Finite Group Databasebeta

← #13140636

Subgroups — #13140636

Order 1728.

754 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 69 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#11722664C211yesmaximal, characteristic, complemented
2#64149⋊C3C211yesmaximal, characteristic, complemented
3#11721071C212yesmaximal, complemented
4#63660⋊C3C211yesmaximal, characteristic, complemented
5#11720328C212yesmaximal, complemented
6#10591510—31nomaximal
7#10591509—31nomaximal
8C2×D4×H(3,𝔽3)C2211yescharacteristic
9C2×#6235—22no—
10#88744—22no—
11D4×#272—22no—
12#88740C2212yes—
13C2×#638⋊C3—22no—
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
15C4×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
16#3748⋊C3C2212yescomplemented
17#88438—22no—
18#88436—22no—
19C3×#1004—31no—
20C3×#1004—31no—
21C3×#1004—61no—
22#64149D311yescharacteristic, complemented
23C3×#940—31no—
24#63825—32no—
25#63823—32no—
26#63660D311yescharacteristic, complemented
27C3×#833—31no—
28#63569—32no—
29#63568—32no—
30C23×H(3,𝔽3)D412yes—
31C22×#272—22no—
32C22×(C3⋊D3)⋊C3—21no—
33D4×H(3,𝔽3)—22no—
34C2×C4×H(3,𝔽3)C2311yescharacteristic
35#6235—24no—
36#6235—22no—
37C2×#1084—22no—
38C4⋊#272D412yescomplemented
39C4×#272—22no—
40#6229D412yescomplemented
41#638⋊C3—44no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C4⋊((C3⋊D3)⋊C3)—22no—
44(C3⋊(C3⋊C8))⋊C3—22no—
45#5241—91no—
46#5077—91no—
47#4747—91no—
48C2×C32×D4—21no—
49C2×C32×D4C2×D311yescharacteristic
50C2×C32×D4C2×D311yescharacteristic
51C2×#638—22no—
52C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
53C2×C3×#67—64no—
54C2×C3×#67—62no—
55C2×C3×#67—62no—
56C2×C3×#67—62no—
57C3×#237—62no—
58C3×#237—62no—
59C3×#237—64no—
60C3×D3×D4—62no—
61C3×D3×D4—62no—
62C3×D3×D4—64no—
63C3×#235—32no—
64C3×#235—32no—
65C3×#235—62no—
66C2×C3×D12—31no—
67C4×C3⋊Dic3C2×D311yescharacteristic, complemented
68#3748C2×D312yescomplemented
69C3×C4×Dic3—31no—
70C3×#208—32no—
71#3717—62no—
72#3715—62no—
73#3714—62no—
74#3713—62no—
75C22×H(3,𝔽3)—22no—
76C22×H(3,𝔽3)—42no—
77C22×H(3,𝔽3)C2×D411yescharacteristic
78C2×#272—22no—
79C2×#272—82no—
80C2×(C3⋊D3)⋊C3—42no—
81C2×(C3⋊D3)⋊C3—41no—
82C4×H(3,𝔽3)C2×D412yes—
83#1084—22no—
84(C3⋊Dic3)⋊C3—22no—
85C3×H(5,𝔽2)—91no—
86#1004—31no—
87#1004—31no—
88#1004—61no—
89C3×#127—91no—
90#949—92no—
91#940—31no—
92#940—91no—
93#932—92no—
94#908—92no—
95C3×#100—91no—
96#837—92no—
97#833—31no—
98#833—91no—
99#825—92no—
100#805—92no—
101C23×C32—22no—
102C23×C32D1212yes—
103C23×C32#6712yes—
104C22×C3⋊D3—21no—
105C22×C3×D3—64no—
106C22×C3×D3—62no—
107C22×C3×D3—62no—
108C22×C3×D3—61no—
109C32×D4—24no—
110C32×D4—22no—
111C32×D4—22no—
112C2×C32×C4—21no—
113C2×C32×C4C22×D311yescharacteristic
114C2×C32×C4C22×D311yescharacteristic
115#638—44no—
116C2×C3⋊Dic3—21no—
117C4⋊(C3⋊D3)—22no—
118C3×#67—64no—
119C3×#67—68no—
