Finite Group Databasebeta

← #13155541

Subgroups — #13155541

Order 1728.

614 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 112 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
1#11723452C211yesmaximal, characteristic
2#11722071C211yesmaximal, characteristic, complemented
3#11722024C212yesmaximal
4C3×Dic3⋊(C3⋊C8)C212yesmaximal, complemented
5C3×#63553C211yesmaximal, characteristic, complemented
6#10590648—32nomaximal
7#10590257—31nomaximal
8C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
9C3×C4×C3⋊Dic3C2211yescharacteristic
10C2×C3×C3⋊(C3⋊C8)C2212yes—
11C32×C4×Dic3C2211yescharacteristic
12C2×C32×C3⋊C8C2212yes—
13#88806C412yes—
14#88804C412yes—
15#64081—31no—
16#63874—32no—
17#63874—61no—
18#63647—32no—
19C3×Dic3⋊C8—32no—
20C3×C8×Dic3—32no—
21C3×C4⋊(C3⋊C8)—32no—
22Dic3⋊(C3⋊C8)D312yescomplemented
23#63557—32no—
24#63557—32no—
25#63557—62no—
26#63555—31no—
27#63554—32no—
28#63553D311yescharacteristic, complemented
29C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
30C2×C33×C4C2311yescharacteristic
31C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
32C4×C3⋊(C3⋊D3)—22no—
33C2×C3×C3⋊Dic3C2×C412yes—
34C2×C32×Dic3C2×C412yes—
35C3×C3⋊(C3⋊C8)—22no—
36C32×C3⋊C8—22no—
37#4840—92no—
38#4690—91no—
39C2×C4×C3⋊D3—31no—
40C2×C4×C3⋊D3—32no—
41C2×C4×C3⋊D3—61no—
42C2×C4×C3⋊D3—62no—
43C2×C4×C3⋊D3—121no—
44C2×C32×C8—32no—
45C32×C42—31no—
46#3752—32no—
47#3750—32no—
48C4×C3⋊Dic3C2×D311yescharacteristic
49C2×C3⋊(C3⋊C8)C2×D312yes—
50C3×C4×Dic3—21no—
51C3×C4×Dic3—32no—
52C3×C4×Dic3—61no—
53C3×C4×Dic3C2×D312yes—
54C2×C3×C3⋊C8—22no—
55C2×C3×C3⋊C8—32no—
56C2×C3×C3⋊C8—32no—
57C2×C3×C3⋊C8—62no—
58C2×C3×C3⋊C8C2×D312yes—
59C2×C3×C3⋊C8C2×D312yes—
60#3724—32no—
61#3724—32no—
62#3724—62no—
63#3722—32no—
64#3722—32no—
65#3722—62no—
66C22×C33C22×C411yescharacteristic
67C2×C3⋊(C3⋊D3)—21no—
68C2×C3⋊(C3⋊D3)—22no—
69C33×C4#4112yes—
70C3⋊(C3⋊Dic3)—21no—
71C3×C3⋊Dic3—22no—
72C32×Dic3—22no—
73#872—92no—
74#872—91no—
75#872—181no—
76C3×C4⋊C8—92no—
77C3×C4×C8—91no—
78(C2×D3)⋊C8—92no—
79(C2×D3)⋊C8—92no—
80(C2×D3)⋊C8—92no—
81(C2×D3)⋊C8—182no—
82#815—92no—
83Dic3⋊C8—32no—
84C8×Dic3—32no—
85C4⋊(C3⋊C8)—32no—
86#803—91no—
87C22×C3⋊D3—32no—
88C22×C3⋊D3—31no—
89C22×C3⋊D3—61no—
90C22×C3⋊D3—62no—
91C22×C3⋊D3—121no—
92C2×C32×C4—22no—
93C2×C32×C4—21no—
94C2×C32×C4—32no—
95C2×C32×C4—41no—
96C2×C32×C4C22×D311yescharacteristic
97C2×C32×C4C22×D312yes—
98C2×C3⋊Dic3—32no—
99C2×C3⋊Dic3—31no—
100C2×C3⋊Dic3—61no—
101C2×C3⋊Dic3—62no—
102C2×C3⋊Dic3—121no—
103C2×C3⋊Dic3C4×D312yes—
