Finite Group Databasebeta

← #13156277

Subgroups — #13156277

Order 1728.

624 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 110 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
1C4⋊#6296C211yesmaximal, characteristic, complemented
2D12×C3⋊Dic3C211yesmaximal, characteristic
3#11722055C212yesmaximal, complemented
4C3×#63572C211yesmaximal, characteristic, complemented
5C3×#63563C212yesmaximal
6#10591494—31nomaximal
7#10590762—32nomaximal
8C2×C32×D12C2211yescharacteristic
9C2×#6296—21no—
10C2×D3×C3⋊Dic3—21no—
11C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
12C3×C4⋊(C3⋊Dic3)C2212yes—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C2×C32×C3⋊C8C2212yes—
15#88815—21no—
16(C2×D3)⋊(C3⋊Dic3)—21no—
17D4×C3⋊Dic3—31no—
18#63919—31no—
19#63918—32no—
20Dic3×D12—32no—
21Dic3×D12—61no—
22#63645—32no—
23C3×#832—32no—
24C3×#821—32no—
25C3×Dic3⋊C8—32no—
26#63575—32no—
27#63575—32no—
28#63575—62no—
29#63572D311yescharacteristic, complemented
30#63563D312yes—
31C22×C32×D3—21no—
32C2×C33×C4C2311yescharacteristic
33C2×C3⋊(C3⋊Dic3)—21no—
34C2×C3×C3⋊Dic3—22no—
35C2×C3×C3⋊Dic3D411yescharacteristic
36C2×C3×C3⋊Dic3D411yescharacteristic
37C32×D12—21no—
38C32×D12Q812yescomplemented
39#6296—42no—
40D3×C3⋊Dic3—42no—
41C32×C3⋊C8—22no—
42#4846—91no—
43#4759—92no—
44C2×C32×D4—31no—
45C22×C3⋊Dic3—61no—
46C2×C3×D12—21no—
47C2×C3×D12C2×D312yes—
48C2×#627—62no—
49C2×#627—61no—
50C2×D3×Dic3—62no—
51C2×D3×Dic3—62no—
52C2×C32×C8—32no—
53#3757—61no—
54C4⋊(C3⋊Dic3)—31no—
55C4⋊(C3⋊Dic3)—32no—
56C4⋊(C3⋊Dic3)—61no—
57C4⋊(C3⋊Dic3)—62no—
58C4⋊(C3⋊Dic3)—121no—
59C4⋊(C3⋊Dic3)C2×D312yes—
60C4×C3⋊Dic3C2×D311yescharacteristic
61C3×C4⋊Dic3—32no—
62C3×C4⋊Dic3—32no—
63C3×C4⋊Dic3—62no—
64C3×C4×Dic3—32no—
65C3×C4×Dic3—61no—
66C2×C3×C3⋊C8—22no—
67C2×C3×C3⋊C8C2×D312yes—
68C2×C3×C3⋊C8C2×D312yes—
69#3724—61no—
70Dic3⋊Dic3—62no—
71(C2×D3)⋊Dic3—62no—
72(C2×D3)⋊Dic3—62no—
73C22×C33C2×D411yescharacteristic
74C2×C32×D3—22no—
75C2×C32×D3—41no—
76C2×C32×D3—41no—
77C33×C4C2×Q811yescharacteristic
78C33×C4#4111yescharacteristic
79C3⋊(C3⋊Dic3)—41no—
80C3×C3⋊Dic3—21no—
81C3×C3⋊Dic3—42no—
82C3×C3⋊Dic3D812yes—
83D4×Dic3—92no—
84D4×Dic3—181no—
85#888—92no—
86C4×D12—91no—
87#869—91no—
88C3×C4⋊C8—91no—
89C3×#102—92no—
90#832—32no—
91#821—32no—
92#818—92no—
93#818—92no—
94#818—182no—
95Dic3⋊C8—32no—
96#807—92no—
97C23×C32—61no—
98C22×C3×D3—22no—
99C22×C3×D3—22no—
100C32×D4—32no—
101C32×D4—61no—
102C2×C32×C4—22no—
103C2×C32×C4—21no—
104C2×C32×C4—41no—
105C2×C32×C4C22×D312yes—
106C2×C32×C4C22×D311yescharacteristic
107C2×C3⋊Dic3—22no—
108C2×C3⋊Dic3—61no—
109C2×C3⋊Dic3—62no—
110C2×C3⋊Dic3—62no—
111C2×C3⋊Dic3—122no—
112C2×C3⋊Dic3—122no—
113C2×C3⋊Dic3—122no—
114C2×C3⋊Dic3D1211yescharacteristic
115C2×C3⋊Dic3D1211yescharacteristic
