Finite Group Databasebeta

← #77165

Subgroups — #77165

Order 384.

522 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#394C211yesmaximal, characteristic, complemented
2#5223C211yesmaximal, characteristic, complemented
3#5223C211yesmaximal, characteristic, complemented
4#4994C211yesmaximal, characteristic, complemented
5#4992C211yesmaximal, characteristic, complemented
6#4961C211yesmaximal, characteristic, complemented
7#4960C211yesmaximal, characteristic, complemented
8#4960C211yesmaximal, characteristic, complemented
9#4959C211yesmaximal, characteristic, complemented
10#4956C211yesmaximal, characteristic, complemented
11#4956C211yesmaximal, characteristic, complemented
12#4953C211yesmaximal, characteristic, complemented
13#4952C211yesmaximal, characteristic, complemented
14#4665C211yesmaximal, characteristic, complemented
15C3⋊#401C211yesmaximal, characteristic, complemented
16#2825—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C22×#67C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×D12C2211yescharacteristic
23C22×C4×D3C2211yescharacteristic
24C2×C3×C4⋊C4C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2211yescharacteristic
27C2×C3×C22⋊C4C2211yescharacteristic
28C2×C3×C22⋊C4C2211yescharacteristic
29C2×#217C2211yescharacteristic
30C2×#217C2211yescharacteristic
31C2×#217C2211yescharacteristic
32C2×#217C2211yescharacteristic
33C2×(C2×D3)⋊C4C2211yescharacteristic
34C2×(C2×D3)⋊C4C2211yescharacteristic
35C2×(C2×D3)⋊C4C2211yescharacteristic
36C2×(C2×D3)⋊C4C2211yescharacteristic
37C2×(C2×D3)⋊C4C2211yescharacteristic
38C2×(C2×D3)⋊C4C2211yescharacteristic
39C2×(C2×D3)⋊C4C2211yescharacteristic
40C2×C4⋊Dic3C2211yescharacteristic
41C2×Dic3⋊C4C2211yescharacteristic
42C3×#95C2211yescharacteristic
43C3⋊#95C2211yescharacteristic
44C3⋊#95C2211yescharacteristic
45C3⋊#95C2211yescharacteristic
46C3⋊#95C2211yescharacteristic
47C3⋊#95C2211yescharacteristic
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52#401—31no—
53#397—31no—
54#397—31no—
55#397—31no—
56#396—31no—
57#394—31no—
58#394—31no—
59#394D311yescharacteristic, complemented
60#392—31no—
61#392—31no—
62#392—31no—
63#392—31no—
64#386—31no—
65#386—31no—
66#386—31no—
67C24×C3C2311yescharacteristic
68C24×C3C2311yescharacteristic
69C23×D3C2311yescharacteristic
70C23×D3C2311yescharacteristic
71C2×C3×D4—41no—
72C2×C3×D4—41no—
73C2×C3×D4—41no—
74C22×C3×C4C2311yescharacteristic
75C22×C3×C4C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C2×#67—22no—
80C2×#67—22no—
81C2×#67—41no—
82C2×#67—41no—
83C2×#67—41no—
84C2×#67—41no—
85C2×#67—41no—
86C2×#67—41no—
87C2×#67—41no—
88C2×#67—41no—
89C2×#67—41no—
90C2×#67—41no—
91C22×Dic3C2311yescharacteristic
92C22×Dic3C2311yescharacteristic
93C22×Dic3C2311yescharacteristic
94C22×Dic3C2311yescharacteristic
95C22×Dic3C2311yescharacteristic
96C22×Dic3C2311yescharacteristic
97C2×D12—22no—
98C2×D12—41no—
99C2×D12—41no—
100C2×C4×D3—41no—
101C2×C4×D3—41no—
102C2×C4×D3—41no—
103C3×C4⋊C4—41no—
104C3×C22⋊C4—41no—
105C3×C22⋊C4—41no—
106C3×C22⋊C4—41no—
107C3×C22⋊C4—41no—
108#217—41no—
109#217—41no—
110#217—41no—
