784C214yesmaximal, complemented
5#10593321—31nomaximal
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C2×C3×#634—21no—
8C2×#6296C412yescomplemented
9C2×#6294—21no—
10#89040—21no—
11#89040—21no—
12#89040C412yescomplemented
13#89040C2212yes—
14C3×C22⋊#157—21no—
15C3×C4⋊#157C2212yes—
16C3×C4×#157—21no—
17C3×#3748—21no—
18#88809—21no—
19#88808C2212yes—
20C2×#3797—31no—
21C2×D3×Dic6—61no—
22#64286D311yescharacteristic
23#63818—31no—
24D3×#208—61no—
25#63730—34no—
26C22×C3×C3⋊D3D411yescharacteristic
27C2×C3×#157—22no—
28C2×C33×C4D411yescharacteristic
29C2×C3×C3⋊Dic3—22no—
30C2×C3×C3⋊Dic3C2×C411yescharacteristic
31C3×C4×C3⋊D3C2×C411yescharacteristic
32C3×C4×C3⋊D3D411yescharacteristic
33C3×C4×C3⋊D3D411yescharacteristic
34C3×C4×C3⋊D3C2311yescharacteristic
35C3×#634—22no—
36C3×#634—22no—
37#6296—22no—
38#6296C2×C414yes—
39#6294—22no—
40#6294—22no—
41C3×C3⋊(C3⋊C8)—22no—
42#5004—91nomaximal
43C2×C4×C3⋊D3C2×D311yescharacteristic
44C2×#634—21no—
45C2×C3×C4×D3—61no—
46C2×C3×C4×D3—61no—
47C2×C3×Dic6—61no—
48C2×C3×Dic6—61no—
49C2×C3×Dic6—121no—
50C2×#627—32no—
51C2×#627—62no—
52C2×#624—61no—
53C2×D3×Dic3—62no—
54#3797—32no—
55#3797—32no—
56#3797—61no—
57#3797—61no—
58D3×Dic6—62no—
59D3×Dic6—62no—
60D3×Dic6—62no—
61D3×Dic6—62no—
62C22⋊#157—21no—
63C4⋊#157C2×D312yes—
64C4×#157—21no—
65#3748—21no—
66C3×#208—61no—
67C3×#208—61no—
68C3×#208—121no—
69#3712—61no—
70#3711—62no—
71#3710—32no—
72D3×C3⋊C8—62no—
73C22×C33C22⋊C411yescharacteristic
74C2×C3×C3⋊D3—22no—
75C2×C3×C3⋊D3C22⋊C411yescharacteristic
76C2×C3×C3⋊D3C2×D411yescharacteristic
77C3×#157—42no—
78C33×C4C22⋊C411yescharacteristic
79C33×C4C2×D411yescharacteristic
80C3×C3⋊Dic3—24no—
81C3×C3⋊Dic3C22⋊C411yescharacteristic
82C3×C3⋊Dic3C22×C411yescharacteristic
83C2×D3×Q8—181no—
84C22×Dic6—91no—
85C3×#117—91no—
86C2×#208—91no—
87D3×M16—181no—
88#810—94no—
89C22×C3⋊D3#6711yescharacteristic
90C22×C3×D3—61no—
91C22×C3×D3—61no—
92C2×#157—22no—
93C2×C32×C4—21no—
94C2×C32×C4—21no—
95C2×C32×C4—41no—
96C2×C32×C4—41no—
97C2×C32×C4D1211yescharacteristic
98C2×C3⋊Dic3—22no—
99C2×C3⋊Dic3C4×D311yescharacteristic
100C4×C3⋊D3C4×D311yescharacteristic
101C4×C3⋊D3D1211yescharacteristic
102C4×C3⋊D3#6711yescharacteristic
103C4×C3⋊D3C22×D311yescharacteristic
104#634—22no—
105#634—22no—
106C2×C3×Dic3—61no—
107C2×C3×Dic3—61no—
108C2×C3×Dic3—62no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—122no—
111C3×C4×D3—61no—
112C3×C4×D3—61no—
113C3×C4×D3—61no—
114C3×C4×D3—61no—
115C3×C4×D3—61no—
116C3×C4×D3—61no—
117C3×C4×D3—61no—
118C3×C4×D3—61no—
119C3×Dic6—62no—
120C3×Dic6—62no—
121C3×Dic6—62no—
