Finite Group Databasebeta

← #13166339

Subgroups — #13166339

Order 1728.

960 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 960 conjugacy classes of subgroups — too many to draw. Only its 88 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×C4×D3×C3⋊D3C211yesmaximal, characteristic
2C2×#88989C211yesmaximal, characteristic, complemented
3C2×C3×#3788C211yesmaximal, characteristic, complemented
4#11722726C212yesmaximal, complemented
5(C2×D3)⋊#622C212yesmaximal, complemented
6#10593803—31nomaximal
7C22×D3×C3⋊D3C411yescharacteristic
8C2×C4×C3⋊(C3⋊D3)C411yescharacteristic
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C32×C4×D3C411yescharacteristic
11C2×#6289C2211yescharacteristic
12C2×Dic3×C3⋊D3C411yescharacteristic
13C2×D3×C3⋊Dic3C2211yescharacteristic
14C4×D3×C3⋊D3—22no—
15C4×D3×C3⋊D3—22no—
16C2×C3⋊#622C2212yescomplemented
17#88989—22no—
18C2×C3×#622C2212yescomplemented
19C3×#3788—22no—
20C22×C4×C3⋊D3—31no—
21C2×C4×D32—62no—
22C2×#3788—31no—
23C2×#3788D311yescharacteristic, complemented
24C22⋊#622—32no—
25#63778—32no—
26C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
27C22×C3×C3⋊D3C2×C411yescharacteristic
28C22×C32×D3C2×C411yescharacteristic
29C2×D3×C3⋊D3—22no—
30C2×D3×C3⋊D3—22no—
31C2×D3×C3⋊D3—22no—
32C2×C33×C4C2×C411yescharacteristic
33C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
34C4×C3⋊(C3⋊D3)—22no—
35C2×C3×C3⋊Dic3C2311yescharacteristic
36C3×C4×C3⋊D3D412yes—
37C3×C4×C3⋊D3D412yes—
38C2×C32×Dic3C2×C411yescharacteristic
39C32×C4×D3—22no—
40#6289—21no—
41#6289—21no—
42Dic3×C3⋊D3—22no—
43D3×C3⋊Dic3—21no—
44D3×C3⋊Dic3—21no—
45C3⋊#622—22no—
46C3×#622—22no—
47#5126—91nomaximal
48C23×C3⋊D3—31no—
49C22×D32—62no—
50C22×C32×C4—31no—
51C22×C3⋊Dic3—31no—
52C2×C4×C3⋊D3—31no—
53C2×C4×C3⋊D3—31no—
54C2×C4×C3⋊D3—31no—
55C2×C4×C3⋊D3—62no—
56C2×C4×C3⋊D3—62no—
57C2×C4×C3⋊D3—62no—
58C2×C4×C3⋊D3—122no—
59C2×C4×C3⋊D3C2×D311yescharacteristic
60C2×C3×C4×D3—22no—
61C2×C3×C4×D3—62no—
62C2×#624—62no—
63C2×D3×Dic3—62no—
64C2×D3×Dic3—62no—
65C4×D32—68no—
66C4×D32—68no—
67C2×#622—32no—
68C2×#622C2×D312yescomplemented
69#3788—22no—
70#3788—62no—
71C22×C33C22×C411yescharacteristic
72C2×C3⋊(C3⋊D3)—21no—
73C2×C3⋊(C3⋊D3)—21no—
74C2×C3⋊(C3⋊D3)M1612yescomplemented
75C2×C3×C3⋊D3—21no—
76C2×C3×C3⋊D3—21no—
77C2×C3×C3⋊D3C22⋊C412yes—
78C2×C32×D3—21no—
79C2×C32×D3—21no—
80C2×C32×D3M1612yescomplemented
81D3×C3⋊D3—28no—
82C33×C4C22⋊C412yes—
83C3⋊(C3⋊Dic3)—21no—
84C3×C3⋊Dic3C2×D411yescharacteristic
85C3×C3⋊Dic3C2×D411yescharacteristic
86C32×Dic3—21no—
87C22×C4×D3—91no—
88C22×C4×D3—182no—
89C2×C3×M16—91no—
90C2×#208—91no—
91(C2×D3)⋊C8—92no—
92(C2×D3)⋊C8—92no—
93C23×C32—31no—
94C22×C3⋊D3—31no—
95C22×C3⋊D3—62no—
96C22×C3⋊D3—62no—
97C22×C3⋊D3—62no—
98C22×C3⋊D3—62no—
99C22×C3⋊D3—122no—
100C22×C3⋊D3C4×D311yescharacteristic
101C22×C3×D3—22no—
102C22×C3×D3—62no—
103C2×D32—68no—
104C2×D32—68no—
105C2×D32—68no—
106C2×C32×C4—22no—
107C2×C32×C4—31no—
108C2×C32×C4—42no—
109C2×C32×C4—62no—
