Finite Group Databasebeta

← #13167481

Subgroups — #13167481

Order 1728.

596 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 107 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×C3×C4⋊#157C211yesmaximal, characteristic
2C2×#88940C212yesmaximal
3#11723689C211yesmaximal, characteristic
4C4×#6294C211yesmaximal, characteristic
5#11722791C212yesmaximal
6#10593412—31nomaximal
7C22×C3×#157C2212yes—
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×#6294C2212yes—
10C2×#6294C2211yescharacteristic
11C2×#6294C2211yescharacteristic
12C3×C4⋊#157C414yescomplemented
13#88940—24no—
14C3×C4⋊(C3⋊Dic3)—21no—
15C3×C4×C3⋊Dic3—21no—
16#88815—21no—
17#88813—21no—
18C2×C4⋊#157D311yescharacteristic
19C2×#3773—32no—
20#63900—31no—
21D3×C4⋊Dic3—61no—
22C4×#624—31no—
23C4×D3×Dic3—61no—
24#63737—32no—
25C22×C3×C3⋊D3C2311yescharacteristic
26C2×C3×#157—22no—
27C2×C3×#157—22no—
28C2×C3×#157C2×C414yes—
29C2×C33×C4D411yescharacteristic
30C2×C3×C3⋊Dic3—21no—
31C2×C3×C3⋊Dic3—21no—
32C2×C3×C3⋊Dic3—22no—
33C2×C3×C3⋊Dic3D411yescharacteristic
34C3×C4×C3⋊D3C2×C412yes—
35C3×C4×C3⋊D3Q812yes—
36#6294—21no—
37#6294—21no—
38#6294—22no—
39#6294—22no—
40#6294D412yes—
41#6294Q812yes—
42#4984—91nomaximal
43C22×#157C2×D312yes—
44C2×C4×C3⋊D3C2×D311yescharacteristic
45C2×C3×C4×D3—61no—
46C2×C3×C4×D3—61no—
47C2×#624—31no—
48C2×#624—31no—
49C2×#624—32no—
50C2×D3×Dic3—61no—
51C2×D3×Dic3—61no—
52C2×D3×Dic3—62no—
53C4⋊#157Dic314yescomplemented
54#3773—64no—
55C4⋊(C3⋊Dic3)—21no—
56C4×C3⋊Dic3—21no—
57C3×C4⋊Dic3—61no—
58C3×C4⋊Dic3—61no—
59C3×C4⋊Dic3—121no—
60C3×C4×Dic3—61no—
61C3×C4×Dic3—61no—
62C3×C4×Dic3—121no—
63#3724—61no—
64Dic3⋊Dic3—62no—
65Dic32—61no—
66Dic32—61no—
67Dic32—61no—
68C22×C33C2×D411yescharacteristic
69C2×C3×C3⋊D3C22×C411yescharacteristic
70C2×C3×C3⋊D3C2×D411yescharacteristic
71C2×C3×C3⋊D3C2×Q811yescharacteristic
72C2×C3×C3⋊D3C2×Q811yescharacteristic
73C2×C3×C3⋊D3#4111yescharacteristic
74C2×C3×C3⋊D3#4111yescharacteristic
75C3×#157—24no—
76C33×C4C4⋊C412yescomplemented
77C3×C3⋊Dic3—22no—
78C3×C3⋊Dic3—22no—
79C3×C3⋊Dic3—42no—
80C3×C3⋊Dic3C4⋊C412yescomplemented
81C2×C3×C4⋊C4—91no—
82C2×C4⋊Dic3—91no—
83C2×C4⋊Dic3—92no—
84C2×C4×Dic3—91no—
85D3×C4⋊C4—181no—
86C42×D3—181no—
87C3⋊#95—92no—
88C22×C3⋊D3C22×D311yescharacteristic
89C22×C3×D3—61no—
90C22×C3×D3—61no—
91C2×#157—22no—
92C2×#157—22no—
93C2×#157C2×Dic314yes—
94C2×C32×C4—21no—
95C2×C32×C4—21no—
96C2×C32×C4—41no—
97C2×C32×C4—41no—
98C2×C32×C4D1211yescharacteristic
99C2×C3⋊Dic3—21no—
100C2×C3⋊Dic3—21no—
101C2×C3⋊Dic3—22no—
102C2×C3⋊Dic3D1211yescharacteristic
103C4×C3⋊D3Dic612yes—
104C4×C3⋊D3C2×Dic312yes—
105C2×C3×Dic3—61no—
106C2×C3×Dic3—61no—
107C2×C3×Dic3—61no—
108C2×C3×Dic3—61no—
109C2×C3×Dic3—61no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—121no—
114C2×C3×Dic3—121no—
115C2×C3×Dic3—122no—
116C3×C4×D3—62no—
117C3×C4×D3—62no—
118C3×C4×D3—62no—
119C3×C4×D3—62no—
120#624—32no—
121#624—32no—
122#624—61no—
123#624—61no—
124#624—62no—
125#624—62no—
126D3×Dic3—62no—
127D3×Dic3—62no—
