Finite Group Databasebeta

← #78217

Subgroups — #78217

Order 384.

484 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#400C211yesmaximal, characteristic, complemented
2#4994C211yesmaximal, characteristic, complemented
3#4991C211yesmaximal, characteristic, complemented
4#4990C211yesmaximal, characteristic, complemented
5#4988C211yesmaximal, characteristic, complemented
6#4675C211yesmaximal, characteristic, complemented
7#4674C211yesmaximal, characteristic, complemented
8#4672C211yesmaximal, characteristic, complemented
9#4672C211yesmaximal, characteristic, complemented
10#4671C211yesmaximal, characteristic
11#4670C211yesmaximal, characteristic, complemented
12#4666C211yesmaximal, characteristic, complemented
13#4665C211yesmaximal, characteristic, complemented
14#4665C211yesmaximal, characteristic, complemented
15#4652C211yesmaximal, characteristic, complemented
16#2648—31nomaximal, Sylow
17C22×D12C2211yescharacteristic
18C22×D12C2211yescharacteristic
19C22×D12C2211yescharacteristic
20C22×C4×D3C2211yescharacteristic
21C2×C3×C4⋊C4C2211yescharacteristic
22C2×C3×C4⋊C4C2211yescharacteristic
23C2×(C2×D3)⋊C4C2211yescharacteristic
24C2×(C2×D3)⋊C4C2211yescharacteristic
25C2×(C2×D3)⋊C4C2211yescharacteristic
26C2×(C2×D3)⋊C4C2211yescharacteristic
27C2×(C2×D3)⋊C4C2211yescharacteristic
28C2×(C2×D3)⋊C4C2211yescharacteristic
29C2×(C2×D3)⋊C4C2211yescharacteristic
30C2×(C2×D3)⋊C4C2211yescharacteristic
31C2×(C2×D3)⋊C4C2211yescharacteristic
32C2×(C2×D3)⋊C4C2211yescharacteristic
33C2×(C2×D3)⋊C4C2211yescharacteristic
34C2×(C2×D3)⋊C4C2211yescharacteristic
35C2×(C2×D3)⋊C4C2211yescharacteristic
36C2×(C2×D3)⋊C4C2211yescharacteristic
37C2×C4⋊Dic3C2211yescharacteristic
38C2×Dic3⋊C4C2211yescharacteristic
39C2×Dic3⋊C4C2211yescharacteristic
40C2×C4×Dic3C2211yescharacteristic
41C2×C4×Dic3C2211yescharacteristic
42C3×#95C2211yescharacteristic
43C3×#95C2211yescharacteristic
44C3×#95C2211yescharacteristic
45C3×#95C2211yescharacteristic
46C3×#95C2211yescharacteristic
47C3⋊#95C2211yescharacteristic
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52#400D311yescharacteristic, complemented
53#397—31no—
54#397—31no—
55#396—31no—
56#394—31no—
57#394—31no—
58#394—31no—
59#392—31no—
60#390—31no—
61#388—31no—
62#388—31no—
63#387—31no—
64#386—31no—
65#386—31no—
66C4×C22⋊C4—31no—
67C23×D3C2311yescharacteristic
68C23×D3C2311yescharacteristic
69C23×D3C2311yescharacteristic
70C22×C3×C4C2311yescharacteristic
71C22×C3×C4C2311yescharacteristic
72C22×C3×C4C2311yescharacteristic
73C22×C3×C4C2311yescharacteristic
74C22×C3×C4C2311yescharacteristic
75C22×C3×C4C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×Dic3C2311yescharacteristic
78C22×Dic3C2311yescharacteristic
79C22×Dic3C2311yescharacteristic
80C22×Dic3C2311yescharacteristic
81C22×Dic3C2311yescharacteristic
82C2×D12—22no—
83C2×D12—22no—
84C2×D12—22no—
85C2×D12—41no—
86C2×D12—41no—
87C2×D12—41no—
88C2×D12—41no—
89C2×D12—41no—
90C2×D12—41no—
91C2×C4×D3—22no—
92C2×C4×D3—41no—
93C2×C4×D3—41no—
94C3×C4⋊C4—41no—
95C3×C4⋊C4—41no—
