Finite Group Databasebeta

← #10592771⋊C3 #13142986

Subgroups — #10592771⋊C3 #13142986

Order 1728.

598 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 106 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
1H(3,𝔽3)×#127C211yesmaximal, characteristic, complemented
2C2×#3753⋊C3C212yesmaximal, complemented
3C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
4#63659⋊C3C212yesmaximal, complemented
5C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic, complemented
6#10592771C311yesmaximal, characteristic, complemented
7C3×#5072—31nomaximal
8C2×D4×H(3,𝔽3)—21no—
9C2×D4×H(3,𝔽3)—21no—
10C2×D4×H(3,𝔽3)C2212yes—
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×C4×(C3⋊D3)⋊C3—21no—
13C42×H(3,𝔽3)C2211yescharacteristic
14#3753⋊C3—24no—
15#3752⋊C3—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
18C32×#127—21no—
19C32×#127D311yescharacteristic, complemented
20C32×#127C2×C311yescharacteristic, complemented
21C2×#3753C2×C312yescomplemented
22C4×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
23C2×C3×#213—32no—
24C3×C4×D12—31no—
25#63659C2×C312yescomplemented
26C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
27C3×#832—32no—
28C3×C4⋊(C3⋊C8)—31no—
29C23×H(3,𝔽3)—22no—
30C22×(C3⋊D3)⋊C3—21no—
31D4×H(3,𝔽3)—22no—
32D4×H(3,𝔽3)—22no—
33D4×H(3,𝔽3)—42no—
34D4×H(3,𝔽3)—42no—
35C2×C4×H(3,𝔽3)D411yescharacteristic
36C2×C4×H(3,𝔽3)D411yescharacteristic
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×(C3⋊Dic3)⋊C3—21no—
39C4⋊((C3⋊D3)⋊C3)—21no—
40C4⋊((C3⋊D3)⋊C3)D412yescomplemented
41C4×(C3⋊D3)⋊C3—42no—
42(C3⋊(C3⋊C8))⋊C3—22no—
43C3×#459—91no—
44#5072—31no—
45#5072—91no—
46C2×C32×D4—22no—
47C2×C32×D4—22no—
48C2×C32×D4—22no—
49C2×C32×D4—21no—
50C2×C32×D4—21no—
51C2×C32×D4—21no—
52C2×C32×D4—21no—
53C2×C32×D4C2×D312yes—
54C2×C32×D4C22×C312yes—
55C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
56C2×C4×C3⋊D3—21no—
57C2×C3×D12—31no—
58C2×C3×C4×D3—61no—
59C32×C42—21no—
60C32×C42C2×D311yescharacteristic
61C32×C42C22×C311yescharacteristic
62#3753—24no—
63#3752—21no—
64C4⋊(C3⋊Dic3)C22×C311yescharacteristic
65C2×C3⋊(C3⋊C8)C22×C312yes—
66C3×#213—64no—
67C3×(C2×D3)⋊C4—61no—
68C3×C4⋊Dic3—31no—
69C2×C3×C3⋊C8—32no—
70C22×H(3,𝔽3)—42no—
71C22×H(3,𝔽3)—42no—
72C22×H(3,𝔽3)—42no—
73C22×H(3,𝔽3)C2×D411yescharacteristic
74C2×(C3⋊D3)⋊C3—22no—
75C2×(C3⋊D3)⋊C3—41no—
76C2×(C3⋊D3)⋊C3—41no—
77C4×H(3,𝔽3)—21no—
78C4×H(3,𝔽3)D812yescomplemented
79C4×H(3,𝔽3)C2×D411yescharacteristic
80C4×H(3,𝔽3)#4111yescharacteristic
81(C3⋊Dic3)⋊C3—41no—
82C2×C3×D8—92no—
83C3×#127—31no—
84C3×#127—31no—
85C3×#127—61no—
86C3×#127C3×D311yescharacteristic
87C3×C4×D4—91no—
88C2×#213—32no—
89C2×#213—92no—
90C4×D12—31no—
91C4×D12—91no—
92C3×C4⋊C8—91no—
93C3×#102—92no—
94#832—32no—
95#832—92no—
96C4⋊(C3⋊C8)—31no—
97C4⋊(C3⋊C8)—91no—
98C23×C32—24no—
99C23×C32—22no—
100C23×C32—22no—
101C22×C3⋊D3—21no—
102C22×C3×D3—61no—
103C32×D4—22no—
104C32×D4—22no—
105C32×D4—22no—
106C32×D4—22no—
107C32×D4—42no—
108C32×D4—42no—
109C32×D4—44no—
110C32×D4—42no—
111C32×D4—42no—
112C32×D4—44no—
113C32×D4—42no—
114C32×D4—42no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4#6711yescharacteristic
119C2×C32×C4#6711yescharacteristic
120C2×C32×C4C3×D411yescharacteristic
