Finite Group Databasebeta

← #13137419

Subgroups — #13137419

Order 1728.

531 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 531 conjugacy classes of subgroups — too many to draw. Only its 83 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#11721721C211yesmaximal, characteristic, complemented
2#64084⋊C3C211yesmaximal, characteristic, complemented
3C2×#88439C211yesmaximal, characteristic, complemented
4C2×#88438C211yesmaximal, characteristic, complemented
5C2×#88437C211yesmaximal, characteristic, complemented
6C2×#88436C211yesmaximal, characteristic
7C4×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic, complemented
8#10590370—31nomaximal
9#10590353—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×#634⋊C3C2211yescharacteristic
14C42×H(3,𝔽3)C2211yescharacteristic
15#88542—22no—
16#3752⋊C3—22no—
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88439—22no—
20#88438—22no—
21#88437—22no—
22#88436—22no—
23#64084D311yescharacteristic, complemented
24C3×#875—31no—
25C3×#875—31no—
26C3×#875—31no—
27C3×#875—61no—
28C2×#3719—31no—
29C2×#3718—31no—
30C2×#3717—31no—
31C2×#3716—31no—
32C2×#3715—31no—
33C2×#3715—31no—
34C2×#3714—31no—
35C2×#3713—31no—
36C4×C3⋊(C3⋊C8)D311yescharacteristic, complemented
37C3×C4×C3⋊C8—31no—
38C22×#272—21no—
39C22×(C3⋊D3)⋊C3—21no—
40C2×C4×H(3,𝔽3)D412yes—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×#1084—21no—
43C4⋊#272—21no—
44C4⋊#272—21no—
45#6229—21no—
46#6229—21no—
47C2×(C3⋊Dic3)⋊C3—21no—
48C4⋊((C3⋊D3)⋊C3)—21no—
49C4⋊((C3⋊D3)⋊C3)—21no—
50#634⋊C3—21no—
51#634⋊C3—21no—
52(C3⋊(C3⋊C8))⋊C3D412yes—
53(C3⋊(C3⋊C8))⋊C3D412yes—
54#5059—91no—
55#5059—91no—
56#4696—91no—
57C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
58C2×#634C2×D311yescharacteristic
59C2×C3×D12—31no—
60C2×C3×D12—31no—
61C2×C3×D12—31no—
62C2×C3×D12—61no—
63C2×C3×Dic6—31no—
64C2×C3×Dic6—31no—
65C2×C3×Dic6—31no—
66C2×C3×Dic6—61no—
67C32×C42—21no—
68C32×C42C2×D311yescharacteristic
69C32×C42C2×D311yescharacteristic
70#3752—22no—
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
73C3×(C2×D3)⋊C4—64no—
74C3×(C2×D3)⋊C4—62no—
75C3×(C2×D3)⋊C4—62no—
76C3×(C2×D3)⋊C4—62no—
77C2×C3×C3⋊C8—31no—
78C2×C3×C3⋊C8—31no—
79#3719—62no—
80#3718—62no—
81#3717—62no—
82#3716—62no—
83#3715—62no—
84#3715—62no—
85#3714—62no—
86#3713—62no—
87C22×H(3,𝔽3)C2×D411yescharacteristic
88C2×#272—41no—
89C2×#272—42no—
90C2×(C3⋊D3)⋊C3—41no—
91C2×(C3⋊D3)⋊C3—42no—
92C4×H(3,𝔽3)—22no—
93C4×H(3,𝔽3)C2×D411yescharacteristic
94C4×H(3,𝔽3)C2×D411yescharacteristic
95#1084—41no—
96(C3⋊Dic3)⋊C3—41no—
97C3×#124—91no—
98C3×#124—91no—
99C2×#216—91no—
100C2×#216—91no—
101C2×#215—91no—
102C2×#215—91no—
103C2×#214—91no—
104C2×#214—91no—
105C2×#213—91no—
106C2×#213—91no—
107C2×Dic12—91no—
108C2×D24—91no—
109C2×#204—91no—
110C2×#204—91no—
111#875—31no—
112#875—31no—
113#875—31no—
114#875—61no—
115#875—91no—
116C3×C4×C8—91no—
117C4×C3⋊C8—31no—
118C4×C3⋊C8—91no—
119C22×C3⋊D3—21no—
120C22×C3×D3—62no—
121C22×C3×D3—61no—
122C22×C3×D3—61no—
