Finite Group Databasebeta

← #13167517

Subgroups — #13167517

Order 1728.

452 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 59 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×#88943C212yesmaximal, complemented
2C2×#88942C212yesmaximal
3C4⋊#6296C211yesmaximal, characteristic, complemented
4#11723672C211yesmaximal, characteristic
5C3×#63777C211yesmaximal, characteristic, complemented
6#10593443—31nomaximal
7C2×#6296C2212yes—
8C2×#6296C2211yescharacteristic
9C2×#6295C2211yescharacteristic
10C2×C3×#622C2212yes—
11#88943—24no—
12#88942—24no—
13C3×#3752—21no—
14C3×C4⋊(C3⋊Dic3)—21no—
15C3×C4×C3⋊Dic3C2211yescharacteristic
16#88815—21no—
17#88814—21no—
18C2×#3776—32no—
19C2×#3775—32no—
20#63919—31no—
21#63918—61no—
22#63915—31no—
23#63852—61no—
24#63777D311yescharacteristic, complemented
25C22×C3×C3⋊D3—21no—
26C2×C33×C4D411yescharacteristic
27C2×C3×C3⋊Dic3—22no—
28C2×C3×C3⋊Dic3—21no—
29C2×C3×C3⋊Dic3D411yescharacteristic
30C2×C3×C3⋊Dic3C2311yescharacteristic
31#6296—22no—
32#6296—22no—
33#6296—22no—
34#6295—22no—
35C3×#622D414yes—
36#4715—91nomaximal
37C2×#627—31no—
38C2×#627—32no—
39C2×#627—61no—
40C2×#627—62no—
41C2×#626—61no—
42C2×#625—31no—
43C2×#622C2×D312yes—
44#3776—64no—
45#3775—64no—
46#3752—21no—
47C4⋊(C3⋊Dic3)—21no—
48C4×C3⋊Dic3C2×D311yescharacteristic
49C3×(C2×D3)⋊C4—61no—
50C3×(C2×D3)⋊C4—61no—
51C3×(C2×D3)⋊C4—121no—
52C3×C4⋊Dic3—61no—
53C3×C4⋊Dic3—61no—
54C3×C4⋊Dic3—121no—
55C3×C4×Dic3—61no—
56C3×C4×Dic3—61no—
57#3724—61no—
58Dic3⋊Dic3—62no—
59#3722—61no—
60(C2×D3)⋊Dic3—61no—
61(C2×D3)⋊Dic3—61no—
62C22×C33C2×D411yescharacteristic
63C2×C3×C3⋊D3—41no—
64C2×C3×C3⋊D3—41no—
65C2×C3×C3⋊D3—41no—
66C33×C4—21no—
67C3×C3⋊Dic3—21no—
68C3×C3⋊Dic3—42no—
69C3×C3⋊Dic3—41no—
70C3×C3⋊Dic3C2×D412yes—
71C2×Dic12—92no—
72C2×#204—92no—
73#888—181no—
74#885—181no—
75#875—91no—
76#869—91no—
77C3×#97—91no—
78C22×C3⋊D3—21no—
79C22×C3×D3—61no—
80C22×C3×D3—61no—
81C22×C3×D3—121no—
82C2×C32×C4—21no—
83C2×C32×C4—21no—
84C2×C32×C4—41no—
85C2×C32×C4—41no—
86C2×C32×C4D1211yescharacteristic
87C2×C3⋊Dic3—21no—
88C2×C3⋊Dic3—22no—
89C2×C3⋊Dic3D1211yescharacteristic
90C2×C3⋊Dic3C22×D311yescharacteristic
91C2×C3×Dic3—61no—
92C2×C3×Dic3—61no—
93C2×C3×Dic3—61no—
94C2×C3×Dic3—61no—
95C2×C3×Dic3—61no—
96C2×C3×Dic3—62no—
97C2×C3×Dic3—61no—
98C2×C3×Dic3—62no—
99C2×C3×Dic3—122no—
100C2×C3×Dic3—121no—
101#627—62no—
102#627—62no—
103#627—62no—
104#627—122no—
105#627—122no—
106#627—122no—
107#626—122no—
108#625—62no—
109#622D1214yes—
110#497—271noSylow
111C2×C33#12711yescharacteristic
112C2×C33#13711yescharacteristic
113C2×C33#13711yescharacteristic
114C3×C3⋊D3—81no—
115C2×#67—181no—
116C2×D12—91no—
117C2×Dic6—92no—
118C2×Dic6—91no—
119C2×Dic6—181no—
120C2×Dic6—182no—
121C2×C3×C8—92no—
122C3×C4⋊C4—181no—
123C3×C22⋊C4—181no—
124C3×C42—91no—
125#217—181no—
126(C2×D3)⋊C4—61no—
127(C2×D3)⋊C4—61no—
128(C2×D3)⋊C4—121no—
129(C2×D3)⋊C4—181no—
130(C2×D3)⋊C4—181no—
131C4⋊Dic3—61no—
132C4⋊Dic3—61no—
133C4⋊Dic3—121no—
134C4⋊Dic3—181no—
135Dic3⋊C4—182no—
136C4×Dic3—61no—
137C4×Dic3—61no—
138Dic12—184no—
139#204—184no—
140C22×C32—21no—
141C22×C32—21no—
142C22×C32—41no—
143C22×C32—41no—
144C22×C32C2×D1211yescharacteristic
145C2×C3⋊D3—41no—
146C2×C3⋊D3—41no—
147C2×C3⋊D3—41no—
148C2×C3×D3—121no—
149C2×C3×D3—121no—
150C2×C3×D3—121no—
151C2×C3×D3—122no—
152C2×C3×D3—121no—
153C2×C3×D3—121no—
154C2×C3×D3—121no—
155C2×C3×D3—241no—
156C2×C3×D3—241no—
157C32×C4—22no—
158C32×C4—21no—
159C32×C4—41no—
160C32×C4—42no—
161C32×C4—81no—
