Finite Group Databasebeta

← #13166446

Subgroups — #13166446

Order 1728.

752 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 64 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×#89017C211yesmaximal, characteristic
2#11724261C212yesmaximal, complemented
3#11723099C211yesmaximal, characteristic, complemented
4C3×#63774C211yesmaximal, characteristic, complemented
5#11722728C212yesmaximal, complemented
6#10593832—31nomaximal
7C2×C4×C3⋊(C3⋊D3)C411yescharacteristic, complemented
8C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
9C2×C32×Dic6C411yescharacteristic, complemented
10C2×#6292C411yescharacteristic, complemented
11C2×#6292C2211yescharacteristic
12C2×Dic3×C3⋊D3C411yescharacteristic, complemented
13C2×Dic3×C3⋊D3C2211yescharacteristic
14#89017—22no—
15#89017—22no—
16#88994C2212yes—
17C3×C22⋊#157C2212yes—
18Dic3⋊#157—22no—
19Dic3×#157—22no—
20C2×#3832—31no—
21C2×#3796—62no—
22#64286—32no—
23#63774—31no—
24#63774—32no—
25#63774D311yescharacteristic, complemented
26C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
27C22×C3×C3⋊D3C2×C411yescharacteristic
28C22×C3×C3⋊D3C2311yescharacteristic
29C2×C3⋊#157—22no—
30C2×C3×#157—22no—
31C2×C33×C4C2×C411yescharacteristic
32C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
33C4×C3⋊(C3⋊D3)—21no—
34C4×C3⋊(C3⋊D3)—21no—
35C3×C4⋊(C3⋊D3)—22no—
36C2×C32×Dic3C2×C411yescharacteristic
37C2×C32×Dic3C2×C411yescharacteristic
38C32×Dic6—22no—
39#6292—22no—
40#6292—22no—
41Dic3×C3⋊D3—21no—
42Dic3×C3⋊D3—21no—
43Dic3×C3⋊D3D412yes—
44Dic3×C3⋊D3D412yes—
45#4740—91nomaximal
46C2×C32×Q8—31no—
47#3832—62no—
48#3832—62no—
49C2×C4⋊(C3⋊D3)—31no—
50C2×C4⋊(C3⋊D3)—31no—
51C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
52C2×C4×C3⋊D3—31no—
53C2×C4×C3⋊D3—31no—
54C2×C4×C3⋊D3—31no—
55C2×C4×C3⋊D3—62no—
56C2×C4×C3⋊D3—122no—
57C2×C3×D12—62no—
58C2×C3×Dic6—22no—
59C2×#626—62no—
60C2×#626—62no—
61C2×D3×Dic3—62no—
62C2×D3×Dic3—62no—
63#3796—68no—
64#3796—68no—
65C22⋊#157—32no—
66C22⋊#157C2×D312yes—
67C4⋊#157—62no—
68C4×#157—62no—
69C22×C33C22×C411yescharacteristic
70C2×C3⋊(C3⋊D3)—21no—
71C2×C3⋊(C3⋊D3)—41no—
72C2×C3×C3⋊D3—21no—
73C2×C3×C3⋊D3—41no—
74C2×C3×C3⋊D3—41no—
75C2×C3×C3⋊D3C2×D412yes—
76C3⋊#157—42no—
77C3×#157—42no—
78C33×C4—21no—
79C3⋊(C3⋊Dic3)C22⋊C411yescharacteristic
80C3⋊(C3⋊Dic3)C22⋊C411yescharacteristic
81C32×Dic3—21no—
82C32×Dic3C22⋊C412yes—
83C2×#239—182no—
84C2×#237—91no—
85#877—92no—
86C3×#99—91no—
87#834—91no—
88#806—92no—
89C22×C3⋊D3—31no—
90C22×C3⋊D3—62no—
91C22×C3⋊D3—122no—
92C22×C3⋊D3C4×D311yescharacteristic
93C22×C3⋊D3C22×D311yescharacteristic
94C22×C3×D3—62no—
95C22×C3×D3—62no—
96C2×#157—22no—
97C2×#157—62no—
98C32×Q8—62no—
