Finite Group Databasebeta

← #77730

Subgroups — #77730

Order 384.

472 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#397C211yesmaximal, characteristic, complemented
2#4991C211yesmaximal, characteristic, complemented
3#4986C211yesmaximal, characteristic, complemented
4#4961C211yesmaximal, characteristic, complemented
5#4961C211yesmaximal, characteristic, complemented
6#4960C211yesmaximal, characteristic
7#4959C211yesmaximal, characteristic, complemented
8#4956C211yesmaximal, characteristic, complemented
9#4947C211yesmaximal, characteristic, complemented
10#4674C211yesmaximal, characteristic, complemented
11#4673C211yesmaximal, characteristic, complemented
12#4673C211yesmaximal, characteristic, complemented
13#4668C211yesmaximal, characteristic, complemented
14#4665C211yesmaximal, characteristic, complemented
15#4660C211yesmaximal, characteristic, complemented
16#2824—31nomaximal, Sylow
17C22×#67C2211yescharacteristic
18C22×#67C2211yescharacteristic
19C22×D12C2211yescharacteristic
20C22×C4×D3C2211yescharacteristic
21C22×Dic6C2211yescharacteristic
22C2×C3×C4⋊C4C2211yescharacteristic
23C2×C3×C22⋊C4C2211yescharacteristic
24C2×C3×C22⋊C4C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×#217C2211yescharacteristic
27C2×#217C2211yescharacteristic
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×C4⋊Dic3C2211yescharacteristic
39C2×Dic3⋊C4C2211yescharacteristic
40C3×#95C2211yescharacteristic
41C3×#95C2211yescharacteristic
42C3×#95C2211yescharacteristic
43C3⋊#95C2211yescharacteristic
44C3⋊#95C2211yescharacteristic
45C3⋊#95C2211yescharacteristic
46C3⋊#95C2211yescharacteristic
47C3⋊#95C2211yescharacteristic
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52#400—31no—
53#400—31no—
54#397—31no—
55#397D311yescharacteristic, complemented
56#394—31no—
57#394—31no—
58#394—31no—
59#394—31no—
60#393—31no—
61#393—31no—
62#393—31no—
63#392—31no—
64#386—31no—
65#386—31no—
66#385—31no—
67C24×C3C2311yescharacteristic
68C23×D3C2311yescharacteristic
69C23×D3C2311yescharacteristic
70C22×C3×C4C2311yescharacteristic
71C22×C3×C4C2311yescharacteristic
72C22×C3×C4C2311yescharacteristic
73C22×C3×C4C2311yescharacteristic
74C22×C3×C4C2311yescharacteristic
75C22×C3×C4C2311yescharacteristic
76C2×#67—22no—
77C2×#67—41no—
78C2×#67—41no—
79C2×#67—41no—
80C2×#67—41no—
81C2×#67—41no—
82C22×Dic3C2311yescharacteristic
83C22×Dic3C2311yescharacteristic
84C22×Dic3C2311yescharacteristic
85C22×Dic3C2311yescharacteristic
86C22×Dic3C2311yescharacteristic
87C22×Dic3C2311yescharacteristic
88C2×D12—22no—
89C2×D12—41no—
90C2×D12—41no—
91C2×C4×D3—41no—
92C2×C4×D3—41no—
93C2×C4×D3—41no—
94C2×Dic6—41no—
95C2×Dic6—41no—
96C2×Dic6—41no—
97C3×C4⋊C4—41no—
98C3×C22⋊C4—41no—
99C3×C22⋊C4—41no—
100C3×C22⋊C4—41no—
101#217—41no—
102#217—41no—
103(C2×D3)⋊C4—22no—
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—
110(C2×D3)⋊C4—41no—
111(C2×D3)⋊C4—41no—
112C4⋊Dic3—41no—
113C4⋊Dic3—41no—
114Dic3⋊C4—41no—
115C22×Q8—31no—
116C22×D4—31no—
117C22×D4—31no—
118C22×D4—31no—
119C23×C4—31no—
120C2×C4⋊C4—31no—
121C2×C4⋊C4—31no—
122C2×C4⋊C4—31no—
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⋊C4C2×D311yescharacteristic
136C2×C22⋊C4C2×D311yescharacteristic
137C2×C22⋊C4C2×D311yescharacteristic
138#95—31no—
139#95—31no—
140#95—31no—
141#95—31no—
142#95—31no—
143#95—31no—
144#95—31no—
145#95—31no—
146#95—31no—
147#95C2×D311yescharacteristic
148#95C2×D311yescharacteristic
149#95C2×D311yescharacteristic
150C23×C3—21no—
151C23×C3—21no—
152C23×C3—21no—
153C23×C3—21no—
154C23×C3—21no—
155C23×C3—21no—
156C23×C3—21no—
157C23×C3C2411yescharacteristic
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—
190#67—44no—
191#67—44no—
192C2×Dic3—21no—
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—21no—
208C2×Dic3—21no—
209C2×Dic3—21no—
210D12—44no—
211C4×D3—44no—
212Dic6—44no—
213C24—31no—
214C24—31no—
215C24C22×D311yescharacteristic