122C3×Dic6—62no—
123C3×Dic6—122no—
124C3×Dic6—122no—
125#627—34no—
126#627—64no—
127#627—64no—
128#627—62no—
129#624—62no—
130#624—62no—
131D3×Dic3—64no—
132D3×Dic3—64no—
133C3⋊(C3⋊C8)—22no—
134C3×C3⋊C8—62no—
135C3×C3⋊C8—62no—
136C3×C3⋊C8—122no—
137#419—271noSylow
138C2×C33—21no—
139C2×C33C2×C22⋊C411yescharacteristic
140C3×C3⋊D3—21no—
141C3×C3⋊D3#13712yes—
142C2×C3×Q8—181no—
143C22×C3×C4—91no—
144C22×Dic3—181no—
145D3×Q8—182no—
146D3×Q8—182no—
147D3×Q8—182no—
148D3×Q8—182no—
149C2×C4×D3—61no—
150C2×C4×D3—61no—
151C2×C4×D3—182no—
152C2×Dic6—61no—
153C2×Dic6—61no—
154C2×Dic6—92no—
155C2×Dic6—92no—
156C2×Dic6—92no—
157C2×Dic6—121no—
158C2×Dic6—182no—
159C2×Dic6—181no—
160C2×Dic6—181no—
161C3×M16—181no—
162C3×C4⋊C4—92no—
163C3×C22⋊C4—181no—
164C3×C42—181no—
165#208—61no—
166#208—61no—
167#208—121no—
168#208—181no—
169C2×C3⋊C8—92no—
170ΓL(1,𝔽25)—182no—
171D3×C8—182no—
172C22×C32—21no—
173C22×C32—21no—
174C22×C32—41no—
175C22×C32—41no—
176C22×C32(C2×D3)⋊C411yescharacteristic
177C2×C3⋊D3—22no—
178C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
179C2×C3⋊D3C2×#6711yescharacteristic
180C2×C3×D3—61no—
181C2×C3×D3—61no—
182C2×C3×D3—61no—
183C2×C3×D3—61no—
184C2×C3×D3—64no—
185C2×C3×D3—122no—
186#157—42no—
187C32×C4—21no—
188C32×C4—21no—
189C32×C4—21no—
190C32×C4—21no—
191C32×C4—41no—
192C32×C4—41no—
193C32×C4—41no—
194C32×C4—41no—
195C32×C4(C2×D3)⋊C411yescharacteristic
196C32×C4C2×D1211yescharacteristic
197C3⋊Dic3—24no—
198C3⋊Dic3(C2×D3)⋊C411yescharacteristic
199C3⋊Dic3C2×C4×D311yescharacteristic
200C3×Dic3—61no—
201C3×Dic3—61no—
202C3×Dic3—61no—
203C3×Dic3—61no—
204C3×Dic3—64no—
205C3×Dic3—64no—
206C3×Dic3—124no—
207C22×Q8—271no—
208C2×M16—271no—
209#117—91no—
210#103—274no—
211C33#41911yescharacteristic, complemented, Sylow
212C23×C3—91no—
213C22×D3—61no—
214C22×D3—61no—
215C3×Q8—182no—
216C3×Q8—182no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—91no—
222C2×C3×C4—91no—
223C2×C3×C4—91no—
224C2×C3×C4—91no—
225C2×C3×C4—91no—
226C2×C3×C4—182no—
227C2×C3×C4—182no—
228C2×C3×C4GO+(4,𝔽2)11yescharacteristic
229C2×Dic3—61no—
230C2×Dic3—61no—
231C2×Dic3—62no—
232C2×Dic3—62no—
233C2×Dic3—122no—
234C2×Dic3—182no—
235C2×Dic3—182no—
236C4×D3—61no—
237C4×D3—61no—
238C4×D3—61no—
239C4×D3—61no—
240C4×D3—61no—
241C4×D3—61no—
242C4×D3—61no—
243C4×D3—61no—
244C4×D3—184no—
245C4×D3—184no—
246Dic6—62no—
247Dic6—62no—
248Dic6—62no—
249Dic6—62no—
250Dic6—94no—
251Dic6—122no—
252Dic6—122no—
253Dic6—184no—
254Dic6—182no—
255Dic6—184no—
256C3×C8—182no—
257C3⋊C8—62no—
258C3⋊C8—62no—
259C3⋊C8—122no—
260C2×C32—22no—
261C2×C32—21no—