96(C2×D3)⋊C4—22no—
97(C2×D3)⋊C4—22no—
98(C2×D3)⋊C4—22no—
99(C2×D3)⋊C4—22no—
100(C2×D3)⋊C4—41no—
101(C2×D3)⋊C4—41no—
102(C2×D3)⋊C4—41no—
103(C2×D3)⋊C4—41no—
104(C2×D3)⋊C4—41no—
105(C2×D3)⋊C4—41no—
106(C2×D3)⋊C4—41no—
107(C2×D3)⋊C4—41no—
108(C2×D3)⋊C4—41no—
109(C2×D3)⋊C4—41no—
110C4⋊Dic3—41no—
111Dic3⋊C4—22no—
112Dic3⋊C4—41no—
113C4×Dic3—41no—
114C4×Dic3—41no—
115C22×D4—31no—
116C22×D4—31no—
117C22×D4—31no—
118C23×C4—31no—
119C2×C4⋊C4—31no—
120C2×C4⋊C4—31no—
121C2×C4⋊C4—31no—
122C2×C4⋊C4C2×D311yescharacteristic
123C2×C4⋊C4C2×D311yescharacteristic
124C2×C22⋊C4—31no—
125C2×C22⋊C4—31no—
126C2×C22⋊C4—31no—
127C2×C22⋊C4—31no—
128C2×C22⋊C4—31no—
129C2×C22⋊C4—31no—
130C2×C22⋊C4—31no—
131C2×C22⋊C4—31no—
132C2×C22⋊C4—31no—
133C2×C22⋊C4—31no—
134C2×C22⋊C4—31no—
135C2×C22⋊C4—31no—
136C2×C22⋊C4—31no—
137C2×C22⋊C4—31no—
138C2×C42—31no—
139C2×C42—31no—
140#95—31no—
141#95—31no—
142#95—31no—
143#95—31no—
144#95—31no—
145#95C2×D311yescharacteristic
146#95C2×D311yescharacteristic
147#95C2×D311yescharacteristic
148#95C2×D311yescharacteristic
149#95C2×D311yescharacteristic
150C23×C3C2411yescharacteristic
151C22×D3—21no—
152C22×D3—21no—
153C22×D3—21no—
154C22×D3—21no—
155C22×D3—21no—
156C22×D3—21no—
157C22×D3—21no—
158C22×D3—21no—
159C22×D3—21no—
160C22×D3—21no—
161C22×D3—21no—
162C22×D3—21no—
163C22×D3—21no—
164C22×D3—21no—
165C22×D3—21no—
166C22×D3—21no—
167C22×D3—21no—
168C22×D3—21no—
169C22×D3—21no—
170C22×D3—21no—
171C22×D3#4112yes—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4#4112yes—
193C2×Dic3—21no—
194C2×Dic3—21no—
195C2×Dic3—21no—
196C2×Dic3—21no—
197C2×Dic3—21no—
198C2×Dic3—21no—
199C2×Dic3—21no—
200C2×Dic3—21no—
201C2×Dic3—21no—
202C2×Dic3—21no—
203C2×Dic3—21no—
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3—21no—
207C2×Dic3#4112yes—
208D12—44no—
209D12—44no—
210D12—44no—
211C4×D3—44no—
212C24—31no—
213C24—31no—
214C24—31no—
215C2×D4—62no—
216C2×D4—62no—
217C2×D4—62no—
218C2×D4—121no—
219C2×D4—121no—
220C2×D4—121no—
221C2×D4—121no—
222C2×D4—121no—
223C2×D4—121no—
224C22×C4—31no—
225C22×C4—31no—
226C22×C4—31no—
227C22×C4—31no—
228C22×C4—31no—
229C22×C4—62no—
230C22×C4—121no—
231C22×C4—121no—
232C22×C4C22×D311yescharacteristic
233C22×C4C22×D311yescharacteristic
234C22×C4C22×D311yescharacteristic
235C22×C4C22×D311yescharacteristic
236C22×C4C22×D311yescharacteristic
237C22×C4C22×D311yescharacteristic
238C22×C4C22×D311yescharacteristic
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—62no—
242C4⋊C4—121no—
243C4⋊C4—121no—
244C22⋊C4—62no—
245C22⋊C4—62no—
246C22⋊C4—62no—
247C22⋊C4—62no—
248C22⋊C4—121no—
249C22⋊C4—121no—
250C22⋊C4—121no—
251C22⋊C4—121no—
252C22⋊C4—121no—
253C22⋊C4—121no—
254C22⋊C4—121no—
255C22⋊C4—121no—
256C22⋊C4—121no—
257C22⋊C4—121no—
258C42—121no—
259C42—121no—
260C22×C3C2×#4111yescharacteristic
261C22×C3C2×#4111yescharacteristic
262C22×C3C2×#4111yescharacteristic
263C22×C3H(5,𝔽2)11yescharacteristic
264C22×C3H(5,𝔽2)11yescharacteristic
265C22×C3H(5,𝔽2)11yescharacteristic
266C22×C3H(5,𝔽2)11yescharacteristic
267C2×D3—22no—