111#217—41no—
112(C2×D3)⋊C4—41no—
113(C2×D3)⋊C4—41no—
114(C2×D3)⋊C4—41no—
115(C2×D3)⋊C4—41no—
116(C2×D3)⋊C4—41no—
117(C2×D3)⋊C4—41no—
118(C2×D3)⋊C4—41no—
119C4⋊Dic3—41no—
120Dic3⋊C4—41no—
121C22×D4—31no—
122C22×D4—31no—
123C22×D4—31no—
124C22×D4—31no—
125C22×D4—31no—
126C22×D4C2×D311yescharacteristic
127C23×C4—31no—
128C2×C4⋊C4—31no—
129C2×C4⋊C4—31no—
130C2×C4⋊C4C2×D311yescharacteristic
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×C22⋊C4—31no—
139C2×C22⋊C4—31no—
140C2×C22⋊C4—31no—
141C2×C22⋊C4—31no—
142C2×C22⋊C4C2×D311yescharacteristic
143C2×C22⋊C4C2×D311yescharacteristic
144C2×C22⋊C4C2×D311yescharacteristic
145C2×C22⋊C4C2×D311yescharacteristic
146#95—31no—
147#95—31no—
148#95—31no—
149#95—31no—
150#95—31no—
151#95—31no—
152#95—31no—
153#95—31no—
154#95—31no—
155#95C2×D311yescharacteristic
156C23×C3—21no—
157C23×C3—21no—
158C23×C3—21no—
159C23×C3—21no—
160C23×C3—21no—
161C23×C3—21no—
162C23×C3—21no—
163C23×C3—21no—
164C23×C3—21no—
165C23×C3—21no—
166C23×C3—21no—
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—21no—
170C23×C3C2411yescharacteristic
171C22×D3—21no—
172C22×D3—21no—
173C22×D3—21no—
174C22×D3—21no—
175C22×D3—21no—
176C22×D3—21no—
177C22×D3—21no—
178C22×D3—21no—
179C22×D3—21no—
180C22×D3—21no—
181C22×D3—21no—
182C22×D3—21no—
183C22×D3—21no—
184C22×D3#4112yes—
185C3×D4—44no—
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—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201#67—44no—
202#67—44no—
203#67—44no—
204#67—44no—
205C2×Dic3—21no—
206C2×Dic3—21no—
207C2×Dic3—21no—
208C2×Dic3—21no—
209C2×Dic3—21no—
210C2×Dic3—21no—
211C2×Dic3—21no—
212C2×Dic3—21no—
213C2×Dic3—21no—
214C2×Dic3—21no—
215C2×Dic3—21no—
216C2×Dic3—21no—
217C2×Dic3—21no—
218C2×Dic3—21no—
219C2×Dic3—21no—
220C2×Dic3—21no—
221C2×Dic3—21no—
222C2×Dic3—21no—
223D12—44no—
224C4×D3—44no—
225C24—31no—
226C24—31no—
227C24C22×D311yescharacteristic
228C24C22×D311yescharacteristic
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—62no—
233C2×D4—62no—
234C2×D4—62no—
235C2×D4—121no—
236C2×D4—121no—
237C2×D4—121no—
238C2×D4—121no—
239C2×D4—121no—
240C2×D4—121no—
241C2×D4—121no—
242C2×D4—121no—
243C2×D4—121no—
244C2×D4—121no—
245C2×D4—121no—
246C2×D4—121no—
247C22×C4—31no—
248C22×C4—31no—
249C22×C4—31no—
250C22×C4—31no—
251C22×C4—31no—
252C22×C4—31no—
253C22×C4—121no—
254C22×C4—121no—
255C22×C4—121no—
256C22×C4C22×D311yescharacteristic
257C22×C4C22×D311yescharacteristic
258C22×C4C22×D311yescharacteristic
259C22×C4C22×D311yescharacteristic
260C22×C4C22×D311yescharacteristic
261C4⋊C4—41no—
262C4⋊C4—121no—
263C4⋊C4—121no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—121no—
269C22⋊C4—121no—
270C22⋊C4—121no—
271C22⋊C4—121no—
272C22⋊C4—121no—
273C22⋊C4—121no—
274C22⋊C4—121no—
275C22⋊C4—121no—
276C22⋊C4—121no—
277C22⋊C4—121no—
278C22⋊C4—121no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3—41no—
284C22×C3—41no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3—41no—
288C22×C3—41no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3C2×#4111yescharacteristic
294C22×C3H(5,𝔽2)11yescharacteristic