104C4×C3⋊D3—62no—
105C4×C3⋊D3—64no—
106C4×C3⋊D3—62no—
107C4×C3⋊D3—62no—
108C4×C3⋊D3—122no—
109C4×C3⋊D3—122no—
110C4×C3⋊D3—124no—
111C2×C3×Dic3—22no—
112C2×C3×Dic3—32no—
113C2×C3×Dic3—32no—
114C2×C3×Dic3—62no—
115C2×C3×Dic3C4×D312yes—
116C2×C3×Dic3C4×D312yes—
117C32×C8—62no—
118C3⋊(C3⋊C8)—22no—
119C3×C3⋊C8—22no—
120C3×C3⋊C8—22no—
121C3×C3⋊C8—42no—
122C3×C3⋊C8—62no—
123C3×C3⋊C8—62no—
124C3×C3⋊C8—122no—
125#433—271noSylow
126C2×C33#11711yescharacteristic
127C2×C33#13112yes—
128C3⋊(C3⋊D3)—41no—
129C2×C4×D3—91no—
130C2×C4×D3—92no—
131C2×C4×D3—181no—
132C2×C4×D3—182no—
133C2×C4×D3—361no—
134C2×C3×C8—32no—
135C2×C3×C8—32no—
136C2×C3×C8—62no—
137C2×C3×C8—92no—
138C3×C42—32no—
139C3×C42—61no—
140C3×C42—91no—
141(C2×D3)⋊C4—92no—
142(C2×D3)⋊C4—92no—
143(C2×D3)⋊C4—92no—
144(C2×D3)⋊C4—182no—
145Dic3⋊C4—92no—
146Dic3⋊C4—92no—
147Dic3⋊C4—92no—
148Dic3⋊C4—182no—
149C4×Dic3—32no—
150C4×Dic3—61no—
151C4×Dic3D3211yescharacteristic
152C2×C3⋊C8—32no—
153C2×C3⋊C8—32no—
154C2×C3⋊C8—62no—
155C2×C3⋊C8D3212yes—
156C22×C32—22no—
157C22×C32—21no—
158C22×C32—41no—
159C22×C32C2×C4×D311yescharacteristic
160C22×C32C2×C4×D312yes—
161C2×C3⋊D3—61no—
162C2×C3⋊D3—62no—
163C2×C3⋊D3—62no—
164C2×C3⋊D3—64no—
165C2×C3⋊D3—62no—
166C2×C3⋊D3—62no—
167C2×C3⋊D3—62no—
168C2×C3⋊D3—122no—
169C2×C3⋊D3—122no—
170C2×C3⋊D3—122no—
171C2×C3⋊D3—122no—
172C2×C3⋊D3—124no—
173C32×C4—22no—
174C32×C4—22no—
175C32×C4—22no—
176C32×C4—42no—
177C32×C4—62no—
178C32×C4#23512yes—
179C32×C4#23712yes—
180C32×C4#23912yes—
181C3⋊Dic3—22no—
182C3⋊Dic3—61no—
183C3⋊Dic3—62no—
184C3⋊Dic3—62no—
185C3⋊Dic3—122no—
186C3⋊Dic3—122no—
187C3×Dic3—22no—
188C3×Dic3—22no—
189C3×Dic3—42no—
190C3×Dic3—62no—
191C3×Dic3—62no—
192C3×Dic3—122no—
193#117—271no—
194C4⋊C8—92no—
195C22⋊C8—272no—
196#97—271no—
197C4×C8—91no—
198C33#43311yescharacteristic, complemented, Sylow
199C22×D3—92no—
200C22×D3—91no—
201C22×D3—181no—
202C22×D3—182no—
203C22×D3—361no—
204C2×C3×C4—22no—
205C2×C3×C4—21no—
206C2×C3×C4—32no—
207C2×C3×C4—32no—
208C2×C3×C4—41no—
209C2×C3×C4—62no—
210C2×C3×C4—92no—
211C2×C3×C4C2×D3211yescharacteristic
212C2×C3×C4C2×D3212yes—
213C2×Dic3—32no—
214C2×Dic3—32no—
215C2×Dic3—62no—
216C2×Dic3—92no—
217C2×Dic3—91no—
218C2×Dic3—181no—
219C2×Dic3—182no—
220C2×Dic3—361no—
221C2×Dic3#62412yes—
222C4×D3—184no—
223C4×D3—182no—
224C4×D3—182no—
225C4×D3—182no—
226C4×D3—362no—
227C4×D3—362no—
228C4×D3—364no—
229C3×C8—62no—
230C3×C8—62no—
231C3×C8—122no—
232C3×C8—182no—
233C3⋊C8—22no—
234C3⋊C8—62no—
235C3⋊C8—62no—
236C3⋊C8—122no—