123C22×C3×D3—61no—
124C2×C32×C4—22no—
125C2×C32×C4—21no—
126C2×C32×C4D1212yes—
127C2×C32×C4#6712yes—
128C2×C32×C4C22×D311yescharacteristic
129C2×C32×C4C22×D311yescharacteristic
130C2×C3⋊Dic3—21no—
131C4⋊(C3⋊D3)—21no—
132C4⋊(C3⋊D3)—21no—
133#634—21no—
134#634—21no—
135C2×C3×Dic3—62no—
136C2×C3×Dic3—61no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—61no—
139C3×D12—61no—
140C3×D12—61no—
141C3×D12—61no—
142C3×D12—61no—
143C3×D12—61no—
144C3×D12—61no—
145C3×D12—121no—
146C3×D12—121no—
147C3×Dic6—61no—
148C3×Dic6—61no—
149C3×Dic6—61no—
150C3×Dic6—61no—
151C3×Dic6—61no—
152C3×Dic6—61no—
153C3×Dic6—121no—
154C3×Dic6—121no—
155C3⋊(C3⋊C8)D1212yes—
156C3⋊(C3⋊C8)D1212yes—
157C3×C3⋊C8—32no—
158C3×C3⋊C8—32no—
159#495—271noSylow
160C2×H(3,𝔽3)#12711yescharacteristic
161C2×H(3,𝔽3)#13512yes—
162#272—81no—
163(C3⋊D3)⋊C3—81no—
164C2×C3×Q8—91no—
165C2×C3×Q8—91no—
166C2×C3×D4—91no—
167C2×C3×D4—91no—
168C2×D12—31no—
169C2×D12—31no—
170C2×D12—31no—
171C2×D12—61no—
172C2×D12—91no—
173C2×Dic6—31no—
174C2×Dic6—31no—
175C2×Dic6—31no—
176C2×Dic6—61no—
177C2×Dic6—91no—
178C2×C3×C8—91no—
179C2×C3×C8—91no—
180C3×C22⋊C4—182no—
181C3×C22⋊C4—182no—
182C3×C42—31no—
183C3×C42—31no—
184C3×C42—61no—
185C3×C42D3211yescharacteristic
186#216—182no—
187#216—182no—
188#215—182no—
189#215—182no—
190#214—182no—
191#214—182no—
192#213—182no—
193#213—182no—
194(C2×D3)⋊C4—62no—
195(C2×D3)⋊C4—62no—
196(C2×D3)⋊C4—64no—
197(C2×D3)⋊C4—62no—
198(C2×D3)⋊C4—182no—
199C2×C3⋊C8—31no—
200C2×C3⋊C8—31no—
201C2×C3⋊C8—91no—
202C2×C3⋊C8—91no—
203Dic12—182no—
204D24—182no—
205#204—182no—
206#204—182no—
207C22×C32—21no—
208C22×C32C2×D1211yescharacteristic
209C22×C32C2×#6711yescharacteristic
210C2×C3⋊D3—41no—
211C2×C3⋊D3—42no—
212C2×C3×D3—122no—
213C2×C3×D3—124no—
214C2×C3×D3—121no—
215C2×C3×D3—121no—
216C2×C3×D3—122no—
217C2×C3×D3—122no—
218C2×C3×D3—121no—
219C2×C3×D3—122no—
220C32×C4—22no—
221C32×C4—22no—
222C32×C4—21no—
223C32×C4—21no—
224C32×C4—42no—
225C32×C4C2×D1211yescharacteristic
226C32×C4C2×D1211yescharacteristic
227C32×C4D3×D411yescharacteristic
228C32×C4D3×D411yescharacteristic
229C3⋊Dic3—41no—
230C3×Dic3—122no—
231C3×Dic3—121no—
232C3×Dic3—121no—
233C3×Dic3—121no—
234C2×Q16—271no—
235C2×SD16—271no—
236C2×SD16—271no—
237C2×D8—271no—
238#124—91no—
239#124—91no—
240C4×C8—91no—
241H(3,𝔽3)#49511yescharacteristic, complemented, Sylow
242C23×C3—181no—
243C23×C3—181no—
244C22×D3—61no—
245C22×D3—61no—
246C22×D3—62no—
247C22×D3—61no—
248C22×D3—181no—
249C3×Q8—181no—
250C3×Q8—181no—
251C3×Q8—181no—
252C3×Q8—181no—
253C3×D4—181no—
254C3×D4—181no—
255C3×D4—181no—
256C3×D4—181no—
257C2×C3×C4—32no—
258C2×C3×C4—32no—
259C2×C3×C4—31no—
260C2×C3×C4—31no—
261C2×C3×C4—62no—
262C2×C3×C4—61no—
263C2×C3×C4—181no—
264C2×C3×C4—181no—
265C2×C3×C4#62612yes—
266C2×C3×C4C2×D3211yescharacteristic
267C2×Dic3—61no—
268C2×Dic3—61no—
269C2×Dic3—62no—
270C2×Dic3—61no—
271C2×Dic3—181no—
272D12—61no—
273D12—61no—
274D12—61no—
275D12—61no—
276D12—61no—
277D12—61no—
278D12—121no—
279D12—121no—