295C22×C3H(5,𝔽2)11yescharacteristic
296C22×C3H(5,𝔽2)11yescharacteristic
297C22×C3H(5,𝔽2)11yescharacteristic
298C22×C3H(5,𝔽2)11yescharacteristic
299C22×C3H(5,𝔽2)11yescharacteristic
300C2×D3—22no—
301C2×D3—22no—
302C2×D3—22no—
303C2×D3—41no—
304C2×D3—41no—
305C2×D3—41no—
306C2×D3—41no—
307C2×D3—41no—
308C2×D3—41no—
309C2×D3—41no—
310C2×D3—41no—
311C2×D3—41no—
312C2×D3—41no—
313C2×D3—41no—
314C3×C4—41no—
315C3×C4—41no—
316C3×C4—41no—
317C3×C4—41no—
318C3×C4—41no—
319Dic3—41no—
320Dic3—41no—
321Dic3—41no—
322Dic3—41no—
323Dic3—41no—
324Dic3—41no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—32no—
340C23—61no—
341C23—61no—
342C23—61no—
343C23—61no—
344C23—61no—
345C23—61no—
346C23—61no—
347C23—61no—
348C23—61no—
349C23—61no—
350C23—61no—
351C23—61no—
352C23—61no—
353C23C23×D311yescharacteristic
354D4—44no—
355D4—124no—
356D4—124no—
357D4—124no—
358D4—124no—
359D4—124no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—61no—
376C2×C4—61no—
377C2×C4—61no—
378C2×C4—61no—
379C2×C4—61no—
380C2×C4—61no—
381C2×C4—61no—
382C2×C4—61no—
383C2×C4—61no—
384C2×C4—61no—
385C2×C4—61no—
386C2×C4—61no—
387C2×C4—61no—
388C2×C4—61no—
389C2×C4—61no—
390C2×C4—61no—
391C2×C4—61no—
392C2×C4—61no—
393C2×C4—124no—
394C2×C3—81no—
395C2×C3—81no—
396C2×C3#53811yescharacteristic
397C2×C3#53811yescharacteristic
398C2×C3#53811yescharacteristic
399C2×C3#56011yescharacteristic
400C2×C3#56011yescharacteristic
401C2×C3#56011yescharacteristic
402C2×C3#56011yescharacteristic
403D3—42no—
404D3—81no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—62no—
420C22—62no—
421C22—62no—
422C22—121no—
423C22—121no—
424C22—121no—
425C22—121no—
426C22—121no—
427C22—121no—
428C22—121no—
429C22—121no—
430C22—121no—
431C22—121no—
432C22—121no—
433C22#100411yescharacteristic
434C22#100411yescharacteristic
435C22#100411yescharacteristic
436C22#100411yescharacteristic
437C22D3×#4111yescharacteristic
438C22#100911yescharacteristic
439C22#100911yescharacteristic
440C4—41no—
441C4—41no—
442C4—41no—
443C4—41no—
444C4—41no—
445C4—121no—
446C4—121no—
447C4—121no—
448C4—121no—
449C4—121no—
450C4—121no—
451C3#282511yescharacteristic, complemented, Sylow
452C2—81no—
453C2—81no—
454C2—122no—
455C2—241no—
456C2#559511yescharacteristic
457C2#561311yescharacteristic
458C2#562011yescharacteristic
459C2#562011yescharacteristic
460C2#565711yescharacteristic
461C2#566611yescharacteristic
462C2#567811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
hypercenter#394D3
hypocenterC3#2825
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C3×#394C2semidirect
2. #5223C2semidirect
3. #5223C2semidirect
4. #4994C2semidirect
5. #4992C2semidirect
6. #4961C2semidirect
7. #4960C2semidirect
8. #4960C2semidirect
9. #4959C2semidirect
10. #4956C2semidirect
11. #4956C2semidirect
12. #4953C2semidirect
13. #4952C2semidirect
14. #4665C2semidirect
15. C3⋊#401C2semidirect
59. #394D3semidirect
451. C3#2825semidirect

Maximal subgroups

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

subgroupindex
C3×#3942
#52232
#49942
#49922
#49612
#49602
#49592
#49562
#49532
#49522
#46652
C3⋊#4012
#28253