Finite Group Databasebeta

← #13139950

Subgroups — #13139950

Order 1728.

629 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 629 conjugacy classes of subgroups — too many to draw. Only its 131 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×#3828⋊C3C211yesmaximal, characteristic
2#11721826C211yesmaximal, characteristic, complemented
3C22×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic, complemented
4#11720309C212yesmaximal, complemented
5#11720308C212yesmaximal, complemented
6#10591105—31nomaximal
7#10591083—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C2×#638⋊C3—21no—
10#3828⋊C3—22no—
11#3828⋊C3C414yescomplemented
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C4×(C3⋊D3)⋊C3—21no—
14C2×#634⋊C3C2211yescharacteristic
15#88552—21no—
16C4⋊#1084C2212yes—
17C4×#1084—21no—
18C2×(C3⋊(C3⋊C8))⋊C3—22no—
19C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
20C2×#3828D311yescharacteristic
21C2×C3×#235—31no—
22C22×C3⋊(C3⋊C8)D311yescharacteristic, complemented
23C3×#926—31no—
24C3×#926—31no—
25C3×#926—61no—
26C22×C3×C3⋊C8—31no—
27#63567—32no—
28#63565—32no—
29#63564—32no—
30#63562—32no—
31C23×H(3,𝔽3)D411yescharacteristic
32C22×(C3⋊D3)⋊C3—21no—
33C2×C4×H(3,𝔽3)C2×C412yes—
34C2×C4×H(3,𝔽3)D411yescharacteristic
35C2×C4×H(3,𝔽3)D412yes—
36C2×C4×H(3,𝔽3)C2311yescharacteristic
37C2×#1084—22no—
38#638⋊C3—24no—
39C2×(C3⋊Dic3)⋊C3—21no—
40C4⋊((C3⋊D3)⋊C3)—21no—
41C4⋊((C3⋊D3)⋊C3)C2×C412yes—
42C4×(C3⋊D3)⋊C3—24no—
43#634⋊C3—21no—
44#634⋊C3C2×C412yes—
45(C3⋊(C3⋊C8))⋊C3—22no—
46#5233—91no—
47#5113—91no—
48#5002—91no—
49C22×C32×C4—21no—
50C22×C32×C4C2×D311yescharacteristic
51C22×C32×C4C2×D311yescharacteristic
52C2×#638—21no—
53#3828—22no—
54#3828Dic314yescomplemented
55C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
56C2×C4×C3⋊D3—21no—
57C2×#634C2×D311yescharacteristic
58C2×C3×#67—61no—
59C3×#235—34no—
60C3×#235—62no—
61C2×C3×D12—31no—
62C2×C3×C4×D3—61no—
63C2×C3×Dic6—31no—
64C2×C3⋊(C3⋊C8)—22no—
65C2×C3⋊(C3⋊C8)C2×D312yes—
66C3×#217—62no—
67C3×#217—61no—
68C3×#217—61no—
69C3×C4⋊Dic3—32no—
70C3×C4⋊Dic3—32no—
71C3×C4⋊Dic3—62no—
72C3×C4×Dic3—62no—
73C3×C4×Dic3—61no—
74C3×C4×Dic3—61no—
75C2×C3×C3⋊C8—32no—
76C2×C3×C3⋊C8—62no—
77C22×H(3,𝔽3)—22no—
78C22×H(3,𝔽3)C22⋊C412yes—
79C22×H(3,𝔽3)C2×D411yescharacteristic
80C2×(C3⋊D3)⋊C3—22no—
81C2×(C3⋊D3)⋊C3—42no—
82C4×H(3,𝔽3)C22⋊C412yes—
83C4×H(3,𝔽3)C22×C411yescharacteristic
84C4×H(3,𝔽3)C2×D411yescharacteristic
85#1084—42no—
86(C3⋊Dic3)⋊C3—22no—
87C2×C3×#41—91no—
88C2×#235—31no—
89C2×#235—91no—
90C22×C3×C8—91no—
91C3×#117—91no—
92#926—31no—
93#926—31no—
94#926—61no—
95C22×C3⋊C8—31no—
96C22×C3⋊C8—91no—
97#835—92no—
98#832—92no—
99#821—92no—
100#816—92no—
101#810—92no—
102#809—92no—
103C23×C32—21no—
104C23×C32D1211yescharacteristic
105C23×C32#6711yescharacteristic
106C22×C3⋊D3—21no—
107C22×C3×D3—61no—
108C2×C32×C4—22no—
109C2×C32×C4—22no—
110C2×C32×C4—21no—
111C2×C32×C4—21no—
112C2×C32×C4C4×D312yes—
113C2×C32×C4D1211yescharacteristic