280D12—181no—
281D12—181no—
282Dic6—61no—
283Dic6—61no—
284Dic6—61no—
285Dic6—61no—
286Dic6—61no—
287Dic6—61no—
288Dic6—121no—
289Dic6—121no—
290Dic6—181no—
291Dic6—181no—
292C3×C8—92no—
293C3×C8—92no—
294C3⋊C8—32no—
295C3⋊C8—32no—
296C3⋊C8—92no—
297C3⋊C8—92no—
298C2×C32—22no—
299C2×C32—21no—
300C2×C32#87411yescharacteristic
301C2×C32#90412yes—
302C2×C32#94011yescharacteristic
303C2×C32#95012yes—
304C3⋊D3—81no—
305C3×D3—242no—
306C3×D3—241no—
307C3×D3—241no—
308C3×D3—241no—
309C2×Q8—91no—
310C2×Q8—91no—
311C2×D4—91no—
312C2×D4—91no—
313Q16—542no—
314SD16—542no—
315SD16—542no—
316D8—542no—
317C2×C8—91no—
318C2×C8—91no—
319C22⋊C4—182no—
320C22⋊C4—182no—
321C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
322C22×C3—31no—
323C22×C3—31no—
324C22×C3—61no—
325C22×C3—361no—
326C22×C3—362no—
327C22×C3—362no—
328C22×C3—361no—
329C22×C3C2×#62611yescharacteristic
330C2×D3—121no—
331C2×D3—121no—
332C2×D3—122no—
333C2×D3—122no—
334C2×D3—122no—
335C2×D3—124no—
336C2×D3—122no—
337C2×D3—121no—
338C2×D3—361no—
339C2×D3—362no—
340C3×C4—22no—
341C3×C4—31no—
342C3×C4—31no—
343C3×C4—31no—
344C3×C4—31no—
345C3×C4—62no—
346C3×C4—62no—
347C3×C4—61no—
348C3×C4—61no—
349C3×C4—122no—
350C3×C4—361no—
351C3×C4—361no—
352C3×C4D3×D1211yescharacteristic
353C3×C4D3×D1211yescharacteristic
354Dic3—121no—
355Dic3—121no—
356Dic3—122no—
357Dic3—121no—
358Dic3—361no—
359C32—21no—
360C32#469611yescharacteristic, complemented
361C32#505911yescharacteristic, complemented
362C23—181no—
363C23—181no—
364Q8—181no—
365Q8—181no—
366Q8—181no—
367Q8—181no—
368D4—181no—
369D4—181no—
370D4—181no—
371D4—181no—
372C2×C4—181no—
373C2×C4—181no—
374C2×C4#620012yes—
375C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
376C8—92no—
377C8—92no—
378C2×C3—32no—
379C2×C3—32no—
380C2×C3—31no—
381C2×C3—31no—
382C2×C3—62no—
383C2×C3—61no—
384C2×C3—721no—
385C2×C3—721no—
386C2×C3#6382912yes—
387C2×C3#6391111yescharacteristic
388D3—241no—
389D3—241no—
390D3—242no—
391D3—241no—
392D3—721no—
393C22—362no—
394C22—362no—
395C22—361no—
396C22—361no—
397C22C2×#620011yescharacteristic
398C4—22no—
399C4—361no—
400C4—361no—
401C4#8865411yescharacteristic
402C4#8865411yescharacteristic
403C3—31no—
404C3—31no—
405C3—61no—
406C3#1059037011yescharacteristic
407C2—721no—
408C2—721no—
409C2#1172107312yes—
410C2#1172115811yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radicalC2×#88436C2
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#495
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721721C2semidirect
2. #64084⋊C3C2semidirect
3. C2×#88439C2semidirect
4. C2×#88438C2semidirect
5. C2×#88437C2semidirect
7. C4×(C3⋊(C3⋊C8))⋊C3C2semidirect
23. #64084D3semidirect
36. C4×C3⋊(C3⋊C8)D3semidirect
241. H(3,𝔽3)#495semidirect
360. C32#4696semidirect
361. C32#5059semidirect

Maximal subgroups

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

subgroupindex
#117217212
#64084⋊C32
C2×#884392
C2×#884382
C2×#884372
C2×#884362
C4×(C3⋊(C3⋊C8))⋊C32
#105903703
#105903533