Finite Group Databasebeta

← #13166623

Subgroups — #13166623

Order 1728.

1144 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 1144 conjugacy classes of subgroups — too many to draw. Only its 94 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×#6289C211yesmaximal, characteristic
2C2×#88924C212yesmaximal, complemented
3#11723112C211yesmaximal, characteristic, complemented
4C3×C22⋊#622C211yesmaximal, characteristic, complemented
5#11722726C212yesmaximal, complemented
6#10593800—31nomaximal
7C23×C3⋊(C3⋊D3)C411yescharacteristic
8C22×C3×C3⋊Dic3C2211yescharacteristic
9C22×C32×Dic3C411yescharacteristic
10C2×#6289—24no—
11C2×#6289C411yescharacteristic
12C2×#6289C411yescharacteristic
13C2×#6289C2211yescharacteristic
14C2×#6289C2211yescharacteristic
15C2×C3⋊#622C2212yescomplemented
16C2×C3×#622C2212yescomplemented
17#88924—24no—
18C22×C4×C3⋊D3—31no—
19C22×#624—62no—
20C2×#3787—32no—
21C22⋊#622—31no—
22C22⋊#622D311yescharacteristic, complemented
23#63778—32no—
24C23×C33C2×C411yescharacteristic
25C22×C3⋊(C3⋊D3)—22no—
26C22×C3⋊(C3⋊D3)—22no—
27C22×C3⋊(C3⋊D3)C814yescomplemented
28C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
29C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
30C2×C3×C3⋊Dic3—22no—
31C2×C3×C3⋊Dic3C2×C411yescharacteristic
32C2×C3×C3⋊Dic3C2311yescharacteristic
33C2×C32×Dic3—22no—
34C2×C32×Dic3C2×C411yescharacteristic
35C2×C32×Dic3C2×C411yescharacteristic
36#6289—22no—
37#6289—22no—
38#6289—22no—
39#6289D414yes—
40C3⋊#622—22no—
41C3×#622—22no—
42D3×C22⋊C8—91nomaximal
43C23×C3⋊D3—31no—
44C23×C3⋊D3—62no—
45C23×C3⋊D3—122no—
46C22×C32×C4—31no—
47C22×C3⋊Dic3C2×D311yescharacteristic
48C2×C4×C3⋊D3—31no—
49C2×C4×C3⋊D3—31no—
50C2×C4×C3⋊D3—31no—
51C2×C4×C3⋊D3—31no—
52C2×C4×C3⋊D3—64no—
53C22×C3×Dic3—22no—
54C22×C3×Dic3—62no—
55C2×#624—62no—
56C2×#624—62no—
57C2×#624—62no—
58C2×#624—62no—
59C2×#624—616no—
60C2×#622—32no—
61C2×#622C2×D312yescomplemented
62#3787—64no—
63C22×C33—21no—
64C22×C33—21no—
65C22×C33C2×C812yescomplemented
66C22×C33C22×C411yescharacteristic
67C2×C3⋊(C3⋊D3)—21no—
68C2×C3⋊(C3⋊D3)—21no—
69C2×C3⋊(C3⋊D3)—28no—
70C2×C3⋊(C3⋊D3)C22⋊C412yes—
71C2×C3⋊(C3⋊D3)C2×C812yes—
72C2×C3⋊(C3⋊D3)C2×C812yescomplemented
73C2×C3⋊(C3⋊D3)M1612yes—
74C3×C3⋊Dic3—21no—
75C3×C3⋊Dic3C2×D412yes—
76C32×Dic3—21no—
77C32×Dic3C22⋊C412yes—
78C22×C4×D3—91no—
79C22×C4×D3—182no—
80C2×D3×C8—92no—
81C3×C22⋊C8—91no—
82#830—91no—
83(C2×D3)⋊C8—92no—
84C23×C32—22no—
85C23×C32—42no—
86C23×C32C4×D311yescharacteristic
87C22×C3⋊D3—31no—
88C22×C3⋊D3—34no—
89C22×C3⋊D3—31no—
90C22×C3⋊D3—68no—
91C22×C3⋊D3—62no—
92C22×C3⋊D3—62no—
93C22×C3⋊D3—62no—
94C22×C3⋊D3—62no—
95C22×C3⋊D3—68no—
96C22×C3⋊D3—68no—
97C22×C3⋊D3—128no—
98C22×C3⋊D3—128no—
99C22×C3⋊D3—128no—
100C22×C3⋊D3—122no—
101C22×C3⋊D3—122no—
102C2×C32×C4—31no—
103C2×C32×C4—31no—
104C2×C32×C4—62no—
105C2×C3⋊Dic3—22no—
106C2×C3⋊Dic3C4×D311yescharacteristic
107C2×C3⋊Dic3C22×D311yescharacteristic