110C2×C32×C4C4×D311yescharacteristic
111C2×C3⋊Dic3—31no—
112C2×C3⋊Dic3—62no—
113C2×C3⋊Dic3—61no—
114C2×C3⋊Dic3—61no—
115C2×C3⋊Dic3—122no—
116C2×C3⋊Dic3C22×D311yescharacteristic
117C4×C3⋊D3—62no—
118C4×C3⋊D3—62no—
119C4×C3⋊D3—61no—
120C4×C3⋊D3—61no—
121C4×C3⋊D3—68no—
122C4×C3⋊D3—128no—
123C4×C3⋊D3D1212yes—
124C4×C3⋊D3#6712yes—
125C2×C3×Dic3—22no—
126C2×C3×Dic3—62no—
127C3×C4×D3—28no—
128C3×C4×D3—64no—
129C3×C4×D3—64no—
130#624—64no—
131#624—64no—
132D3×Dic3—68no—
133D3×Dic3—64no—
134D3×Dic3—64no—
135#622—22no—
136#622—62no—
137#407—271noSylow
138C2×C33C2×C22⋊C411yescharacteristic
139C2×C33C2×M1611yescharacteristic
140C2×C33C2×M1611yescharacteristic
141C3⋊(C3⋊D3)—22no—
142C3×C3⋊D3—22no—
143C32×D3—22no—
144C23×D3—91no—
145C23×D3—182no—
146C22×C3×C4—62no—
147C22×C3×C4—91no—
148C22×Dic3—91no—
149C22×Dic3—182no—
150C2×C4×D3—62no—
151C2×C4×D3—91no—
152C2×C4×D3—91no—
153C2×C4×D3—91no—
154C2×C4×D3—182no—
155C2×C4×D3—182no—
156C2×C4×D3—182no—
157C2×C4×D3—182no—
158C2×C4×D3—182no—
159C2×C4×D3—188no—
160C2×C4×D3—188no—
161C2×C4×D3—362no—
162C2×C4×D3#15711yescharacteristic
163C3×M16—182no—
164C2×C3×C8—92no—
165#208—182no—
166C2×C3⋊C8—92no—
167C22×C32—22no—
168C22×C32—32no—
169C22×C32—42no—
170C22×C32—61no—
171C22×C32—61no—
172C22×C32C2×C4×D311yescharacteristic
173C2×C3⋊D3—21no—
174C2×C3⋊D3—21no—
175C2×C3⋊D3—32no—
176C2×C3⋊D3—64no—
177C2×C3⋊D3—68no—
178C2×C3⋊D3—61no—
179C2×C3⋊D3—61no—
180C2×C3⋊D3—64no—
181C2×C3⋊D3—64no—
182C2×C3⋊D3—124no—
183C2×C3⋊D3—124no—
184C2×C3⋊D3—124no—
185C2×C3⋊D3(C2×D3)⋊C412yes—
186C2×C3×D3—24no—
187C2×C3×D3—24no—
188C2×C3×D3—24no—
189C2×C3×D3—64no—
190C2×C3×D3—64no—
191C2×C3×D3—64no—
192D32—632no—
193C32×C4—24no—
194C32×C4—44no—
195C32×C4—61no—
196C32×C4(C2×D3)⋊C412yes—
197C3⋊Dic3—61no—
198C3⋊Dic3—64no—
199C3⋊Dic3—124no—
200C3⋊Dic3C2×D1211yescharacteristic
201C3⋊Dic3C2×#6711yescharacteristic
202C3×Dic3—24no—
203C3×Dic3—62no—
204C3×Dic3—62no—
205C23×C4—271no—
206C2×M16—91no—
207C2×M16—271no—
208C22⋊C8—272no—
209C22⋊C8—272no—
210C33#40711yescharacteristic, complemented, Sylow
211C23×C3—62no—
212C23×C3—91no—
213C22×D3—62no—
214C22×D3—91no—
215C22×D3—182no—
216C22×D3—182no—
217C22×D3—182no—
218C22×D3—188no—
219C22×D3—188no—
220C22×D3—188no—
221C22×D3—182no—
222C22×D3—362no—
223C22×D3C2×#15711yescharacteristic
224C2×C3×C4—22no—
225C2×C3×C4—42no—
226C2×C3×C4—62no—
227C2×C3×C4—68no—
228C2×C3×C4—91no—
229C2×C3×C4—92no—
230C2×C3×C4—92no—
231C2×C3×C4C2×#15711yescharacteristic
232C2×Dic3—62no—
233C2×Dic3—91no—
234C2×Dic3—182no—
235C2×Dic3—182no—
236C2×Dic3—184no—
237C2×Dic3—184no—
238C2×Dic3—362no—
239C2×Dic3C2×#15711yescharacteristic
240C4×D3—22no—
241C4×D3—64no—
242C4×D3—64no—
243C4×D3—181no—
244C4×D3—181no—
245C4×D3—184no—
246C4×D3—184no—
247C4×D3—188no—
248C4×D3—188no—
249C4×D3—181no—
250C4×D3—181no—
251C4×D3—182no—
252C4×D3—368no—
253C3×C8—182no—
254C3⋊C8—182no—
255C2×C32—22no—
256C2×C32—22no—
257C2×C32—22no—
258C2×C32—42no—
259C2×C32—42no—
