Finite Group Databasebeta

← #13161962

Subgroups — #13161962

Order 1728.

614 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C3×#64145C211yesmaximal, characteristic
2#11723680C211yesmaximal, characteristic, complemented
3C3×#63659C212yesmaximal
4#11722076C212yesmaximal, complemented
5#11722070C211yesmaximal, characteristic, complemented
6#10591478—32nomaximal
7#10591403—31nomaximal
8C2×C33×D4C2211yescharacteristic
9C2×C3×#634C2211yescharacteristic
10C3×#3757—22no—
11C3×C4⋊(C3⋊Dic3)C2212yes—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C2×C3×C3⋊(C3⋊C8)C2212yes—
14#88815—22no—
15#64145D311yescharacteristic
16C3×#936—32no—
17C3×#936—61no—
18#63872—31no—
19#63869—32no—
20#63659D312yes—
21C3×#832—32no—
22C3×#832—32no—
23C3×#832—62no—
24#63567—32no—
25#63566—32no—
26#63565—32no—
27#63553—31no—
28Dic3×C3⋊C8—32no—
29C23×C33—21no—
30C33×D4—22no—
31C2×C33×C4C2311yescharacteristic
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3—22no—
34C2×C3×C3⋊Dic3D412yes—
35C3×#634—22no—
36C3×C3⋊(C3⋊C8)—22no—
37#5056—91no—
38#4768—92no—
39C2×C32×D4—22no—
40C2×C32×D4—21no—
41C2×C32×D4—41no—
42C2×C32×D4C2×D312yes—
43C2×C32×D4C2×D311yescharacteristic
44C2×#634C2×D311yescharacteristic
45C2×C3×Dic6—32no—
46C2×C3×Dic6—61no—
47#3757—22no—
48C4⋊(C3⋊Dic3)C2×D312yes—
49C4×C3⋊Dic3C2×D311yescharacteristic
50C2×C3⋊(C3⋊C8)C2×D312yes—
51C3×#217—62no—
52C3×#217—62no—
53C3×#217—64no—
54C3×C4⋊Dic3—32no—
55C3×C4⋊Dic3—32no—
56C3×C4⋊Dic3—62no—
57C3×C4×Dic3—32no—
58C3×C4×Dic3—61no—
59C2×C3×C3⋊C8—32no—
60C2×C3×C3⋊C8—32no—
61C2×C3×C3⋊C8—62no—
62#3724—62no—
63Dic3⋊Dic3—62no—
64Dic3⋊Dic3—62no—
65C22×C33—41no—
66C22×C33—42no—
67C22×C33C2×D411yescharacteristic
68C33×C4#4112yes—
69C3×C3⋊Dic3—22no—
70C3×C3⋊Dic3—41no—
71C3×C3⋊Dic3—42no—
72C3×#124—91no—
73#936—32no—
74#936—61no—
75#890—92no—
76#870—91no—
77C3×#102—92no—
78#835—92no—
79#832—32no—
80#832—32no—
81#832—62no—
82#816—92no—
83C8×Dic3—92no—
84#810—92no—
85C4×C3⋊C8—91no—
86C23×C32—21no—
87C23×C32—22no—
88C23×C32—22no—
89C23×C32—42no—
90C23×C32—42no—
91C32×D4—22no—
92C32×D4—24no—
93C32×D4—22no—
94C32×D4—22no—
95C32×D4—42no—
96C32×D4—42no—
97C32×D4—44no—
98C2×C32×C4—22no—
99C2×C32×C4—21no—
100C2×C32×C4—41no—
101C2×C32×C4C22×D312yes—
102C2×C32×C4C22×D311yescharacteristic
103C2×C3⋊Dic3—21no—
104C2×C3⋊Dic3—22no—
105C2×C3⋊Dic3#6712yes—
106#634—22no—
107C2×C3×Dic3—32no—
108C2×C3×Dic3—32no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—62no—
114C2×C3×Dic3—122no—
115C3×Dic6—62no—
116C3×Dic6—62no—
117C3×Dic6—122no—
118C3⋊(C3⋊C8)—22no—
119C3×C3⋊C8—62no—
120C3×C3⋊C8—62no—
121C3×C3⋊C8—122no—
122#488—271noSylow
123C2×C33—81no—
124C2×C33#12411yescharacteristic
125C2×C33#13512yes—
126C2×C3×Q8—91no—
127C2×C3×D4—22no—
128C2×C3×D4—21no—
129C2×C3×D4—41no—
130C2×C3×D4D3211yescharacteristic
131C2×C3×D4D3212yes—
132C2×Dic6—32no—
133C2×Dic6—61no—
134C2×C3×C8—92no—
135C3×C4⋊C4—92no—
136C3×C22⋊C4—182no—
137C3×C42—91no—
138#217—62no—
139#217—62no—
140#217—64no—
141C4⋊Dic3—32no—
142C4⋊Dic3—32no—
143C4⋊Dic3—62no—
144C4⋊Dic3—182no—
145Dic3⋊C4—182no—
146Dic3⋊C4—182no—
147C4×Dic3—32no—
148C4×Dic3—61no—
149C2×C3⋊C8—32no—
150C2×C3⋊C8—32no—
151C2×C3⋊C8—62no—
152C22×C32—22no—
153C22×C32—21no—
154C22×C32—41no—
155C22×C32—44no—
156C22×C32—42no—
157C22×C32—44no—
158C22×C32—42no—
159C22×C32—41no—
160C22×C32—42no—
161C22×C32—44no—
162C22×C32—42no—
163C22×C32—42no—
164C22×C32—82no—
165C22×C32—82no—
166C22×C32—84no—
167C22×C32D3×D412yes—
168C22×C32C2×#6711yescharacteristic
169C32×C4—22no—