116C2×C3×Dic3—32no—
117C2×C3×Dic3—32no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—62no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—122no—
123C3×D12—22no—
124C3×D12—22no—
125C3×D12—22no—
126C3×D12Dic614yescomplemented
127#627—124no—
128#627—122no—
129D3×Dic3—124no—
130D3×Dic3—124no—
131C32×C8—62no—
132C3×C3⋊C8—24no—
133C3×C3⋊C8—22no—
134C3×C3⋊C8—22no—
135#474—271noSylow
136C2×C33C22⋊Q811yescharacteristic
137C2×C33C2×D811yescharacteristic
138C2×C33#13711yescharacteristic
139C32×D3—42no—
140C2×C3×D4—32no—
141C2×C3×D4—61no—
142C22×Dic3—182no—
143C22×Dic3—182no—
144C2×D12D3211yescharacteristic
145C2×C4×D3—181no—
146C2×Dic6—182no—
147C2×Dic6—181no—
148C2×C3×C8—32no—
149C2×C3×C8—32no—
150C2×C3×C8—62no—
151C3×C4⋊C4—92no—
152C3×C42—91no—
153#217—182no—
154#217—182no—
155(C2×D3)⋊C4—181no—
156C4⋊Dic3—32no—
157C4⋊Dic3—32no—
158C4⋊Dic3—62no—
159C4⋊Dic3—92no—
160C4⋊Dic3—91no—
161C4⋊Dic3—181no—
162C4⋊Dic3—182no—
163C4⋊Dic3—181no—
164C4⋊Dic3—361no—
165Dic3⋊C4—182no—
166C4×Dic3—32no—
167C4×Dic3—61no—
168C2×C3⋊C8D3212yes—
169C22×C32—22no—
170C22×C32—21no—
171C22×C32—41no—
172C22×C32—62no—
173C22×C32—121no—
174C22×C32—121no—
175C22×C32C2×D1211yescharacteristic
176C22×C32D3×D412yes—
177C2×C3×D3—24no—
178C2×C3×D3—24no—
179C2×C3×D3—42no—
180C2×C3×D3—42no—
181C2×C3×D3—42no—
182C2×C3×D3—42no—
183C32×C4—22no—
184C32×C4—22no—
185C32×C4—21no—
186C32×C4—21no—
187C32×C4—41no—
188C32×C4—41no—
189C32×C4C2×Dic612yes—
190C32×C4#23712yes—
191C32×C4#23711yescharacteristic
192C32×C4D3×Q811yescharacteristic
193C3⋊Dic3—21no—
194C3⋊Dic3—42no—
195C3⋊Dic3—121no—
196C3⋊Dic3—122no—
197C3⋊Dic3—122no—
198C3⋊Dic3—242no—
199C3⋊Dic3—242no—
200C3⋊Dic3D2412yes—
201C3×Dic3—34no—
202C3×Dic3—62no—
203C3×Dic3—62no—
204C3×Dic3—121no—
205C3×Dic3—122no—
206C3×Dic3—122no—
207C3×Dic3—242no—
208C4⋊Q8—271no—
209C4×D4—271no—
210#107—272no—
211C4⋊C8—91no—
212#102—92no—
213C33#47411yescharacteristic, complemented, Sylow
214C23×C3—62no—
215C23×C3—62no—
216C22×D3—21no—
217C3×D4—34no—
218C3×D4—62no—
219C3×D4—62no—
220C3×D4—62no—
221C2×C3×C4—21no—
222C2×C3×C4—22no—
223C2×C3×C4—41no—
224C2×C3×C4—91no—
225C2×C3×C4—91no—
226C2×C3×C4—182no—
227C2×C3×C4C2×D3212yes—
228C2×C3×C4C2×D3211yescharacteristic
229C2×Dic3—32no—
230C2×Dic3—32no—
231C2×Dic3—61no—
232C2×Dic3—61no—
233C2×Dic3—62no—
234C2×Dic3—62no—
235C2×Dic3—122no—
236C2×Dic3—182no—
237C2×Dic3—182no—
238C2×Dic3—181no—
239C2×Dic3—362no—
240C2×Dic3—362no—
241C2×Dic3—364no—
242C2×Dic3—364no—
243D12—21no—
244D12#62712yescomplemented
245C4×D3—362no—
246Dic6—364no—
247Dic6—362no—
248C3×C8—64no—
249C3×C8—62no—
250C3×C8—62no—
251C3⋊C8—22no—
252C2×C32—22no—
253C2×C32—22no—
254C2×C32—22no—
255C2×C32—21no—
256C2×C32—21no—
257C2×C32—21no—
258C2×C32—41no—