108C4×C3⋊D3—34no—
109C4×C3⋊D3—62no—
110C4×C3⋊D3—62no—
111C4×C3⋊D3—62no—
112C2×C3×Dic3—28no—
113C2×C3×Dic3—22no—
114C2×C3×Dic3—22no—
115C2×C3×Dic3—62no—
116C2×C3×Dic3—62no—
117C2×C3×Dic3—68no—
118#624—68no—
119#624—68no—
120#624—68no—
121#624—68no—
122#622—22no—
123#622—62no—
124C2×C22⋊C8—271noSylow
125C2×C33—22no—
126C2×C33C2×C22⋊C411yescharacteristic
127C2×C33C22×C811yescharacteristic
128C2×C33C2×M1611yescharacteristic
129C3⋊(C3⋊D3)—22no—
130C3⋊(C3⋊D3)C22⋊C814yescomplemented
131C23×D3—91no—
132C23×D3—182no—
133C23×D3—362no—
134C22×C3×C4—62no—
135C22×C3×C4—91no—
136C22×Dic3—62no—
137C22×Dic3#15711yescharacteristic
138C2×C4×D3—91no—
139C2×C4×D3—91no—
140C2×C4×D3—91no—
141C2×C4×D3—91no—
142C2×C4×D3—182no—
143C2×C4×D3—182no—
144C2×C4×D3—182no—
145C2×C4×D3—182no—
146C2×C4×D3—184no—
147C2×C4×D3—1816no—
148C2×C3×C8—92no—
149C2×C3⋊C8—92no—
150D3×C8—184no—
151C22×C32—21no—
152C22×C32—21no—
153C22×C32—24no—
154C22×C32—24no—
155C22×C32—22no—
156C22×C32—24no—
157C22×C32—44no—
158C22×C32—44no—
159C22×C32—44no—
160C22×C32—42no—
161C22×C32D3×C812yescomplemented
162C22×C32C2×C4×D311yescharacteristic
163C2×C3⋊D3—32no—
164C2×C3⋊D3—32no—
165C2×C3⋊D3—32no—
166C2×C3⋊D3—32no—
167C2×C3⋊D3—64no—
168C2×C3⋊D3—64no—
169C2×C3⋊D3—64no—
170C2×C3⋊D3—64no—
171C2×C3⋊D3—68no—
172C2×C3⋊D3—61no—
173C2×C3⋊D3—61no—
174C2×C3⋊D3—64no—
175C2×C3⋊D3—632no—
176C2×C3⋊D3—64no—
177C2×C3⋊D3—124no—
178C2×C3⋊D3—124no—
179C2×C3⋊D3—1232no—
180C2×C3⋊D3—124no—
181C2×C3⋊D3—124no—
182C2×C3⋊D3—124no—
183C2×C3⋊D3—124no—
184C32×C4—32no—
185C32×C4—61no—
186C3⋊Dic3—21no—
187C3⋊Dic3D3×D412yes—
188C3×Dic3—24no—
189C3×Dic3—24no—
190C3×Dic3—64no—
191C3×Dic3—64no—
192C23×C4—271no—
193C22×C8—272no—
194C22⋊C8—91no—
195C22⋊C8—272no—
196C22⋊C8—271no—
197C33C2×C22⋊C811yescharacteristic, complemented, Sylow
198C23×C3—22no—
199C23×C3—42no—
200C23×C3C2×#15711yescharacteristic
201C22×D3—91no—
202C22×D3—91no—
203C22×D3—94no—
204C22×D3—182no—
205C22×D3—182no—
206C22×D3—188no—
207C22×D3—182no—
208C22×D3—182no—
209C22×D3—188no—
210C22×D3—188no—
211C22×D3—362no—
212C22×D3—362no—
213C22×D3—368no—
214C22×D3—368no—
215C22×D3—368no—
216C2×C3×C4—62no—
217C2×C3×C4—62no—
218C2×C3×C4—68no—
219C2×C3×C4—91no—
220C2×C3×C4—91no—
221C2×C3×C4—182no—
222C2×Dic3—22no—
223C2×Dic3—62no—
224C2×Dic3—62no—
225C2×Dic3—68no—
226C2×Dic3C2×#15711yescharacteristic
227C2×Dic3C2×#15711yescharacteristic
228C4×D3—94no—
229C4×D3—188no—
230C4×D3—182no—
231C4×D3—182no—
232C4×D3—188no—
233C4×D3—188no—
234C4×D3—188no—
235C4×D3—182no—
236C3×C8—182no—
237C3⋊C8—182no—
238C2×C32—22no—
239C2×C32—28no—
240C2×C32—22no—
241C2×C32—22no—
242C2×C32—22no—
243C2×C32—48no—
244C2×C32—42no—
245C2×C32—42no—
246C2×C32—42no—
247C2×C32D3×C22⋊C411yescharacteristic
248C2×C32C2×D3×C811yescharacteristic
249C2×C32D3×M1611yescharacteristic
250C3⋊D3—34no—
251C3⋊D3—68no—
252C3⋊D3—62no—
253C3⋊D3—68no—
254C3⋊D3—128no—
255C3⋊D3—128no—
256C24—271no—
257C22×C4—31no—
258C22×C4—91no—
