Finite Group Databasebeta

← #13155515

Subgroups — #13155515

Order 1728.

686 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 134 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×D3×C3⋊Dic3C211yesmaximal, characteristic
2#11722070C211yesmaximal, characteristic, complemented
3#11722050C212yesmaximal, complemented
4C3×(C2×D3)⋊(C3⋊C8)C212yesmaximal
5C3×#63553C211yesmaximal, characteristic, complemented
6#10591061—31nomaximal
7#10590655—32nomaximal
8C2×C32×C4×D3C2211yescharacteristic
9C2×D3×C3⋊Dic3C411yescharacteristic
10C2×D3×C3⋊Dic3C411yescharacteristic
11C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C2×C3×C3⋊(C3⋊C8)C2212yes—
14C2×C32×C3⋊C8C2212yes—
15Dic3×C3⋊Dic3C411yescharacteristic
16Dic3×C3⋊Dic3C411yescharacteristic
17C2×C4×C3⋊Dic3—31no—
18C4×D3×Dic3—32no—
19C4×D3×Dic3—61no—
20#63641—32no—
21C3×#830—32no—
22C3×(C2×D3)⋊C8—32no—
23C3×C8×Dic3—32no—
24#63572—32no—
25#63572—32no—
26#63572—62no—
27(C2×D3)⋊(C3⋊C8)D312yes—
28#63553—31no—
29#63553D311yescharacteristic, complemented
30Dic3×C3⋊C8—32no—
31C22×C32×D3C2×C411yescharacteristic
32C2×C33×C4C2311yescharacteristic
33C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
34C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
35C2×C3×C3⋊Dic3C2×C411yescharacteristic
36C2×C3×C3⋊Dic3C2×C411yescharacteristic
37C2×C32×Dic3C2×C411yescharacteristic
38C32×C4×D3—22no—
39D3×C3⋊Dic3—22no—
40D3×C3⋊Dic3C814yescomplemented
41C3×C3⋊(C3⋊C8)—22no—
42C32×C3⋊C8—22no—
43#4718—92no—
44#4685—91no—
45C22×C32×C4—31no—
46C22×C3⋊Dic3—31no—
47C22×C3⋊Dic3—31no—
48C2×C3×C4×D3—21no—
49C2×C3×C4×D3C2×D312yes—
50C2×D3×Dic3—32no—
51C2×D3×Dic3—32no—
52C2×D3×Dic3—61no—
53C2×D3×Dic3—61no—
54C2×C32×C8—32no—
55C4×C3⋊Dic3—31no—
56C4×C3⋊Dic3—32no—
57C4×C3⋊Dic3—31no—
58C4×C3⋊Dic3—31no—
59C4×C3⋊Dic3—62no—
60C4×C3⋊Dic3—61no—
61C4×C3⋊Dic3—121no—
62C4×C3⋊Dic3C2×D311yescharacteristic
63C2×C3⋊(C3⋊C8)C2×D312yes—
64C3×C4×Dic3—32no—
65C3×C4×Dic3—61no—
66C2×C3×C3⋊C8—22no—
67C2×C3×C3⋊C8—32no—
68C2×C3×C3⋊C8—32no—
69C2×C3×C3⋊C8—62no—
70C2×C3×C3⋊C8C2×D312yes—
71C2×C3×C3⋊C8C2×D312yes—
72Dic32—32no—
73Dic32—32no—
74Dic32—61no—
75Dic32—61no—
76C22×C33C22×C411yescharacteristic
77C2×C32×D3—21no—
78C2×C32×D3—21no—
79C2×C32×D3C2×C812yes—
80C33×C4#4112yes—
81C3⋊(C3⋊Dic3)—21no—
82C3⋊(C3⋊Dic3)C2×C812yes—
83C3×C3⋊Dic3C2×C812yes—
84C3×C3⋊Dic3M1612yes—
85C32×Dic3—21no—
86C2×C4×Dic3—92no—
87C2×C4×Dic3—181no—
88C42×D3—91no—
89C3×C22⋊C8—92no—
90C3×C4×C8—91no—
91#830—32no—
92(C2×D3)⋊C8—32no—
93Dic3⋊C8—92no—
94Dic3⋊C8—92no—
95Dic3⋊C8—92no—
96Dic3⋊C8—182no—
97C8×Dic3—32no—
98C8×Dic3—92no—
99C4×C3⋊C8—91no—
100C23×C32—31no—
101C22×C3×D3—21no—
102C22×C3×D3C4×D312yes—
103C2×C32×C4—22no—
104C2×C32×C4—21no—
105C2×C32×C4—31no—
106C2×C32×C4—41no—
107C2×C32×C4—62no—
108C2×C32×C4C22×D311yescharacteristic
109C2×C32×C4C22×D312yes—
110C2×C3⋊Dic3—32no—
111C2×C3⋊Dic3—31no—
112C2×C3⋊Dic3—34no—
113C2×C3⋊Dic3—32no—
114C2×C3⋊Dic3—31no—
115C2×C3⋊Dic3—61no—
116C2×C3⋊Dic3—61no—