99C2×C32×C4—22no—
100C2×C32×C4—31no—
101C2×C32×C4—31no—
102C2×C32×C4—42no—
103C2×C32×C4C4×D311yescharacteristic
104C2×C3⋊Dic3—31no—
105C2×C3⋊Dic3—62no—
106C2×C3⋊Dic3—122no—
107C4⋊(C3⋊D3)—22no—
108C4⋊(C3⋊D3)—62no—
109C4⋊(C3⋊D3)—62no—
110C4×C3⋊D3—32no—
111C4×C3⋊D3—32no—
112C4×C3⋊D3—61no—
113C4×C3⋊D3—61no—
114C4×C3⋊D3—61no—
115C4×C3⋊D3—61no—
116C4×C3⋊D3—64no—
117C4×C3⋊D3—64no—
118C4×C3⋊D3—124no—
119C4×C3⋊D3—124no—
120C2×C3×Dic3—22no—
121C2×C3×Dic3—22no—
122C3×D12—68no—
123C3×Dic6—28no—
124#626—68no—
125#626—68no—
126D3×Dic3—64no—
127D3×Dic3—64no—
128D3×Dic3—64no—
129D3×Dic3—64no—
130#410—271noSylow
131C2×C33—21no—
132C2×C33C2×C22⋊C411yescharacteristic
133C3⋊(C3⋊D3)—41no—
134C3×C3⋊D3—22no—
135C3×C3⋊D3—41no—
136C2×C3×Q8—62no—
137C2×C3×D4—91no—
138C2×#67—91no—
139C2×#67—91no—
140C22×Dic3—91no—
141C22×Dic3—91no—
142#239—188no—
143#239—188no—
144#237—182no—
145#237—182no—
146C2×D12—62no—
147C2×D12—182no—
148C2×D12—182no—
149C2×C4×D3—91no—
150C2×C4×D3—182no—
151C2×C4×D3—182no—
152C2×C4×D3—182no—
153C2×C4×D3—362no—
154C2×Dic6#15711yescharacteristic, complemented
155C3×C22⋊C4—92no—
156#217—92no—
157Dic3⋊C4—182no—
158C4×Dic3—182no—
159C22×C32—22no—
160C22×C32—42no—
161C22×C32C2×C4×D311yescharacteristic
162C2×C3⋊D3—21no—
163C2×C3⋊D3—41no—
164C2×C3⋊D3—41no—
165C2×C3⋊D3—61no—
166C2×C3⋊D3—64no—
167C2×C3⋊D3—124no—
168C2×C3⋊D3—124no—
169C2×C3⋊D3—128no—
170C2×C3⋊D3—121no—
171C2×C3⋊D3D3×D412yes—
172C2×C3×D3—64no—
173C2×C3×D3—64no—
174C2×C3×D3—124no—
175C2×C3×D3—124no—
176#157—42no—
177#157—122no—
178C32×C4—21no—
179C32×C4—24no—
180C32×C4—32no—
181C32×C4—44no—
182C32×C4—61no—
183C3⋊Dic3—31no—
184C3⋊Dic3—31no—
185C3⋊Dic3—62no—
186C3⋊Dic3—62no—
187C3⋊Dic3—122no—
188C3⋊Dic3—122no—
189C3×Dic3—24no—
190C3×Dic3—24no—
191C2×#41—271no—
192#117—272no—
193#99—91no—
194#99—272no—
195#99—271no—
196C33#41011yescharacteristic, complemented, Sylow
197C23×C3—91no—
198C23×C3—91no—
199C22×D3—62no—
200C22×D3—62no—
201C22×D3—91no—
202C22×D3—182no—
203C22×D3—362no—
204C3×Q8—68no—
205C3×D4—182no—
206C2×C3×C4—22no—
207C2×C3×C4—42no—
208C2×C3×C4—62no—
209C2×C3×C4—62no—
210C2×C3×C4—182no—
211C2×C3×C4C2×#15711yescharacteristic
212#67—182no—
213#67—182no—
214C2×Dic3—91no—
215C2×Dic3—92no—
216C2×Dic3—92no—
217C2×Dic3—182no—
218C2×Dic3—181no—
219C2×Dic3—181no—
220C2×Dic3—182no—
221C2×Dic3—362no—
222C2×Dic3C2×#15711yescharacteristic
223C2×Dic3C2×#15711yescharacteristic
224D12—68no—
225D12—188no—
226D12—188no—
227C4×D3—184no—
228C4×D3—184no—
229C4×D3—181no—
230C4×D3—181no—
231C4×D3—184no—
232C4×D3—184no—
233C4×D3—184no—
234C4×D3—184no—
235C4×D3—364no—
236C4×D3—364no—
237Dic6—22no—
238C2×C32—21no—
239C2×C32—24no—
240C2×C32—22no—