121C2×C32×C4C3×D411yescharacteristic
122C2×C32×C4C22×D311yescharacteristic
123C2×C32×C4C23×C311yescharacteristic
124C2×C3⋊Dic3—21no—
125C4⋊(C3⋊D3)—21no—
126C4⋊(C3⋊D3)C3×D412yescomplemented
127C4×C3⋊D3—42no—
128C2×C3×Dic3—61no—
129C3×D12—32no—
130C3×D12—61no—
131C3×C4×D3—122no—
132C3⋊(C3⋊C8)—22no—
133C3×C3⋊C8—62no—
134#459—91noSylow
135C2×H(3,𝔽3)—82no—
136C2×H(3,𝔽3)#12111yescharacteristic
137C2×H(3,𝔽3)C2×D811yescharacteristic
138C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
139(C3⋊D3)⋊C3—42no—
140C2×C3×D4—21no—
141C2×C3×D4—21no—
142C2×C3×D4—32no—
143C2×C3×D4—32no—
144C2×C3×D4—61no—
145C2×C3×D4—61no—
146C2×C3×D4—61no—
147C2×C3×D4—61no—
148C2×C3×D4—62no—
149C2×C3×D4—62no—
150C2×C3×D4—62no—
151C2×C3×D4—91no—
152C2×C3×D4C2×C3×D312yes—
153C22×C3×C4—181no—
154C2×D12—31no—
155C2×D12—91no—
156C2×C4×D3—61no—
157C2×C4×D3—181no—
158C3×D8—184no—
159C2×C3×C8—92no—
160C3×C4⋊C4—91no—
161C3×C22⋊C4—181no—
162C3×C42—31no—
163C3×C42—31no—
164C3×C42—61no—
165C3×C42C2×C3×D311yescharacteristic
166#213—64no—
167#213—184no—
168(C2×D3)⋊C4—61no—
169(C2×D3)⋊C4—181no—
170C4⋊Dic3—31no—
171C4⋊Dic3—91no—
172C2×C3⋊C8—32no—
173C2×C3⋊C8—92no—
174C22×C32—21no—
175C22×C32—44no—
176C22×C32—44no—
177C22×C32—44no—
178C22×C32—42no—
179C22×C32—42no—
180C22×C32—42no—
181C22×C32—42no—
182C22×C32—42no—
183C22×C32—42no—
184C22×C32C2×#6711yescharacteristic
185C22×C32C2×C3×D411yescharacteristic
186C2×C3⋊D3—22no—
187C2×C3⋊D3—41no—
188C2×C3⋊D3—41no—
189C2×C3×D3—62no—
190C2×C3×D3—121no—
191C2×C3×D3—121no—
192C32×C4—22no—
193C32×C4—21no—
194C32×C4—21no—
195C32×C4—22no—
196C32×C4—21no—
197C32×C4—21no—
198C32×C4#21312yescomplemented
199C32×C4C3×D812yescomplemented
200C32×C4D3×D411yescharacteristic
201C32×C4#23711yescharacteristic
202C32×C4C2×C3×D411yescharacteristic
203C32×C4C3×#4111yescharacteristic
204C3⋊Dic3—41no—
205C3×Dic3—121no—
206C2×D8—92no—
207#127(C3⋊D3)⋊C311yescharacteristic, complemented
208C4×D4—91no—
209C4⋊C8—91no—
210#102—92no—
211H(3,𝔽3)#45911yescharacteristic, complemented, Sylow
212C23×C3—22no—
213C23×C3—62no—
214C23×C3—62no—
215C23×C3—64no—
216C23×C3—181no—
217C22×D3—61no—
218C22×D3—181no—
219C3×D4—22no—
220C3×D4—22no—
221C3×D4—42no—
222C3×D4—42no—
223C3×D4—62no—
224C3×D4—62no—
225C3×D4—62no—
226C3×D4—62no—
227C3×D4—92no—
228C3×D4—122no—
229C3×D4—122no—
230C3×D4—122no—
231C3×D4—122no—
232C3×D4—122no—
233C3×D4—122no—
234C3×D4—124no—
235C3×D4—124no—
236C3×D4—181no—
237C2×C3×C4—31no—
238C2×C3×C4—31no—
239C2×C3×C4—31no—
240C2×C3×C4—31no—
241C2×C3×C4—31no—
242C2×C3×C4—31no—
243C2×C3×C4—61no—
244C2×C3×C4—61no—
245C2×C3×C4—61no—
246C2×C3×C4—181no—
247C2×C3×C4—362no—
248C2×C3×C4C3×#6711yescharacteristic
249C2×C3×C4C3×#6711yescharacteristic
250C2×C3×C4C22×C3×D311yescharacteristic
251C2×Dic3—61no—
252C2×Dic3—181no—
253D12—32no—
254D12—61no—
255D12—92no—
256D12—181no—
257C4×D3—122no—
258C4×D3—362no—
259C3×C8—182no—
260C3⋊C8—62no—
261C3⋊C8—182no—
262C2×C32—21no—
263C2×C32—21no—
264C2×C32—21no—
265C2×C32—84no—
266C2×C32—82no—
267C2×C32—82no—
268C2×C32C2×#21311yescharacteristic
269C2×C32#93211yescharacteristic
270C2×C32#93811yescharacteristic
271C2×C32C3×#12111yescharacteristic
272C2×C32C2×C3×D811yescharacteristic
273C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