128D3×Dic3—62no—
129D3×Dic3—62no—
130D3×Dic3—122no—
131D3×Dic3—122no—
132#389—271noSylow
133C2×C33C2×C4⋊C411yescharacteristic
134C2×C33#12111yescharacteristic
135C2×C33C22⋊Q811yescharacteristic
136C3×C3⋊D3C4×D411yescharacteristic
137C3×C3⋊D3C4×Q811yescharacteristic
138C3×C3⋊D3#12511yescharacteristic
139C3×C3⋊D3C4⋊Q811yescharacteristic
140C22×C3×C4—91no—
141C22×C3×C4—92no—
142C22×Dic3—91no—
143C22×Dic3—91no—
144C22×Dic3—92no—
145C2×C4×D3—61no—
146C2×C4×D3—61no—
147C2×C4×D3—181no—
148C2×C4×D3—181no—
149C2×C4×D3—182no—
150C3×C4⋊C4—94no—
151C3×C4⋊C4—181no—
152C3×C42—181no—
153C4⋊Dic3—61no—
154C4⋊Dic3—61no—
155C4⋊Dic3—121no—
156C4⋊Dic3—181no—
157C4⋊Dic3—184no—
158Dic3⋊C4—182no—
159C4×Dic3—61no—
160C4×Dic3—61no—
161C4×Dic3—121no—
162C4×Dic3—181no—
163C4×Dic3—181no—
164C4×Dic3—181no—
165C22×C32—21no—
166C22×C32—21no—
167C22×C32—41no—
168C22×C32—41no—
169C22×C32C2×D1211yescharacteristic
170C2×C3⋊D3C2×Dic611yescharacteristic
171C2×C3⋊D3D3×D411yescharacteristic
172C2×C3⋊D3#23711yescharacteristic
173C2×C3⋊D3D3×Q811yescharacteristic
174C2×C3⋊D3#23911yescharacteristic
175C2×C3⋊D3C22×Dic311yescharacteristic
176C2×C3×D3—61no—
177C2×C3×D3—61no—
178C2×C3×D3—61no—
179C2×C3×D3—61no—
180C2×C3×D3—61no—
181C2×C3×D3—61no—
182C2×C3×D3—61no—
183C2×C3×D3—61no—
184C2×C3×D3—61no—
185C2×C3×D3—61no—
186C2×C3×D3—61no—
187C2×C3×D3—61no—
188#157—24no—
189C32×C4—22no—
190C32×C4—22no—
191C32×C4—42no—
192C32×C4—42no—
193C32×C4C4⋊Dic312yescomplemented
194C3⋊Dic3—22no—
195C3⋊Dic3—22no—
196C3⋊Dic3—42no—
197C3⋊Dic3C4⋊Dic312yescomplemented
198C3×Dic3—62no—
199C3×Dic3—62no—
200C3×Dic3—62no—
201C3×Dic3—62no—
202C3×Dic3—62no—
203C3×Dic3—62no—
204C3×Dic3—122no—
205C3×Dic3—122no—
206C3×Dic3—122no—
207C3×Dic3—122no—
208C3×Dic3—242no—
209C2×C4⋊C4—91no—
210C2×C4⋊C4—271no—
211C2×C4⋊C4—272no—
212C2×C42—271no—
213#95—272no—
214C33#38911yescharacteristic, complemented, Sylow
215C23×C3—91no—
216C22×D3—61no—
217C22×D3—61no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—91no—
223C2×C3×C4—92no—
224C2×C3×C4—92no—
225C2×C3×C4—94no—
226C2×C3×C4—181no—
227C2×C3×C4—181no—
228C2×C3×C4—182no—
229C2×C3×C4—182no—
230C2×C3×C4—182no—
231C2×C3×C4GO+(4,𝔽2)11yescharacteristic
232C2×Dic3—61no—
233C2×Dic3—61no—
234C2×Dic3—61no—
235C2×Dic3—61no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—62no—
239C2×Dic3—62no—
240C2×Dic3—92no—
241C2×Dic3—92no—
242C2×Dic3—121no—
243C2×Dic3—121no—
244C2×Dic3—122no—
245C2×Dic3—181no—
246C2×Dic3—181no—
247C2×Dic3—182no—
248C2×Dic3—182no—
249C4×D3—62no—
250C4×D3—62no—
251C4×D3—62no—
252C4×D3—62no—
253C4×D3—182no—
254C4×D3—182no—
255C4×D3—182no—
256C4×D3—182no—
257C4×D3—362no—
258C4×D3—362no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32—21no—
263C2×C32—21no—
264C2×C32—21no—
265C2×C32—41no—
266C2×C32—41no—
267C2×C32—41no—
268C2×C32—41no—
269C2×C32—41no—
270C2×C32—41no—
271C2×C32#89511yescharacteristic
272C2×C32#89711yescharacteristic
273C2×C32C2×C4⋊Dic311yescharacteristic
274C3⋊D3#88811yescharacteristic
275C3⋊D3#89011yescharacteristic