260C2×C32—42no—
261C2×C32—62no—
262C2×C32D3×M1611yescharacteristic
263C2×C32D3×M1611yescharacteristic
264C2×C32C2×(C2×D3)⋊C411yescharacteristic
265C3⋊D3—22no—
266C3⋊D3—62no—
267C3⋊D3—68no—
268C3⋊D3—128no—
269C3×D3—28no—
270C3×D3—68no—
271C24—271no—
272C22×C4—31no—
273C22×C4—91no—
274C22×C4—271no—
275C22×C4—271no—
276C22×C4—271no—
277C22×C4—271no—
278C22×C4—542no—
279C22×C4—542no—
280M16—182no—
281M16—542no—
282C2×C8—92no—
283C2×C8—272no—
284C22×C3—22no—
285C22×C3—42no—
286C22×C3—64no—
287C22×C3—64no—
288C22×C3—64no—
289C22×C3—92no—
290C22×C3—181no—
291C22×C3—181no—
292C22×C3C22×#15711yescharacteristic
293C2×D3—21no—
294C2×D3—21no—
295C2×D3—64no—
296C2×D3—64no—
297C2×D3—64no—
298C2×D3—92no—
299C2×D3—184no—
300C2×D3—188no—
301C2×D3—1832no—
302C2×D3—184no—
303C2×D3—184no—
304C2×D3—181no—
305C2×D3—181no—
306C2×D3—364no—
307C2×D3—364no—
308C2×D3—364no—
309C2×D3#378812yescomplemented
310C3×C4—24no—
311C3×C4—44no—
312C3×C4—64no—
313C3×C4—91no—
314C3×C4—91no—
315C3×C4C22⋊#15712yes—
316Dic3—21no—
317Dic3—62no—
318Dic3—62no—
319Dic3—184no—
320Dic3—181no—
321Dic3—364no—
322C32—22no—
323C32—42no—
324C32#512611yescharacteristic, complemented
325C23—31no—
326C23—91no—
327C23—271no—
328C23—542no—
329C23—542no—
330C23—542no—
331C2×C4—31no—
332C2×C4—62no—
333C2×C4—91no—
334C2×C4—92no—
335C2×C4—92no—
336C2×C4—271no—
337C2×C4—541no—
338C2×C4—541no—
339C2×C4—542no—
340C2×C4—541no—
341C2×C4—541no—
342C2×C4—542no—
343C2×C4D3×#15711yescharacteristic
344C8—182no—
345C8—542no—
346C2×C3—22no—
347C2×C3—22no—
348C2×C3—22no—
349C2×C3—42no—
350C2×C3—42no—
351C2×C3—42no—
352C2×C3—68no—
353C2×C3—182no—
354C2×C3C2×#378811yescharacteristic
355C2×C3#6428311yescharacteristic
356C2×C3C2×C22⋊#15711yescharacteristic
357D3—22no—
358D3—68no—
359D3—188no—
360D3—182no—
361D3—368no—
362C22—32no—
363C22—61no—
364C22—61no—
365C22—92no—
366C22—181no—
367C22—181no—
368C22—272no—
369C22—548no—
370C22—541no—
371C22—541no—
372C22C2×D3×#15711yescharacteristic
373C4—61no—
374C4—91no—
375C4—91no—
376C4—541no—
377C4(C2×D3)⋊#15712yes—
378C3—22no—
379C3—42no—
380C3#1059380311yescharacteristic, complemented
381C2—62no—
382C2—182no—
383C2—542no—
384C2D3×#378811yescharacteristic
385C2#1172425511yescharacteristic
386C2C2×(C2×D3)⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC2×C33×C4C2×C4
prime radicalC22×D3×C3⋊D3C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D3×#157
hypocenterC33#407
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88989C2semidirect
3. C2×C3×#3788C2semidirect
4. #11722726C2semidirect
5. (C2×D3)⋊#622C2semidirect
16. C2×C3⋊#622C22semidirect
18. C2×C3×#622C22semidirect
23. C2×#3788D3semidirect
68. C2×#622C2×D3semidirect
74. C2×C3⋊(C3⋊D3)M16semidirect
80. C2×C32×D3M16semidirect
210. C33#407semidirect
309. C2×D3#3788semidirect
324. C32#5126semidirect
380. C3#10593803semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D3×C3⋊D32
C2×#889892
C2×C3×#37882
#117227262
(C2×D3)⋊#6222
#105938033
#51269