274C3⋊D3—42no—
275C3×D3—122no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—91no—
279C2×D4C2×(C3⋊D3)⋊C312yes—
280C22×C4—181no—
281D8—184no—
282C2×C8—92no—
283C4⋊C4—91no—
284C22⋊C4—181no—
285C42C2×(C3⋊D3)⋊C311yescharacteristic
286C22×C3—31no—
287C22×C3—31no—
288C22×C3—42no—
289C22×C3—42no—
290C22×C3—42no—
291C22×C3—61no—
292C22×C3—122no—
293C22×C3—122no—
294C22×C3—122no—
295C22×C3—122no—
296C22×C3—122no—
297C22×C3—122no—
298C22×C3—124no—
299C22×C3—124no—
300C22×C3—124no—
301C22×C3—182no—
302C22×C3—361no—
303C22×C3—361no—
304C22×C3C2×C3×#6711yescharacteristic
305C2×D3—62no—
306C2×D3—121no—
307C2×D3—121no—
308C2×D3—182no—
309C2×D3—361no—
310C2×D3—361no—
311C3×C4—21no—
312C3×C4—31no—
313C3×C4—31no—
314C3×C4—31no—
315C3×C4—31no—
316C3×C4—32no—
317C3×C4—32no—
318C3×C4—61no—
319C3×C4—61no—
320C3×C4—62no—
321C3×C4—61no—
322C3×C4—61no—
323C3×C4—62no—
324C3×C4—361no—
325C3×C4C3×#21312yes—
326C3×C4C3×D3×D411yescharacteristic
327C3×C4C3×#23711yescharacteristic
328Dic3—121no—
329Dic3—361no—
330C32—21no—
331C32#507211yescharacteristic, complemented
332C32C3×#45911yescharacteristic, complemented
333C23—22no—
334C23—181no—
335D4—22no—
336D4—22no—
337D4—42no—
338D4—42no—
339D4—92no—
340D4—181no—
341C2×C4—181no—
342C2×C4—362no—
343C2×C4#638⋊C311yescharacteristic
344C2×C4#638⋊C311yescharacteristic
345C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
346C8—182no—
347C2×C3—31no—
348C2×C3—31no—
349C2×C3—31no—
350C2×C3—31no—
351C2×C3—31no—
352C2×C3—31no—
353C2×C3—61no—
354C2×C3—61no—
355C2×C3—61no—
356C2×C3—82no—
357C2×C3—242no—
358C2×C3—242no—
359C2×C3—244no—
360C2×C3—362no—
361C2×C3C2×C3×#21311yescharacteristic
362C2×C3C3×#93211yescharacteristic
363C2×C3C3×#93811yescharacteristic
364D3—122no—
365D3—362no—
366C22—42no—
367C22—42no—
368C22—42no—
369C22—182no—
370C22—361no—
371C22—361no—
372C22C2×#638⋊C311yescharacteristic
373C4—21no—
374C4—361no—
375C4#3753⋊C312yescomplemented
376C4D4×(C3⋊D3)⋊C311yescharacteristic
377C4#3830⋊C311yescharacteristic
378C3—31no—
379C3—31no—
380C3—61no—
381C3C3×#507211yescharacteristic
382C2—82no—
383C2—362no—
384C2C2×#3753⋊C311yescharacteristic
385C2#64141⋊C311yescharacteristic
386C2#64147⋊C311yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#127C2
socleC22×C3C2×C3×#67
hypercenter#127(C3⋊D3)⋊C3
hypocenterC32C3×#459
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#127C2semidirect
2. C2×#3753⋊C3C2semidirect
3. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
4. #63659⋊C3C2semidirect
5. C4⋊((C3⋊(C3⋊C8))⋊C3)C2semidirect
6. #10592771C3semidirect
19. C32×#127D3semidirect
20. C32×#127C2×C3semidirect
21. C2×#3753C2×C3semidirect
22. C4×C4⋊(C3⋊D3)C2×C3semidirect
25. #63659C2×C3semidirect
26. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
40. C4⋊((C3⋊D3)⋊C3)D4semidirect
78. C4×H(3,𝔽3)D8semidirect
126. C4⋊(C3⋊D3)C3×D4semidirect
198. C32×C4#213semidirect
199. C32×C4C3×D8semidirect
207. #127(C3⋊D3)⋊C3semidirect
211. H(3,𝔽3)#459semidirect
331. C32#5072semidirect
332. C32C3×#459semidirect
375. C4#3753⋊C3semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1272
C2×#3753⋊C32
C4×C4⋊((C3⋊D3)⋊C3)2
#63659⋊C32
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#105927713
C3×#50723