117C2×C3⋊Dic3—62no—
118C2×C3⋊Dic3—62no—
119C2×C3⋊Dic3—62no—
120C2×C3⋊Dic3—121no—
121C2×C3⋊Dic3—121no—
122C2×C3⋊Dic3C4×D311yescharacteristic
123C2×C3⋊Dic3C4×D311yescharacteristic
124C2×C3×Dic3—21no—
125C2×C3×Dic3—32no—
126C2×C3×Dic3—32no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3C4×D312yes—
130C3×C4×D3—24no—
131C3×C4×D3—22no—
132C3×C4×D3—22no—
133D3×Dic3—38no—
134D3×Dic3—64no—
135D3×Dic3—64no—
136D3×Dic3—64no—
137C32×C8—62no—
138C3⋊(C3⋊C8)—22no—
139C3×C3⋊C8—22no—
140C3×C3⋊C8—22no—
141C3×C3⋊C8—42no—
142C3×C3⋊C8—62no—
143C3×C3⋊C8—62no—
144C3×C3⋊C8—122no—
145#431—271noSylow
146C2×C33#11711yescharacteristic
147C2×C33C22×C811yescharacteristic
148C2×C33C2×M1611yescharacteristic
149C32×D3—22no—
150C22×C3×C4—32no—
151C22×C3×C4—61no—
152C22×Dic3—92no—
153C22×Dic3—92no—
154C22×Dic3—181no—
155C22×Dic3—181no—
156C2×C4×D3—91no—
157C2×C4×D3—91no—
158C2×C4×D3D3211yescharacteristic
159C2×C3×C8—32no—
160C2×C3×C8—32no—
161C2×C3×C8—62no—
162C2×C3×C8—92no—
163C3×C42—91no—
164C4×Dic3—32no—
165C4×Dic3—61no—
166C4×Dic3—92no—
167C4×Dic3—92no—
168C4×Dic3—91no—
169C4×Dic3—92no—
170C4×Dic3—91no—
171C4×Dic3—91no—
172C4×Dic3—181no—
173C4×Dic3—181no—
174C4×Dic3—182no—
175C4×Dic3—181no—
176C4×Dic3—361no—
177C2×C3⋊C8—32no—
178C2×C3⋊C8—32no—
179C2×C3⋊C8—62no—
180C2×C3⋊C8D3212yes—
181C22×C32—22no—
182C22×C32—21no—
183C22×C32—32no—
184C22×C32—41no—
185C22×C32—61no—
186C22×C32—61no—
187C22×C32C2×C4×D311yescharacteristic
188C22×C32C2×C4×D312yes—
189C2×C3×D3—22no—
190C2×C3×D3—22no—
191C2×C3×D3—22no—
192C2×C3×D3—22no—
193C2×C3×D3—22no—
194C2×C3×D3D3×C814yes—
195C32×C4—22no—
196C32×C4—22no—
197C32×C4—22no—
198C32×C4—42no—
199C32×C4—61no—
200C32×C4#23512yes—
201C32×C4#23712yes—
202C32×C4#23712yes—
203C3⋊Dic3—34no—
204C3⋊Dic3—32no—
205C3⋊Dic3—62no—
206C3⋊Dic3—64no—
207C3⋊Dic3—61no—
208C3⋊Dic3—62no—
209C3⋊Dic3—62no—
210C3⋊Dic3—122no—
211C3⋊Dic3—122no—
212C3⋊Dic3—122no—
213C3⋊Dic3D3×C812yes—
214C3⋊Dic3ΓL(1,𝔽25)12yes—
215C3×Dic3—22no—
216C3×Dic3—22no—
217C3×Dic3—34no—
218C3×Dic3—34no—
219C3×Dic3—62no—
220C3×Dic3—62no—
221C2×C42—271no—
222C4⋊C8—272no—
223C22⋊C8—92no—
224C4×C8—91no—
225C4×C8—271no—
226C33#43111yescharacteristic, complemented, Sylow
227C23×C3—32no—
228C23×C3—61no—
229C22×D3#62411yescharacteristic
230C2×C3×C4—22no—
231C2×C3×C4—21no—
232C2×C3×C4—32no—
233C2×C3×C4—41no—
234C2×C3×C4—61no—
235C2×C3×C4—64no—
236C2×C3×C4—62no—
237C2×C3×C4—62no—
238C2×C3×C4—91no—
239C2×C3×C4—91no—
240C2×C3×C4C2×D3212yes—
241C2×C3×C4C2×D3211yescharacteristic
242C2×Dic3—32no—
243C2×Dic3—32no—
244C2×Dic3—61no—
245C2×Dic3—61no—
246C2×Dic3—98no—
247C2×Dic3—92no—
248C2×Dic3—91no—
249C2×Dic3—92no—
250C2×Dic3—91no—
251C2×Dic3—184no—
252C2×Dic3—182no—
253C2×Dic3—181no—
254C2×Dic3—181no—
255C2×Dic3—182no—
256C2×Dic3—184no—
257C2×Dic3—184no—
258C2×Dic3—361no—
259C2×Dic3—361no—
260C2×Dic3#62411yescharacteristic
261C4×D3—22no—
262C4×D3—94no—
263C4×D3—182no—
264C3×C8—62no—
265C3×C8—62no—
266C3×C8—122no—
267C3×C8—182no—