114C2×C32×C4C2×Dic312yes—
115C2×C32×C4#6712yes—
116C2×C32×C4#6712yes—
117C2×C32×C4#6711yescharacteristic
118C2×C32×C4C22×D311yescharacteristic
119C2×C32×C4C22×D311yescharacteristic
120#638—24no—
121C2×C3⋊Dic3—21no—
122C4⋊(C3⋊D3)—21no—
123C4⋊(C3⋊D3)C2×Dic312yes—
124C4×C3⋊D3—24no—
125#634—21no—
126#634C2×Dic312yes—
127C3×#67—64no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—64no—
130C2×C3×Dic3—62no—
131C2×C3×Dic3—62no—
132C3×D12—32no—
133C3×D12—61no—
134C3×C4×D3—64no—
135C3×Dic6—32no—
136C3×Dic6—61no—
137C3⋊(C3⋊C8)—22no—
138C3×C3⋊C8—62no—
139#416—271noSylow
140C2×H(3,𝔽3)—22no—
141C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
142C2×H(3,𝔽3)#13512yes—
143(C3⋊D3)⋊C3—42no—
144C3×#41—94no—
145C3×#41—182no—
146C2×C3×Q8—91no—
147C2×C3×D4—91no—
148C2×C3×D4—181no—
149C22×C3×C4—31no—
150C22×C3×C4—31no—
151C22×C3×C4—61no—
152C22×C3×C4—181no—
153C22×C3×C4D3211yescharacteristic
154C2×#67—61no—
155C2×#67—181no—
156#235—34no—
157#235—62no—
158#235—94no—
159#235—182no—
160C2×D12—31no—
161C2×D12—91no—
162C2×C4×D3—61no—
163C2×C4×D3—181no—
164C2×Dic6—31no—
165C2×Dic6—91no—
166C2×C3×C8—92no—
167C2×C3×C8—182no—
168C3×C4⋊C4—92no—
169C3×C22⋊C4—181no—
170C3×C42—181no—
171#217—61no—
172#217—61no—
173#217—62no—
174C4⋊Dic3—32no—
175C4⋊Dic3—32no—
176C4⋊Dic3—62no—
177C4×Dic3—61no—
178C4×Dic3—61no—
179C4×Dic3—62no—
180C2×C3⋊C8—32no—
181C2×C3⋊C8—62no—
182C2×C3⋊C8—92no—
183C2×C3⋊C8—182no—
184C22×C32—22no—
185C22×C32—22no—
186C22×C32—22no—
187C22×C32—21no—
188C22×C32—42no—
189C22×C32(C2×D3)⋊C412yes—
190C22×C32#21712yes—
191C22×C32C2×D1211yescharacteristic
192C22×C32C2×#6711yescharacteristic
193C2×C3⋊D3—22no—
194C2×C3⋊D3—42no—
195C2×C3×D3—62no—
196C2×C3×D3—122no—
197C32×C4—22no—
198C32×C4—21no—
199C32×C4—21no—
200C32×C4(C2×D3)⋊C412yes—
201C32×C4#21712yes—
202C32×C4C2×C4×D311yescharacteristic
203C32×C4C22×Dic311yescharacteristic
204C32×C4C2×#6711yescharacteristic
205C32×C4C2×#6711yescharacteristic
206C3⋊Dic3—22no—
207C3×Dic3—62no—
208C3×Dic3—124no—
209C3×Dic3—122no—
210C3×Dic3—122no—
211C2×#41—91no—
212C22×C8—91no—
213#117—91no—
214#103—272no—
215#102—272no—
216H(3,𝔽3)#41611yescharacteristic, complemented, Sylow
217C23×C3—31no—
218C23×C3—31no—
219C23×C3—61no—
220C23×C3—181no—
221C23×C3#62611yescharacteristic
222C22×D3—61no—
223C22×D3—181no—
224C3×Q8—92no—
225C3×Q8—181no—
226C3×D4—92no—
227C3×D4—181no—
228C3×D4—184no—
229C2×C3×C4—32no—
230C2×C3×C4—32no—
231C2×C3×C4—32no—
232C2×C3×C4—32no—
233C2×C3×C4—31no—
234C2×C3×C4—31no—
235C2×C3×C4—31no—
236C2×C3×C4—31no—
237C2×C3×C4—62no—
238C2×C3×C4—62no—
239C2×C3×C4—61no—
240C2×C3×C4—61no—
241C2×C3×C4—184no—
242C2×C3×C4—181no—
243C2×C3×C4—182no—
244C2×C3×C4D3×Dic312yes—
245C2×C3×C4#62512yes—
246C2×C3×C4#62611yescharacteristic
247C2×C3×C4C2×D3211yescharacteristic
248#67—64no—
249#67—184no—
250C2×Dic3—61no—
251C2×Dic3—62no—
252C2×Dic3—62no—
253C2×Dic3—64no—
254C2×Dic3—181no—
255D12—32no—
256D12—61no—
257D12—92no—
258D12—181no—
259C4×D3—64no—
260C4×D3—184no—