120C3×#67—64no—
121C3×#67—64no—
122C3×#67—62no—
123C3×#67—62no—
124C3×#67—124no—
125C2×C3×Dic3—64no—
126C2×C3×Dic3—62no—
127C2×C3×Dic3—62no—
128C2×C3×Dic3—61no—
129C3×D12—32no—
130C3×D12—32no—
131C3×D12—62no—
132C3×D12—62no—
133C3×C4×D3—64no—
134C3×C4×D3—62no—
135C3×C4×D3—62no—
136C3×Dic6—32no—
137C3×Dic6—32no—
138C3×Dic6—62no—
139C3⋊(C3⋊C8)—22no—
140C3×C3⋊C8—62no—
141#453—271noSylow
142C2×H(3,𝔽3)—21no—
143C2×H(3,𝔽3)—42no—
144C2×H(3,𝔽3)#12011yescharacteristic
145#272—42no—
146(C3⋊D3)⋊C3—81no—
147C3×#41—92no—
148C3×#41—182no—
149C2×C3×D4—31no—
150C2×C3×D4—31no—
151C2×C3×D4—61no—
152C2×C3×D4—91no—
153C2×C3×D4—182no—
154C2×C3×D4—182no—
155C2×C3×D4—182no—
156C2×C3×D4D3211yescharacteristic
157C2×#67—62no—
158C2×#67—62no—
159C2×#67—64no—
160C2×#67—62no—
161C2×#67—182no—
162#237—62no—
163#237—62no—
164#237—64no—
165D3×D4—62no—
166D3×D4—62no—
167D3×D4—64no—
168#235—32no—
169#235—32no—
170#235—62no—
171C2×D12—31no—
172C2×D12—91no—
173C3×M16—92no—
174C3×C42—91no—
175#215—182no—
176#214—182no—
177#213—182no—
178#213—182no—
179C4×Dic3—31no—
180C4×Dic3—91no—
181#208—32no—
182#208—92no—
183D24—182no—
184#204—182no—
185C22×C32—24no—
186C22×C32—22no—
187C22×C32—22no—
188C22×C32—21no—
189C22×C32—42no—
190C22×C32—42no—
191C22×C32—82no—
192C22×C32C2×D1211yescharacteristic
193C22×C32C2×#6711yescharacteristic
194C2×C3⋊D3—42no—
195C2×C3⋊D3—41no—
196C2×C3×D3—64no—
197C2×C3×D3—62no—
198C2×C3×D3—62no—
199C2×C3×D3—122no—
200C2×C3×D3—121no—
201C2×C3×D3—244no—
202C2×C3×D3—242no—
203C2×C3×D3—242no—
204C32×C4—22no—
205C32×C4D3×D412yes—
206C32×C4D3×D412yes—
207C3⋊Dic3—22no—
208C3×Dic3—62no—
209C3×Dic3—64no—
210C3×Dic3—62no—
211C3×Dic3—62no—
212H(5,𝔽2)—91no—
213AGL(1,ℤ/8)—272no—
214#127—91no—
215#104—272no—
216#100—91no—
217H(3,𝔽3)#45311yescharacteristic, complemented, Sylow
218C23×C3—32no—
219C23×C3—32no—
220C23×C3—62no—
221C23×C3—182no—
222C23×C3—181no—
223C23×C3#62612yes—
224C22×D3—62no—
225C22×D3—62no—
226C22×D3—64no—
227C22×D3—61no—
228C22×D3—181no—
229C3×Q8—92no—
230C3×D4—22no—
231C3×D4—62no—
232C3×D4—62no—
233C3×D4—64no—
234C3×D4—92no—
235C3×D4—182no—
236C3×D4—184no—
237C3×D4—182no—
238C3×D4—364no—
239C2×C3×C4—31no—
240C2×C3×C4—31no—
241C2×C3×C4—61no—
242C2×C3×C4—182no—
243C2×C3×C4—181no—
244C2×C3×C4—182no—
245C2×C3×C4C2×D3211yescharacteristic
246#67—62no—
247#67—62no—
248#67—64no—
249#67—64no—
250#67—64no—
251#67—68no—
252#67—124no—
253#67—364no—
254C2×Dic3—62no—
255C2×Dic3—62no—
256C2×Dic3—64no—
257C2×Dic3—61no—
258C2×Dic3—181no—
259D12—32no—
260D12—32no—
261D12—62no—
262D12—62no—
263D12—182no—
264C4×D3—62no—
265C4×D3—62no—
266C4×D3—64no—
267Dic6—32no—
268Dic6—32no—
269Dic6—62no—
270C3×C8—182no—
271C3⋊C8—62no—
272C3⋊C8—182no—
273C2×C32—21no—
274C2×C32—21no—
275C2×C32—21no—
276C2×C32—41no—
277C2×C32—44no—
278C2×C32—42no—
279C2×C32—42no—