241C2×C32—44no—
242C2×C32—42no—
243C2×C32D3×C22⋊C411yescharacteristic
244C3⋊D3—22no—
245C3⋊D3—41no—
246C3⋊D3—124no—
247C3⋊D3—128no—
248C3⋊D3—121no—
249C3×D3—124no—
250C3×D3—124no—
251#41—542no—
252#41—542no—
253C2×Q8—31no—
254C2×D4—91no—
255C2×D4—271no—
256C2×D4—271no—
257C22×C4—271no—
258C22×C4—271no—
259C22×C4—271no—
260C4⋊C4—542no—
261C22⋊C4—92no—
262C22⋊C4—272no—
263C42—542no—
264C22×C3—22no—
265C22×C3—42no—
266C22×C3—92no—
267C22×C3—181no—
268C22×C3—361no—
269C22×C3—361no—
270C22×C3C22×#15711yescharacteristic
271C2×D3—64no—
272C2×D3—64no—
273C2×D3—124no—
274C2×D3—124no—
275C2×D3—184no—
276C2×D3—181no—
277C2×D3—364no—
278C2×D3—364no—
279C2×D3—368no—
280C2×D3—361no—
281C3×C4—24no—
282C3×C4—21no—
283C3×C4—44no—
284C3×C4—64no—
285C3×C4—64no—
286C3×C4—362no—
287Dic3—21no—
288Dic3—91no—
289Dic3—91no—
290Dic3—182no—
291Dic3—182no—
292Dic3—362no—
293Dic3—362no—
294Dic3—362no—
295Dic3C22⋊#15712yes—
296C32—22no—
297C32—42no—
298C32#474011yescharacteristic, complemented
299C23—91no—
300C23—91no—
301C23—271no—
302Q8—62no—
303D4—182no—
304D4—542no—
305D4—542no—
306C2×C4—31no—
307C2×C4—31no—
308C2×C4—182no—
309C2×C4—272no—
310C2×C4—272no—
311C2×C4—271no—
312C2×C4—541no—
313C2×C4—541no—
314C2×C4—541no—
315C2×C4—541no—
316C2×C4—542no—
317C2×C4D3×#15711yescharacteristic
318C2×C3—24no—
319C2×C3—21no—
320C2×C3—22no—
321C2×C3—44no—
322C2×C3—42no—
323C2×C3—182no—
324C2×C3—361no—
325C2×C3C2×C22⋊#15711yescharacteristic
326D3—124no—
327D3—124no—
328D3—364no—
329D3—368no—
330D3—361no—
331C22—92no—
332C22—181no—
333C22—361no—
334C22—361no—
335C22—541no—
336C22—1081no—
337C22C2×D3×#15711yescharacteristic
338C4—21no—
339C4—32no—
340C4—61no—
341C4—271no—
342C4—271no—
343C4—362no—
344C4—1082no—
345C3—22no—
346C3—42no—
347C3#1059383211yescharacteristic, complemented
348C2—21no—
349C2—182no—
350C2—361no—
351C2—1081no—
352C2D3×C22⋊#15711yescharacteristic
subgroupG/H
centerC2D3×C22⋊#157
commutatorC22×C33C22×C4
FrattiniC22C2×D3×#157
FittingC2×C33×C4C2×C4
socleC2×C33C2×C22⋊C4
prime radicalC2×#89017C2
cosocleC22×C3×C3⋊D3C23
hypercenterC2×C4D3×#157
hypocenterC33#410
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11724261C2semidirect
3. #11723099C2semidirect
4. C3×#63774C2semidirect
5. #11722728C2semidirect
7. C2×C4×C3⋊(C3⋊D3)C4semidirect
9. C2×C32×Dic6C4semidirect
10. C2×#6292C4semidirect
12. C2×Dic3×C3⋊D3C4semidirect
25. #63774D3semidirect
154. C2×Dic6#157semidirect
196. C33#410semidirect
298. C32#4740semidirect
347. C3#10593832semidirect

Maximal subgroups

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

subgroupindex
C2×#890172
#117242612
#117230992
C3×#637742
#117227282
#105938323
#47409