Finite Group Databasebeta

← #82902

Subgroups — #82902

Order 384.

582 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#123C211yesmaximal, characteristic, complemented
2C2×C4×#67C211yesmaximal, characteristic, complemented
3#5223C211yesmaximal, characteristic
4#5222C211yesmaximal, characteristic, complemented
5C4×#217C211yesmaximal, characteristic, complemented
6#4990C211yesmaximal, characteristic, complemented
7#4989C211yesmaximal, characteristic, complemented
8#4978C211yesmaximal, characteristic, complemented
9#4976C211yesmaximal, characteristic, complemented
10#4959C211yesmaximal, characteristic, complemented
11#4957C211yesmaximal, characteristic, complemented
12#4956C211yesmaximal, characteristic, complemented
13#4952C211yesmaximal, characteristic, complemented
14#4945C211yesmaximal, characteristic, complemented
15#4944C211yesmaximal, characteristic, complemented
16#2602—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C23×C3×C4C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×C4×D3C2211yescharacteristic
22C3×#123—24no—
23C2×C3×C4⋊C4C2211yescharacteristic
24C2×C3×C4⋊C4C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2211yescharacteristic
27C2×C3×C22⋊C4C2211yescharacteristic
28C2×#217C2211yescharacteristic
29C2×#217C2211yescharacteristic
30C2×#217C2211yescharacteristic
31C2×#217C2211yescharacteristic
32C2×#217C2211yescharacteristic
33C2×#217C2211yescharacteristic
34C4×#67—24no—
35C2×(C2×D3)⋊C4C2211yescharacteristic
36C2×(C2×D3)⋊C4C2211yescharacteristic
37C2×(C2×D3)⋊C4C2211yescharacteristic
38C2×(C2×D3)⋊C4C2211yescharacteristic
39C2×Dic3⋊C4C2211yescharacteristic
40C2×Dic3⋊C4C2211yescharacteristic
41C2×Dic3⋊C4C2211yescharacteristic
42C2×Dic3⋊C4C2211yescharacteristic
43C2×C4×Dic3C2211yescharacteristic
44C2×C4×Dic3C2211yescharacteristic
45C2×C4×Dic3C2211yescharacteristic
46C3⋊#95C2211yescharacteristic
47C3⋊#95C2211yescharacteristic
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C2×#123D311yescharacteristic, complemented
55C2×C4×D4—31no—
56#397—31no—
57#394—31no—
58#394—31no—
59#392—31no—
60#390—31no—
61#389—31no—
62#388—31no—
63#388—31no—
64#388—31no—
65#387—31no—
66#386—31no—
67#385—31no—
68C4×C22⋊C4—31no—
69C24×C3C2311yescharacteristic
70C24×C3C2311yescharacteristic
71C23×D3C2311yescharacteristic
72C2×C3×D4—22no—
73C2×C3×D4—41no—
74C2×C3×D4—41no—
75C22×C3×C4—41no—
76C22×C3×C4—41no—
77C22×C3×C4D414yescomplemented
78C22×C3×C4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C22×C3×C4C2311yescharacteristic
81C22×C3×C4C2311yescharacteristic
82C22×C3×C4C2311yescharacteristic
83C2×#67—22no—
84C2×#67—22no—
85C2×#67—22no—
86C2×#67—22no—
87C2×#67—22no—
88C2×#67—41no—
89C22×Dic3C2311yescharacteristic
90C22×Dic3C2311yescharacteristic
91C22×Dic3C2311yescharacteristic
92C22×Dic3C2311yescharacteristic
93C22×Dic3C2311yescharacteristic
94C22×Dic3C2311yescharacteristic
95C22×Dic3C2311yescharacteristic
96C2×C4×D3—22no—
97C2×C4×D3—41no—
98C2×C4×D3—41no—
99C3×C4⋊C4—22no—
100C3×C4⋊C4—22no—
101C3×C22⋊C4—22no—
102C3×C22⋊C4—22no—
103C3×C22⋊C4—22no—
104#217—22no—
105#217—22no—
106#217—22no—
107#217—22no—
108#217—22no—
109#217—41no—
110(C2×D3)⋊C4—22no—
111(C2×D3)⋊C4—22no—
112(C2×D3)⋊C4—22no—
113(C2×D3)⋊C4—41no—
114Dic3⋊C4—22no—
115Dic3⋊C4—22no—
116Dic3⋊C4—22no—
117Dic3⋊C4—41no—
118C4×Dic3—22no—
119C4×Dic3—41no—
120C4×Dic3—41no—
121C22×D4—31no—
122C22×D4—31no—
123C22×D4C2×D311yescharacteristic
124C23×C4—31no—
125C23×C4C2×D311yescharacteristic
126#123—24no—
127C4×D4—64no—
128C2×C4⋊C4—31no—
129C2×C4⋊C4—31no—
130C2×C4⋊C4—31no—
131C2×C4⋊C4—31no—