162C3⋊Dic3—21no—
163C3⋊Dic3—42no—
164C3⋊Dic3—41no—
165C3⋊Dic3C2×D1212yes—
166C3×Dic3—62no—
167C3×Dic3—62no—
168C3×Dic3—62no—
169C3×Dic3—121no—
170C3×Dic3—122no—
171C3×Dic3—121no—
172C3×Dic3—122no—
173C3×Dic3—121no—
174C3×Dic3—242no—
175C3×Dic3—241no—
176C2×Q16—272no—
177C2×SD16—272no—
178C4⋊Q8—271no—
179#124—271no—
180#97—91no—
181C33#49711yescharacteristic, complemented, Sylow
182C23×C3—181no—
183C22×D3—61no—
184C22×D3—61no—
185C22×D3—121no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—91no—
191C2×C3×C4—91no—
192C2×C3×C4—182no—
193C2×C3×C4—181no—
194C2×C3×C4GO+(4,𝔽2)11yescharacteristic
195#67—362no—
196C2×Dic3—61no—
197C2×Dic3—61no—
198C2×Dic3—61no—
199C2×Dic3—61no—
200C2×Dic3—62no—
201C2×Dic3—61no—
202C2×Dic3—62no—
203C2×Dic3—61no—
204C2×Dic3—122no—
205C2×Dic3—121no—
206D12—182no—
207Dic6—182no—
208Dic6—182no—
209Dic6—182no—
210Dic6—362no—
211Dic6—362no—
212Dic6—362no—
213C3×C8—94no—
214C2×C32—21no—
215C2×C32—21no—
216C2×C32—21no—
217C2×C32—21no—
218C2×C32—21no—
219C2×C32—21no—
220C2×C32—41no—
221C2×C32—41no—
222C2×C32—41no—
223C2×C32—41no—
224C2×C32—41no—
225C2×C32—41no—
226C2×C32#87411yescharacteristic
227C2×C32#90911yescharacteristic
228C2×C32#90911yescharacteristic
229C3⋊D3—81no—
230C3×D3—241no—
231C3×D3—242no—
232C3×D3—241no—
233C2×Q8—271no—
234C2×Q8—272no—
235C2×D4—271no—
236Q16—544no—
237SD16—544no—
238C2×C8—92no—
239C4⋊C4—181no—
240C4⋊C4—541no—
241C22⋊C4—181no—
242C22⋊C4—541no—
243C42—91no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—41no—
247C22×C3—41no—
248C22×C3—361no—
249C22×C3—361no—
250C22×C3—361no—
251C22×C3C2×GO+(4,𝔽2)11yescharacteristic
252C2×D3—121no—
253C2×D3—121no—
254C2×D3—121no—
255C2×D3—122no—
256C2×D3—121no—
257C2×D3—121no—
258C2×D3—121no—
259C2×D3—241no—
260C2×D3—241no—
261C3×C4—22no—
262C3×C4—21no—
263C3×C4—41no—
264C3×C4—42no—
265C3×C4—81no—
266C3×C4—92no—
267C3×C4—181no—
268C3×C4—362no—
269C3×C4—361no—
270Dic3—62no—
271Dic3—62no—
272Dic3—62no—
273Dic3—121no—
274Dic3—122no—
275Dic3—121no—
276Dic3—122no—
277Dic3—121no—
278Dic3—242no—
279Dic3—241no—
280C32—21no—
281C32—21no—
282C32—41no—
283C32—41no—
284C32#471511yescharacteristic, complemented
285C23—181no—
286Q8—542no—
287Q8—542no—
288Q8—542no—
289D4—542no—
290C2×C4—91no—
291C2×C4—91no—
292C2×C4—181no—
293C2×C4—182no—
294C2×C4#157⋊D311yescharacteristic
295C8—94no—
296C2×C3—21no—
297C2×C3—21no—
298C2×C3—21no—
299C2×C3—21no—
300C2×C3—21no—
301C2×C3—21no—
302C2×C3—41no—
303C2×C3—41no—
304C2×C3—41no—
305C2×C3—41no—
306C2×C3—41no—
307C2×C3—41no—
308C2×C3—721no—
309C2×C3#6422611yescharacteristic
310C2×C3#6423111yescharacteristic
311C2×C3#6423911yescharacteristic
312D3—241no—
313D3—242no—
314D3—241no—
315C22—361no—
316C22—361no—
317C22—361no—
318C22C2×#157⋊D311yescharacteristic
319C4—21no—
320C4—92no—
321C4—181no—
322C4—361no—
323C4—362no—
324C3—21no—
325C3—21no—
326C3—41no—
327C3—41no—
328C3#1059344311yescharacteristic, complemented
329C2—721no—
330C2#1172433811yescharacteristic
331C2#1172434411yescharacteristic
332C2#1172435211yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×#6295C22
hypercenterC2×C4#157⋊D3
hypocenterC33#497
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88943C2semidirect
3. C4⋊#6296C2semidirect
5. C3×#63777C2semidirect
24. #63777D3semidirect
181. C33#497semidirect
284. C32#4715semidirect
328. C3#10593443semidirect

Maximal subgroups

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

subgroupindex
C2×#889432
C2×#889422
C4⋊#62962
#117236722
C3×#637772
#105934433
#47159