268C2×D3—22no—
269C2×D3—22no—
270C2×D3—41no—
271C2×D3—41no—
272C2×D3—41no—
273C2×D3—41no—
274C2×D3—41no—
275C2×D3—41no—
276C2×D3—41no—
277C2×D3—41no—
278C2×D3—41no—
279C2×D3—41no—
280C2×D3—41no—
281C2×D3—41no—
282C2×D3—41no—
283C2×D3—41no—
284C2×D3—41no—
285C2×D3—41no—
286C2×D3—41no—
287C2×D3—41no—
288C3×C4—22no—
289C3×C4—41no—
290C3×C4—41no—
291C3×C4—41no—
292C3×C4—41no—
293C3×C4—41no—
294C3×C4—41no—
295Dic3—22no—
296Dic3—41no—
297Dic3—41no—
298Dic3—41no—
299Dic3—41no—
300C23—32no—
301C23—61no—
302C23—61no—
303C23—61no—
304C23—61no—
305C23—61no—
306C23—61no—
307C23—61no—
308C23—61no—
309C23—61no—
310C23—61no—
311C23—61no—
312C23—61no—
313C23—61no—
314C23—61no—
315C23—61no—
316C23—61no—
317C23—61no—
318C23—61no—
319C23—61no—
320C23—61no—
321C23C23×D311yescharacteristic
322D4—124no—
323D4—124no—
324D4—124no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—32no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—61no—
350C2×C4—61no—
351C2×C4—61no—
352C2×C4—61no—
353C2×C4—61no—
354C2×C4—61no—
355C2×C4—61no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—61no—
360C2×C4—124no—
361C2×C4#23912yes—
362C2×C3#52911yescharacteristic
363C2×C3#53811yescharacteristic
364C2×C3#53811yescharacteristic
365C2×C3#53811yescharacteristic
366C2×C3#53811yescharacteristic
367C2×C3#54011yescharacteristic
368C2×C3#54011yescharacteristic
369D3—42no—
370D3—81no—
371D3—81no—
372C22—62no—
373C22—62no—
374C22—62no—
375C22—121no—
376C22—121no—
377C22—121no—
378C22—121no—
379C22—121no—
380C22—121no—
381C22—121no—
382C22—121no—
383C22—121no—
384C22—121no—
385C22—121no—
386C22—121no—
387C22—121no—
388C22—121no—
389C22—121no—
390C22—121no—
391C22—121no—
392C22—121no—
393C22#100411yescharacteristic
394C22C2×#23911yescharacteristic
395C22D3×#4111yescharacteristic
396C22D3×#4111yescharacteristic
397C22#100911yescharacteristic
398C22#100911yescharacteristic
399C22#100911yescharacteristic
400C4—22no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—62no—
408C4—121no—
409C4—121no—
410C4—121no—
411C4—121no—
412C3#264811yescharacteristic, complemented, Sylow
413C2—122no—
414C2—241no—
415C2—241no—
416C2#564411yescharacteristic
417C2#566311yescharacteristic
418C2#567811yescharacteristic
419C2#569111yescharacteristic
420C2#569711yescharacteristic
421C2#570411yescharacteristic
422C2#570411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#400C2
prime radical#4671C2
hypercenter#400D3
hypocenterC3#2648
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C3×#400C2semidirect
2. #4994C2semidirect
3. #4991C2semidirect
4. #4990C2semidirect
5. #4988C2semidirect
6. #4675C2semidirect
7. #4674C2semidirect
8. #4672C2semidirect
9. #4672C2semidirect
11. #4670C2semidirect
12. #4666C2semidirect
13. #4665C2semidirect
14. #4665C2semidirect
15. #4652C2semidirect
52. #400D3semidirect
412. C3#2648semidirect

Maximal subgroups

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

subgroupindex
C3×#4002
#49942
#49912
#49902
#49882
#46752
#46742
#46722
#46712
#46702
#46662
#46652
#46522
#26483