237C2×C32—22no—
238C2×C32—22no—
239C2×C32—22no—
240C2×C32—22no—
241C2×C32—21no—
242C2×C32—41no—
243C2×C32—42no—
244C2×C32#87211yescharacteristic
245C2×C32#89212yes—
246C2×C32#90112yes—
247C2×C32#90112yes—
248C2×C32#90712yes—
249C3⋊D3—122no—
250C3⋊D3—124no—
251C3⋊D3—124no—
252C3⋊D3—121no—
253C3⋊D3—122no—
254C22×C4—271no—
255C2×C8—32no—
256C2×C8—92no—
257C4⋊C4—272no—
258C22⋊C4—272no—
259C42—31no—
260C42—91no—
261C22×C3—21no—
262C22×C3—22no—
263C22×C3—41no—
264C22×C3C4×D3212yes—
265C22×C3C2×#62411yescharacteristic
266C2×D3—182no—
267C2×D3—184no—
268C2×D3—182no—
269C2×D3—182no—
270C2×D3—182no—
271C2×D3—182no—
272C2×D3—181no—
273C2×D3—362no—
274C2×D3—362no—
275C2×D3—362no—
276C2×D3—362no—
277C2×D3—364no—
278C3×C4—22no—
279C3×C4—22no—
280C3×C4—22no—
281C3×C4—42no—
282C3×C4—62no—
283C3×C4—62no—
284C3×C4—122no—
285C3×C4—182no—
286C3×C4#379612yes—
287C3×C4#379912yes—
288C3×C4#380412yes—
289Dic3—22no—
290Dic3—62no—
291Dic3—62no—
292Dic3—122no—
293Dic3—182no—
294Dic3—182no—
295Dic3—181no—
296Dic3—362no—
297Dic3—362no—
298C32—21no—
299C32—22no—
300C32—41no—
301C32#469011yescharacteristic, complemented
302C32#484012yescomplemented
303C23—271no—
304C2×C4—32no—
305C2×C4—92no—
306C2×C4—271no—
307C2×C4—542no—
308C2×C4D3311yescharacteristic
309C8—62no—
310C8—182no—
311C2×C3—22no—
312C2×C3—22no—
313C2×C3—22no—
314C2×C3—21no—
315C2×C3—22no—
316C2×C3—42no—
317C2×C3—41no—
318C2×C3#6380312yes—
319C2×C3#6380412yes—
320C2×C3#6381712yes—
321C2×C3#6383612yes—
322C2×C3#6384911yescharacteristic
323D3—362no—
324D3—364no—
325D3—364no—
326D3—361no—
327D3—362no—
328C22—542no—
329C22—541no—
330C22D3×#62411yescharacteristic
331C4—62no—
332C4—182no—
333C4—541no—
334C4#8896212yes—
335C3—21no—
336C3—22no—
337C3—41no—
338C3#1059025711yescharacteristic, complemented
339C3#1059064812yescomplemented
340C2—1081no—
341C2#1172282412yes—
342C2#1172292011yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D33
commutator, socleC22×C33C22×C4
Fitting, cosocleC2×C33×C4C23
prime radicalC22×C3⋊(C3⋊D3)C2×C4
hypocenterC33#433
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11722071C2semidirect
4. C3×Dic3⋊(C3⋊C8)C2semidirect
5. C3×#63553C2semidirect
22. Dic3⋊(C3⋊C8)D3semidirect
28. #63553D3semidirect
198. C33#433semidirect
301. C32#4690semidirect
302. C32#4840semidirect
338. C3#10590257semidirect
339. C3#10590648semidirect

Maximal subgroups

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

subgroupindex
#117234522
#117220712
#117220242
C3×Dic3⋊(C3⋊C8)2
C3×#635532
#105906483
#105902573