262C2×C32—21no—
263C2×C32—21no—
264C2×C32—41no—
265C2×C32—41no—
266C2×C32—41no—
267C2×C32—42no—
268C2×C32—81no—
269C2×C32C2×(C2×D3)⋊C411yescharacteristic
270C3⋊D3—21no—
271C3⋊D3#95112yes—
272C3×D3—62no—
273C3×D3—62no—
274C3×D3—62no—
275C3×D3—121no—
276C2×Q8—181no—
277C2×Q8—272no—
278C2×Q8—272no—
279C2×Q8—272no—
280C2×Q8—541no—
281C2×Q8—541no—
282C22×C4—91no—
283C22×C4—541no—
284M16—181no—
285M16—541no—
286C2×C8—272no—
287C4⋊C4—92no—
288C22⋊C4—181no—
289C42—181no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3—91no—
295C22×C3—91no—
296C22×C3—182no—
297C22×C3D32⋊C411yescharacteristic
298C2×D3—61no—
299C2×D3—61no—
300C2×D3—61no—
301C2×D3—61no—
302C2×D3—64no—
303C2×D3—122no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—41no—
309C3×C4—41no—
310C3×C4—41no—
311C3×C4—41no—
312C3×C4—91no—
313C3×C4—91no—
314C3×C4—184no—
315C3×C4—362no—
316C3×C4D32⋊C411yescharacteristic
317C3×C4C2×GO+(4,𝔽2)11yescharacteristic
318Dic3—61no—
319Dic3—61no—
320Dic3—61no—
321Dic3—61no—
322Dic3—64no—
323Dic3—64no—
324Dic3—124no—
325C32—21no—
326C32—21no—
327C32—41no—
328C32—41no—
329C32#500411yescharacteristic, complemented
330C23—91no—
331Q8—182no—
332Q8—182no—
333Q8—274no—
334Q8—542no—
335C2×C4—91no—
336C2×C4—91no—
337C2×C4—91no—
338C2×C4—91no—
339C2×C4—91no—
340C2×C4—182no—
341C2×C4—182no—
342C2×C4—542no—
343C2×C4—542no—
344C2×C4#157⋊D311yescharacteristic
345C8—182no—
346C2×C3—22no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—41no—
351C2×C3—42no—
352C2×C3—41no—
353C2×C3—41no—
354C2×C3—81no—
355C2×C3—92no—
356C2×C3—181no—
357C2×C3C2×D32⋊C411yescharacteristic
358D3—62no—
359D3—62no—
360D3—62no—
361D3—121no—
362C22—91no—
363C22—91no—
364C22—182no—
365C22#8893911yescharacteristic
366C4—91no—
367C4—91no—
368C4—184no—
369C4—362no—
370C4#8893911yescharacteristic
371C4C2×#157⋊D311yescharacteristic
372C3—21no—
373C3—21no—
374C3—41no—
375C3—41no—
376C3#1059332111yescharacteristic, complemented
377C2—21no—
378C2—92no—
379C2—181no—
380C2C2×#8893911yescharacteristic
subgroupG/H
centerC2C2×#88939
commutatorC3×C3⋊Dic3C22×C4
FrattiniC4C2×#157⋊D3
FittingC2×C33×C4D4
socleC2×C33C2×C22⋊C4
prime radicalC22×C3×C3⋊D3D4
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4#157⋊D3
hypocenterC33#419
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722784C2semidirect
8. C2×#6296C4semidirect
12. #89040C4semidirect
211. C33#419semidirect
329. C32#5004semidirect
376. C3#10593321semidirect

Maximal subgroups

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

subgroupindex
C2×#890402
C3×#642862
#117236562
#117227842
#105933213
#50049