259C2×C32—41no—
260C2×C32—41no—
261C2×C32—122no—
262C2×C32#87812yes—
263C2×C32#89711yescharacteristic
264C2×C32C2×D2411yescharacteristic
265C2×C32#90911yescharacteristic
266C2×C32D3×D812yes—
267C2×C32#91512yes—
268C3×D3—44no—
269C3×D3—44no—
270C2×Q8—541no—
271C2×D4—31no—
272C22×C4—541no—
273C2×C8—32no—
274C4⋊C4—92no—
275C4⋊C4—271no—
276C4⋊C4—541no—
277C22⋊C4—541no—
278C42—91no—
279C22×C3—21no—
280C22×C3—22no—
281C22×C3—41no—
282C22×C3—64no—
283C22×C3—64no—
284C22×C3—122no—
285C22×C3—122no—
286C22×C3—122no—
287C22×C3—122no—
288C22×C3D3×D1212yes—
289C22×C3#381111yescharacteristic
290C2×D3—22no—
291C2×D3—41no—
292C2×D3—41no—
293C3×C4—21no—
294C3×C4—22no—
295C3×C4—22no—
296C3×C4—21no—
297C3×C4—41no—
298C3×C4—41no—
299C3×C4—92no—
300C3×C4—181no—
301C3×C4—362no—
302C3×C4D3×Dic612yes—
303C3×C4#380512yes—
304C3×C4#380511yescharacteristic
305C3×C4C2×#62711yescharacteristic
306Dic3—34no—
307Dic3—62no—
308Dic3—62no—
309Dic3—121no—
310Dic3—122no—
311Dic3—122no—
312Dic3—242no—
313Dic3—362no—
314Dic3—362no—
315Dic3—361no—
316Dic3—722no—
317Dic3—722no—
318C32—21no—
319C32—22no—
320C32—41no—
321C32#475912yescomplemented
322C32#484611yescharacteristic, complemented
323C23—61no—
324Q8—1082no—
325D4—32no—
326D4—61no—
327C2×C4—91no—
328C2×C4—91no—
329C2×C4—182no—
330C2×C4—541no—
331C2×C4—1082no—
332C2×C4D3311yescharacteristic
333C8—62no—
334C2×C3—21no—
335C2×C3—22no—
336C2×C3—21no—
337C2×C3—22no—
338C2×C3—21no—
339C2×C3—22no—
340C2×C3—41no—
341C2×C3—41no—
342C2×C3—41no—
343C2×C3—124no—
344C2×C3—124no—
345C2×C3D3×D2412yes—
346C2×C3#6380012yes—
347C2×C3#6389312yes—
348C2×C3#6392611yescharacteristic
349C2×C3#6393111yescharacteristic
350C2×C3#6398211yescharacteristic
351D3—42no—
352C22—62no—
353C22—121no—
354C22—121no—
355C22#8895511yescharacteristic
356C4—92no—
357C4—181no—
358C4—362no—
359C4—1081no—
360C4D3×#62711yescharacteristic
361C4#8896311yescharacteristic
362C3—21no—
363C3—22no—
364C3—41no—
365C3#1059076212yescomplemented
366C3#1059149411yescharacteristic, complemented
367C2—122no—
368C2#1172283311yescharacteristic
369C2#1172289511yescharacteristic
370C2#1172298111yescharacteristic
subgroupG/H
centerC22#88955
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypocenterC33#474
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C4⋊#6296C2semidirect
3. #11722055C2semidirect
4. C3×#63572C2semidirect
29. #63572D3semidirect
38. C32×D12Q8semidirect
126. C3×D12Dic6semidirect
213. C33#474semidirect
244. D12#627semidirect
321. C32#4759semidirect
322. C32#4846semidirect
365. C3#10590762semidirect
366. C3#10591494semidirect

Maximal subgroups

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

subgroupindex
C4⋊#62962
D12×C3⋊Dic32
#117220552
C3×#635722
C3×#635632
#105914943
#105907623