268C3⋊C8—22no—
269C3⋊C8—62no—
270C3⋊C8—62no—
271C3⋊C8—122no—
272C2×C32—22no—
273C2×C32—22no—
274C2×C32—21no—
275C2×C32—21no—
276C2×C32—22no—
277C2×C32—21no—
278C2×C32—41no—
279C2×C32—41no—
280C2×C32—41no—
281C2×C32—62no—
282C2×C32#87211yescharacteristic
283C2×C32#87712yes—
284C2×C32C2×D3×C811yescharacteristic
285C2×C32C2×D3×C812yes—
286C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
287C2×C32D3×M1612yes—
288C3×D3—24no—
289C3×D3—24no—
290C22×C4—31no—
291C22×C4—271no—
292C22×C4—271no—
293C2×C8—32no—
294C2×C8—92no—
295C42—91no—
296C42—271no—
297C42—271no—
298C42—271no—
299C22×C3—21no—
300C22×C3—22no—
301C22×C3—34no—
302C22×C3—41no—
303C22×C3—62no—
304C22×C3—62no—
305C22×C3—62no—
306C22×C3—62no—
307C22×C3—62no—
308C22×C3C4×D3212yes—
309C22×C3C2×#62411yescharacteristic
310C2×D3—21no—
311C2×D3—21no—
312C2×D3#371012yes—
313C3×C4—22no—
314C3×C4—22no—
315C3×C4—22no—
316C3×C4—42no—
317C3×C4—62no—
318C3×C4—62no—
319C3×C4—92no—
320C3×C4—92no—
321C3×C4#379512yes—
322C3×C4#380412yes—
323C3×C4#380512yes—
324Dic3—21no—
325Dic3—34no—
326Dic3—34no—
327Dic3—62no—
328Dic3—62no—
329Dic3—94no—
330Dic3—92no—
331Dic3—182no—
332Dic3—184no—
333Dic3—182no—
334Dic3—182no—
335Dic3—181no—
336Dic3—362no—
337Dic3—362no—
338Dic3—362no—
339C32—21no—
340C32—22no—
341C32—41no—
342C32#468511yescharacteristic, complemented
343C32#471812yescomplemented
344C23—31no—
345C2×C4—31no—
346C2×C4—62no—
347C2×C4—91no—
348C2×C4—91no—
349C2×C4—274no—
350C2×C4—271no—
351C2×C4—271no—
352C2×C4—542no—
353C2×C4D3311yescharacteristic
354C8—62no—
355C8—182no—
356C2×C3—21no—
357C2×C3—22no—
358C2×C3—21no—
359C2×C3—22no—
360C2×C3—21no—
361C2×C3—22no—
362C2×C3—41no—
363C2×C3—41no—
364C2×C3—41no—
365C2×C3—64no—
366C2×C3—64no—
367C2×C3D32×C812yes—
368C2×C3D3×ΓL(1,𝔽25)12yes—
369C2×C3C2×#371011yescharacteristic
370C2×C3#6381811yescharacteristic
371C2×C3#6387712yes—
372C2×C3#6395711yescharacteristic
373D3—22no—
374C22—32no—
375C22—61no—
376C22—61no—
377C22D3×#62411yescharacteristic
378C4—61no—
379C4—92no—
380C4—92no—
381C4—272no—
382C4—541no—
383C4#8896112yes—
384C3—21no—
385C3—22no—
386C3—41no—
387C3#1059065512yescomplemented
388C3#1059106111yescharacteristic, complemented
389C2—62no—
390C2D3×#371011yescharacteristic
391C2#1172281811yescharacteristic
392C2#1172291611yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D33
commutatorC2×C33C22×C8
Fitting, cosocleC2×C33×C4C23
socleC22×C33C22×C4
prime radicalC22×C32×D3C2×C4
hypocenterC33#431
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11722070C2semidirect
3. #11722050C2semidirect
5. C3×#63553C2semidirect
29. #63553D3semidirect
40. D3×C3⋊Dic3C8semidirect
226. C33#431semidirect
342. C32#4685semidirect
343. C32#4718semidirect
387. C3#10590655semidirect
388. C3#10591061semidirect

Maximal subgroups

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

subgroupindex
C4×D3×C3⋊Dic32
#117220702
#117220502
C3×(C2×D3)⋊(C3⋊C8)2
C3×#635532
#105910613
#105906553