280C2×C32#88211yescharacteristic
281C2×C32#93711yescharacteristic
282C3⋊D3—81no—
283C3×D3—124no—
284C3×D3—122no—
285C3×D3—122no—
286C3×D3—241no—
287#41—92no—
288#41—182no—
289C2×D4—91no—
290C2×D4—182no—
291C2×D4—182no—
292C2×D4—182no—
293C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
294SD16—542no—
295D8—542no—
296M16—92no—
297C42—91no—
298C22×C3—22no—
299C22×C3—31no—
300C22×C3—31no—
301C22×C3—42no—
302C22×C3—62no—
303C22×C3—62no—
304C22×C3—64no—
305C22×C3—61no—
306C22×C3—122no—
307C22×C3—122no—
308C22×C3—182no—
309C22×C3—242no—
310C22×C3—361no—
311C22×C3—362no—
312C22×C3—722no—
313C22×C3C2×#62611yescharacteristic
314C2×D3—62no—
315C2×D3—62no—
316C2×D3—64no—
317C2×D3—122no—
318C2×D3—121no—
319C2×D3—242no—
320C2×D3—242no—
321C2×D3—244no—
322C2×D3—361no—
323C2×D3—362no—
324C3×C4—32no—
325C3×C4—32no—
326C3×C4—62no—
327C3×C4—182no—
328C3×C4—182no—
329C3×C4#381112yes—
330Dic3—62no—
331Dic3—62no—
332Dic3—62no—
333Dic3—64no—
334Dic3—182no—
335C32—21no—
336C32#474711yescharacteristic, complemented
337C32#507711yescharacteristic, complemented
338C23—182no—
339C23—181no—
340C23#620012yes—
341Q8—92no—
342D4—22no—
343D4—92no—
344D4—184no—
345D4—182no—
346D4—182no—
347D4—364no—
348C2×C4—182no—
349C2×C4—181no—
350C2×C4—182no—
351C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
352C8—182no—
353C2×C3—21no—
354C2×C3—31no—
355C2×C3—31no—
356C2×C3—42no—
357C2×C3—61no—
358C2×C3—61no—
359C2×C3—61no—
360C2×C3—121no—
361C2×C3—122no—
362C2×C3—122no—
363C2×C3—124no—
364C2×C3—362no—
365C2×C3—721no—
366C2×C3#6398111yescharacteristic
367D3—122no—
368D3—122no—
369D3—124no—
370D3—241no—
371D3—721no—
372C22—22no—
373C22—42no—
374C22—182no—
375C22—362no—
376C22—361no—
377C22—722no—
378C22C2×#620011yescharacteristic
379C4—182no—
380C4—182no—
381C4#8867712yes—
382C3—31no—
383C3—31no—
384C3—61no—
385C3#1059151111yescharacteristic
386C2—21no—
387C2—42no—
388C2—362no—
389C2—721no—
390C2#1172132311yescharacteristic
subgroupG/H
centerC2#11721323
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC2×C3#63981
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#453
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #11722664C2semidirect
2. #64149⋊C3C2semidirect
3. #11721071C2semidirect
4. #63660⋊C3C2semidirect
5. #11720328C2semidirect
14. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
15. C4×(C3⋊Dic3)⋊C3C22semidirect
16. #3748⋊C3C22semidirect
22. #64149D3semidirect
26. #63660D3semidirect
38. C4⋊#272D4semidirect
40. #6229D4semidirect
52. C2×C4⋊(C3⋊D3)C2×D3semidirect
67. C4×C3⋊Dic3C2×D3semidirect
68. #3748C2×D3semidirect
217. H(3,𝔽3)#453semidirect
336. C32#4747semidirect
337. C32#5077semidirect

Maximal subgroups

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

subgroupindex
#117226642
#64149⋊C32
#117210712
#63660⋊C32
#117203282
#105915103
#105915093