216C2×Q8—121no—
217C2×Q8—121no—
218C2×Q8—121no—
219C2×D4—62no—
220C2×D4—62no—
221C2×D4—121no—
222C2×D4—121no—
223C2×D4—121no—
224C2×D4—121no—
225C2×D4—121no—
226C2×D4—121no—
227C2×D4—121no—
228C22×C4—31no—
229C22×C4—31no—
230C22×C4—31no—
231C22×C4—31no—
232C22×C4—31no—
233C22×C4—31no—
234C22×C4—121no—
235C22×C4—121no—
236C22×C4—121no—
237C22×C4C22×D311yescharacteristic
238C22×C4C22×D311yescharacteristic
239C22×C4C22×D311yescharacteristic
240C22×C4C22×D311yescharacteristic
241C22×C4C22×D311yescharacteristic
242C22×C4C22×D311yescharacteristic
243C4⋊C4—41no—
244C4⋊C4—121no—
245C4⋊C4—121no—
246C4⋊C4—121no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C22⋊C4—62no—
251C22⋊C4—121no—
252C22⋊C4—121no—
253C22⋊C4—121no—
254C22⋊C4—121no—
255C22⋊C4—121no—
256C22⋊C4—121no—
257C22⋊C4—121no—
258C22⋊C4—121no—
259C22⋊C4—121no—
260C22⋊C4—121no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3—41no—
264C22×C3—41no—
265C22×C3—41no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3C2×#4111yescharacteristic
269C22×C3H(5,𝔽2)11yescharacteristic
270C22×C3H(5,𝔽2)11yescharacteristic
271C22×C3H(5,𝔽2)11yescharacteristic
272C22×C3H(5,𝔽2)11yescharacteristic
273C22×C3H(5,𝔽2)11yescharacteristic
274C22×C32−1+411yescharacteristic
275C2×D3—22no—
276C2×D3—22no—
277C2×D3—22no—
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—
288C2×D3—41no—
289C3×C4—41no—
290C3×C4—41no—
291C3×C4—41no—
292C3×C4—41no—
293C3×C4—41no—
294C3×C4—41no—
295Dic3—41no—
296Dic3—41no—
297Dic3—41no—
298Dic3—41no—
299Dic3—41no—
300Dic3—41no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—32no—
309C23—61no—
310C23—61no—
311C23—61no—
312C23—61no—
313C23—61no—
314C23—61no—
315C23—61no—
316C23—61no—
317C23—61no—
318C23—61no—
319C23—61no—
320C23—61no—
321C23—61no—
322C23C23×D311yescharacteristic
323Q8—124no—
324D4—124no—
325D4—124no—
326D4—124no—
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—61no—
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—61no—
361C2×C4—61no—
362C2×C4—61no—
363C2×C4—124no—
364C2×C3—81no—
365C2×C3#53811yescharacteristic
366C2×C3#53811yescharacteristic
367C2×C3#53911yescharacteristic
368C2×C3#56011yescharacteristic
369C2×C3H(3,𝔽4)11yescharacteristic
370C2×C3#56211yescharacteristic
371C2×C3#56211yescharacteristic
372D3—42no—
373D3—81no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—62no—
382C22—62no—
383C22—62no—
384C22—121no—
385C22—121no—
386C22—121no—
387C22—121no—
388C22—121no—
389C22—121no—
390C22—121no—
391C22—121no—
392C22—121no—
393C22—121no—
394C22—121no—
395C22#100411yescharacteristic
396C22#100411yescharacteristic
397C22D3×#4111yescharacteristic
398C22#100911yescharacteristic
399C22#100911yescharacteristic
400C22#100911yescharacteristic
401C22#101011yescharacteristic
402C4—41no—
403C4—41no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—121no—
409C4—121no—
410C4—121no—
411C4—121no—
412C4—121no—
413C4—121no—
414C3#282411yescharacteristic, complemented, Sylow
415C2—81no—
416C2—122no—
417C2—241no—
418C2#561311yescharacteristic
419C2#566611yescharacteristic
420C2#566711yescharacteristic
421C2#566711yescharacteristic
422C2#568311yescharacteristic
423C2#570411yescharacteristic
424C2#570711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#397C2
prime radical#4960C2
hypercenter#397D3
hypocenterC3#2824
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C3×#397C2semidirect
2. #4991C2semidirect
3. #4986C2semidirect
4. #4961C2semidirect
5. #4961C2semidirect
7. #4959C2semidirect
8. #4956C2semidirect
9. #4947C2semidirect
10. #4674C2semidirect
11. #4673C2semidirect
12. #4673C2semidirect
13. #4668C2semidirect
14. #4665C2semidirect
15. #4660C2semidirect
55. #397D3semidirect
414. C3#2824semidirect

Maximal subgroups

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

subgroupindex
C3×#3972
#49912
#49862
#49612
#49602
#49592
#49562
#49472
#46742
#46732
#46682
#46652
#46602
#28243