170C32×C4—22no—
171C32×C4—22no—
172C32×C4—42no—
173C32×C4#23512yes—
174C32×C4#23712yes—
175C32×C4#23712yes—
176C3⋊Dic3—22no—
177C3⋊Dic3—41no—
178C3⋊Dic3—42no—
179C3×Dic3—62no—
180C3×Dic3—62no—
181C3×Dic3—122no—
182C3×Dic3—122no—
183C3×Dic3—122no—
184C3×Dic3—122no—
185C3×Dic3—122no—
186C3×Dic3—242no—
187#125—271no—
188#124—91no—
189#103—272no—
190#102—92no—
191C4×C8—271no—
192C33#48811yescharacteristic, complemented, Sylow
193C23×C3—22no—
194C23×C3—22no—
195C23×C3—21no—
196C23×C3—42no—
197C23×C3—42no—
198C3×Q8—182no—
199C3×D4—24no—
200C3×D4—22no—
201C3×D4—22no—
202C3×D4—22no—
203C3×D4—42no—
204C3×D4—42no—
205C3×D4—44no—
206C2×C3×C4—22no—
207C2×C3×C4—21no—
208C2×C3×C4—41no—
209C2×C3×C4—92no—
210C2×C3×C4—182no—
211C2×C3×C4—181no—
212C2×C3×C4C2×D3211yescharacteristic
213C2×C3×C4C2×D3212yes—
214C2×Dic3—32no—
215C2×Dic3—32no—
216C2×Dic3—62no—
217C2×Dic3—62no—
218C2×Dic3—62no—
219C2×Dic3—62no—
220C2×Dic3—62no—
221C2×Dic3—122no—
222Dic6—62no—
223Dic6—62no—
224Dic6—122no—
225C3×C8—182no—
226C3⋊C8—62no—
227C3⋊C8—62no—
228C3⋊C8—122no—
229C2×C32—22no—
230C2×C32—22no—
231C2×C32—22no—
232C2×C32—21no—
233C2×C32—22no—
234C2×C32—41no—
235C2×C32—42no—
236C2×C32—82no—
237C2×C32—84no—
238C2×C32—84no—
239C2×C32—82no—
240C2×C32—81no—
241C2×C32#88512yes—
242C2×C32#91212yes—
243C2×C32#91612yes—
244C2×C32#93611yescharacteristic
245C2×C32#95012yes—
246C2×Q8—91no—
247C2×D4#111311yescharacteristic
248C2×C8—92no—
249C4⋊C4—92no—
250C4⋊C4—542no—
251C22⋊C4—182no—
252C42—91no—
253C22×C3—21no—
254C22×C3—22no—
255C22×C3—44no—
256C22×C3—42no—
257C22×C3—44no—
258C22×C3—44no—
259C22×C3—42no—
260C22×C3—42no—
261C22×C3—42no—
262C22×C3—41no—
263C22×C3—42no—
264C22×C3—41no—
265C22×C3—82no—
266C22×C3—82no—
267C22×C3—84no—
268C22×C3D3×#6712yes—
269C22×C3#381111yescharacteristic
270C3×C4—22no—
271C3×C4—22no—
272C3×C4—22no—
273C3×C4—42no—
274C3×C4—182no—
275C3×C4—362no—
276C3×C4—361no—
277C3×C4#380412yes—
278C3×C4#380412yes—
279C3×C4#380512yes—
280Dic3—62no—
281Dic3—62no—
282Dic3—122no—
283Dic3—122no—
284Dic3—122no—
285Dic3—122no—
286Dic3—122no—
287Dic3—242no—
288C32—21no—
289C32—22no—
290C32—41no—
291C32#476812yescomplemented
292C32#505611yescharacteristic, complemented
293C23—21no—
294Q8—182no—
295D4—22no—
296C2×C4—92no—
297C2×C4—181no—
298C2×C4—182no—
299C2×C4C2×#111311yescharacteristic
300C8—182no—
301C2×C3—22no—
302C2×C3—22no—
303C2×C3—22no—
304C2×C3—21no—
305C2×C3—22no—
306C2×C3—42no—
307C2×C3—41no—
308C2×C3—82no—
309C2×C3—84no—
310C2×C3—84no—
311C2×C3—81no—
312C2×C3—82no—
313C2×C3#6393312yes—
314C2×C3#6393512yes—
315C2×C3#6393612yes—
316C2×C3#6395311yescharacteristic
317C2×C3#6395912yes—
318C22—42no—
319C22—41no—
320C22#8904911yescharacteristic
321C4—182no—
322C4—362no—
323C4—361no—
324C4#8904612yes—
325C3—22no—
326C3—21no—
327C3—41no—
328C3#1059140311yescharacteristic, complemented
329C3#1059147812yescomplemented
330C2—81no—
331C2#1172370712yes—
332C2#1172371811yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
Fitting, prime radicalC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4#1113
hypocenterC33#488
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11723680C2semidirect
4. #11722076C2semidirect
5. #11722070C2semidirect
192. C33#488semidirect
291. C32#4768semidirect
292. C32#5056semidirect
328. C3#10591403semidirect
329. C3#10591478semidirect

Maximal subgroups

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

subgroupindex
C3×#641452
#117236802
C3×#636592
#117220762
#117220702
#105914783
#105914033