276C3⋊D3D4×Dic311yescharacteristic
277C3⋊D3Q8×Dic311yescharacteristic
278C3×D3—61no—
279C3×D3—61no—
280C3×D3—61no—
281C3×D3—61no—
282C3×D3—61no—
283C3×D3—61no—
284C3×D3—61no—
285C3×D3—61no—
286C22×C4—91no—
287C22×C4—92no—
288C22×C4—271no—
289C22×C4—271no—
290C22×C4—272no—
291C4⋊C4—94no—
292C4⋊C4—181no—
293C4⋊C4—541no—
294C4⋊C4—544no—
295C42—181no—
296C42—541no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—91no—
302C22×C3—91no—
303C22×C3—91no—
304C22×C3—91no—
305C22×C3—91no—
306C22×C3—91no—
307C22×C3C2×GO+(4,𝔽2)11yescharacteristic
308C2×D3—61no—
309C2×D3—61no—
310C2×D3—61no—
311C2×D3—61no—
312C2×D3—61no—
313C2×D3—61no—
314C2×D3—61no—
315C2×D3—61no—
316C2×D3—61no—
317C2×D3—61no—
318C2×D3—61no—
319C2×D3—61no—
320C3×C4—22no—
321C3×C4—22no—
322C3×C4—42no—
323C3×C4—42no—
324C3×C4—92no—
325C3×C4—182no—
326C3×C4—182no—
327C3×C4—184no—
328C3×C4—362no—
329C3×C4#377312yescomplemented
330Dic3—62no—
331Dic3—62no—
332Dic3—62no—
333Dic3—62no—
334Dic3—62no—
335Dic3—62no—
336Dic3—122no—
337Dic3—122no—
338Dic3—122no—
339Dic3—122no—
340Dic3—242no—
341C32—21no—
342C32—21no—
343C32—41no—
344C32—41no—
345C32#498411yescharacteristic, complemented
346C23—91no—
347C2×C4—91no—
348C2×C4—92no—
349C2×C4—92no—
350C2×C4—94no—
351C2×C4—182no—
352C2×C4—182no—
353C2×C4—181no—
354C2×C4—181no—
355C2×C4—182no—
356C2×C4—272no—
357C2×C4—272no—
358C2×C4—541no—
359C2×C4—541no—
360C2×C4—542no—
361C2×C4—542no—
362C2×C4#157⋊D311yescharacteristic
363C2×C3—21no—
364C2×C3—21no—
365C2×C3—21no—
366C2×C3—21no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3—41no—
370C2×C3—41no—
371C2×C3—41no—
372C2×C3—41no—
373C2×C3—41no—
374C2×C3—41no—
375C2×C3—91no—
376C2×C3—91no—
377C2×C3—91no—
378C2×C3—91no—
379C2×C3D32⋊Q811yescharacteristic
380C2×C3#6423011yescharacteristic
381C2×C3C2×#377311yescharacteristic
382D3—61no—
383D3—61no—
384D3—61no—
385D3—61no—
386D3—61no—
387D3—61no—
388D3—61no—
389D3—61no—
390C22—91no—
391C22—91no—
392C22—91no—
393C22—91no—
394C22—91no—
395C22—91no—
396C22C2×#157⋊D311yescharacteristic
397C4—92no—
398C4—184no—
399C4—182no—
400C4—182no—
401C4—362no—
402C4#8894012yescomplemented
403C3—21no—
404C3—21no—
405C3—41no—
406C3—41no—
407C3#1059341211yescharacteristic, complemented
408C2—91no—
409C2—91no—
410C2—91no—
411C2—91no—
412C2#1172433311yescharacteristic
413C2#1172434311yescharacteristic
414C2C2×#8894011yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutatorC2×C3×C3⋊D3C22×C4
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radical, cosocleC22×C3×C3⋊D3C23
hypercenterC2×C4#157⋊D3
hypocenterC33#389
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
12. C3×C4⋊#157C4semidirect
53. C4⋊#157Dic3semidirect
76. C33×C4C4⋊C4semidirect
80. C3×C3⋊Dic3C4⋊C4semidirect
193. C32×C4C4⋊Dic3semidirect
197. C3⋊Dic3C4⋊Dic3semidirect
214. C33#389semidirect
329. C3×C4#3773semidirect
345. C32#4984semidirect
402. C4#88940semidirect
407. C3#10593412semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊#1572
C2×#889402
#117236892
C4×#62942
#117227912
#105934123
#49849