261Dic6—32no—
262Dic6—61no—
263Dic6—92no—
264Dic6—181no—
265C3×C8—182no—
266C3⋊C8—62no—
267C3⋊C8—182no—
268C2×C32—22no—
269C2×C32—22no—
270C2×C32—22no—
271C2×C32—21no—
272C2×C32—42no—
273C2×C32#90412yes—
274C2×C32C2×(C2×D3)⋊C411yescharacteristic
275C2×C32#95012yes—
276C2×C32C2×#21711yescharacteristic
277C3⋊D3—42no—
278C3×D3—122no—
279#41—94no—
280#41—182no—
281C2×Q8—91no—
282C2×D4—91no—
283C2×D4—181no—
284C22×C4—181no—
285C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
286C2×C8—92no—
287C2×C8—182no—
288C4⋊C4—92no—
289C22⋊C4—181no—
290C42—181no—
291C22×C3—22no—
292C22×C3—31no—
293C22×C3—31no—
294C22×C3—32no—
295C22×C3—32no—
296C22×C3—62no—
297C22×C3—62no—
298C22×C3—61no—
299C22×C3—62no—
300C22×C3—122no—
301C22×C3—182no—
302C22×C3—362no—
303C22×C3(C2×D3)⋊Dic312yes—
304C22×C3C2×#62611yescharacteristic
305C2×D3—62no—
306C2×D3—122no—
307C2×D3—182no—
308C2×D3—362no—
309C3×C4—31no—
310C3×C4—31no—
311C3×C4—31no—
312C3×C4—31no—
313C3×C4—32no—
314C3×C4—32no—
315C3×C4—61no—
316C3×C4—61no—
317C3×C4—62no—
318C3×C4—182no—
319C3×C4—362no—
320C3×C4(C2×D3)⋊Dic312yes—
321C3×C4C2×D3×Dic311yescharacteristic
322C3×C4C2×#62511yescharacteristic
323Dic3—62no—
324Dic3—122no—
325Dic3—122no—
326Dic3—124no—
327Dic3—182no—
328C32—21no—
329C32#511311yescharacteristic, complemented
330C32#523311yescharacteristic, complemented
331C23—181no—
332C23#620011yescharacteristic
333Q8—92no—
334Q8—181no—
335D4—92no—
336D4—184no—
337D4—181no—
338C2×C4—181no—
339C2×C4—184no—
340C2×C4—182no—
341C2×C4((C3⋊D3)⋊C3)⋊C412yes—
342C2×C4#619812yes—
343C2×C4#620011yescharacteristic
344C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
345C8—182no—
346C2×C3—22no—
347C2×C3—31no—
348C2×C3—31no—
349C2×C3—32no—
350C2×C3—32no—
351C2×C3—62no—
352C2×C3—62no—
353C2×C3—61no—
354C2×C3—62no—
355C2×C3—122no—
356C2×C3—362no—
357C2×C3#6382912yes—
358C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
359D3—122no—
360D3—362no—
361C22—22no—
362C22—182no—
363C22—362no—
364C22#8845012yes—
365C22C2×#620011yescharacteristic
366C4—182no—
367C4—362no—
368C4#8845012yes—
369C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
370C4C2×#619811yescharacteristic
371C3—31no—
372C3—31no—
373C3—61no—
374C3#1059108411yescharacteristic
375C2—22no—
376C2—362no—
377C2#1172107312yes—
378C2C2×#8845011yescharacteristic
subgroupG/H
centerC2×C4#6200
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radicalC2×#3828⋊C3C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#416
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. #11721826C2semidirect
3. C22×(C3⋊(C3⋊C8))⋊C3C2semidirect
4. #11720309C2semidirect
5. #11720308C2semidirect
11. #3828⋊C3C4semidirect
22. C22×C3⋊(C3⋊C8)D3semidirect
54. #3828Dic3semidirect
216. H(3,𝔽3)#416semidirect
329. C32#5113semidirect
330. C32#5233semidirect

Maximal subgroups

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

subgroupindex
C2×#3828⋊C32
#117218262
C22×(C3⋊(C3⋊C8))⋊C32
#117203092
#117203082
#105911053
#105910833