259C22×C4—271no—
260C22×C4—271no—
261C22×C4—271no—
262C22×C4—271no—
263C22×C4—544no—
264C2×C8—92no—
265C2×C8—272no—
266C2×C8—544no—
267C22×C3—24no—
268C22×C3—24no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—22no—
272C22×C3—24no—
273C22×C3—44no—
274C22×C3—44no—
275C22×C3—42no—
276C22×C3—44no—
277C22×C3#378712yescomplemented
278C22×C3C22×#15711yescharacteristic
279C2×D3—92no—
280C2×D3—92no—
281C2×D3—92no—
282C2×D3—92no—
283C2×D3—184no—
284C2×D3—184no—
285C2×D3—184no—
286C2×D3—184no—
287C2×D3—188no—
288C2×D3—184no—
289C2×D3—1832no—
290C2×D3—184no—
291C2×D3—181no—
292C2×D3—181no—
293C2×D3—364no—
294C2×D3—364no—
295C2×D3—364no—
296C2×D3—364no—
297C2×D3—3632no—
298C2×D3—364no—
299C2×D3—364no—
300C3×C4—64no—
301C3×C4—64no—
302C3×C4—92no—
303C3×C4—181no—
304Dic3—21no—
305Dic3—64no—
306Dic3—64no—
307Dic3C22⋊#15712yes—
308C32—22no—
309C32—42no—
310C32D3×C22⋊C811yescharacteristic, complemented
311C23—271no—
312C23—271no—
313C23—274no—
314C23—542no—
315C23—542no—
316C23D3×#15711yescharacteristic
317C2×C4—31no—
318C2×C4—31no—
319C2×C4—62no—
320C2×C4—91no—
321C2×C4—91no—
322C2×C4—182no—
323C2×C4—274no—
324C2×C4—542no—
325C2×C4—542no—
326C2×C4—542no—
327C8—182no—
328C8—542no—
329C2×C3—28no—
330C2×C3—22no—
331C2×C3—22no—
332C2×C3—22no—
333C2×C3—22no—
334C2×C3—48no—
335C2×C3—42no—
336C2×C3—42no—
337C2×C3—42no—
338C2×C3C2×#378711yescharacteristic
339C2×C3#6428311yescharacteristic
340C2×C3C2×C22⋊#15711yescharacteristic
341D3—94no—
342D3—188no—
343D3—188no—
344D3—182no—
345D3—368no—
346D3—368no—
347C22—21no—
348C22—21no—
349C22—272no—
350C22—272no—
351C22—272no—
352C22—272no—
353C22—548no—
354C22—541no—
355C22—541no—
356C22#8892412yescomplemented
357C22C2×D3×#15711yescharacteristic
358C4—32no—
359C4—61no—
360C4—92no—
361C4—181no—
362C3—22no—
363C3—42no—
364C3#1059380011yescharacteristic, complemented
365C2—22no—
366C2—274no—
367C2—542no—
368C2C2×#8892411yescharacteristic
369C2#1172426911yescharacteristic
370C2D3×C22⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutatorC2×C33C22×C8
FittingC23×C33C2×C4
socleC22×C33C22×C4
prime radicalC23×C3⋊(C3⋊D3)C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC23D3×#157
hypocenterC33C2×C22⋊C8
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#88924C2semidirect
3. #11723112C2semidirect
4. C3×C22⋊#622C2semidirect
5. #11722726C2semidirect
15. C2×C3⋊#622C22semidirect
16. C2×C3×#622C22semidirect
22. C22⋊#622D3semidirect
27. C22×C3⋊(C3⋊D3)C8semidirect
61. C2×#622C2×D3semidirect
65. C22×C33C2×C8semidirect
72. C2×C3⋊(C3⋊D3)C2×C8semidirect
130. C3⋊(C3⋊D3)C22⋊C8semidirect
161. C22×C32D3×C8semidirect
197. C33C2×C22⋊C8semidirect
277. C22×C3#3787semidirect
310. C32D3×C22⋊C8semidirect
356. C22#88924semidirect
364. C3#10593800semidirect

Maximal subgroups

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

subgroupindex
C22×#62892
C2×#889242
#117231122
C3×C22⋊#6222
#117227262
#105938003
D3×C22⋊C89