132C2×C4⋊C4C2×D311yescharacteristic
133C2×C4⋊C4C2×D311yescharacteristic
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⋊C4—31no—
143C2×C22⋊C4—31no—
144C2×C22⋊C4C2×D311yescharacteristic
145C2×C22⋊C4C2×D311yescharacteristic
146C2×C22⋊C4C2×D311yescharacteristic
147C2×C42—31no—
148C2×C42—31no—
149C2×C42—31no—
150#95—31no—
151#95—31no—
152#95—31no—
153#95—31no—
154#95—31no—
155#95—31no—
156#95—31no—
157#95—31no—
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×C3—21no—
171C23×C3C2×D412yes—
172C23×C3C2411yescharacteristic
173C22×D3—21no—
174C22×D3—21no—
175C22×D3—21no—
176C22×D3—21no—
177C22×D3—21no—
178C22×D3—21no—
179C22×D3#4112yes—
180C3×D4—44no—
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—21no—
193C2×C3×C4—21no—
194C2×C3×C4—44no—
195C2×C3×C4C2×D412yes—
196C2×C3×C4C2×D412yes—
197#67—28no—
198#67—44no—
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—
210C2×Dic3—21no—
211C2×Dic3—21no—
212C2×Dic3—21no—
213C2×Dic3—21no—
214C2×Dic3—21no—
215C2×Dic3—21no—
216C2×Dic3—21no—
217C2×Dic3#4112yes—
218C2×Dic3#4112yes—
219C2×Dic3#4112yes—
220C4×D3—44no—
221C24—31no—
222C24C22×D311yescharacteristic
223C24C22×D311yescharacteristic
224C2×D4—22no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4—62no—
228C2×D4—62no—
229C2×D4—62no—
230C2×D4—62no—
231C2×D4—62no—
232C2×D4—121no—
233C22×C4—31no—
234C22×C4—31no—
235C22×C4—31no—
236C22×C4—31no—
237C22×C4—31no—
238C22×C4—31no—
239C22×C4—31no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—62no—
243C22×C4—121no—
244C22×C4—121no—
245C22×C4#6714yescomplemented
246C22×C4C22×D311yescharacteristic
247C22×C4C22×D311yescharacteristic
248C22×C4C22×D311yescharacteristic
249C22×C4C22×D311yescharacteristic
250C22×C4C22×D311yescharacteristic
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—62no—
254C4⋊C4—62no—
255C4⋊C4—62no—
256C4⋊C4—121no—
257C22⋊C4—22no—
258C22⋊C4—22no—
259C22⋊C4—22no—
260C22⋊C4—62no—
261C22⋊C4—62no—
262C22⋊C4—62no—
263C22⋊C4—62no—
264C22⋊C4—62no—
265C22⋊C4—62no—
266C22⋊C4—62no—
267C22⋊C4—62no—
268C22⋊C4—121no—
269C22⋊C4—121no—
270C42—62no—
271C42—121no—
272C42—121no—
273C22×C3—22no—
274C22×C3—22no—
275C22×C3—22no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—41no—
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×C3C22×D411yescharacteristic
288C22×C3C2×#4111yescharacteristic
289C22×C3C2×#4111yescharacteristic
290C22×C3C2×#4111yescharacteristic
291C22×C3C2×#4111yescharacteristic
292C22×C3H(5,𝔽2)11yescharacteristic
293C22×C3H(5,𝔽2)11yescharacteristic
294C2×D3—22no—
295C2×D3—22no—
296C2×D3—22no—
297C2×D3—41no—
298C2×D3—41no—
299C2×D3—41no—
300C2×D3—41no—
301C3×C4—22no—
302C3×C4—22no—
303C3×C4—41no—
304C3×C4—41no—
305C3×C4—41no—
306Dic3—22no—
307Dic3—22no—
308Dic3—22no—
309Dic3—41no—
310Dic3—41no—
311Dic3—41no—
312Dic3—41no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—32no—
327C23—61no—
328C23—61no—
329C23—61no—
330C23—61no—
331C23—61no—
332C23—61no—
333C23C2×#6712yes—
334C23C23×D311yescharacteristic
335D4—44no—
336D4—68no—
337D4—124no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—32no—
352C2×C4—32no—
353C2×C4—32no—
354C2×C4—44no—
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—61no—
364C2×C4—61no—
365C2×C4—61no—
366C2×C4—61no—
367C2×C4—61no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—61no—
372C2×C4—61no—
373C2×C4—124no—
374C2×C4C2×#6712yes—
375C2×C4C2×#6712yes—
376C2×C3—42no—
377C2×C3—81no—
378C2×C3#52511yescharacteristic
379C2×C3#53211yescharacteristic
380C2×C3#53411yescharacteristic
381C2×C3#55111yescharacteristic
382C2×C3#55311yescharacteristic
383C2×C3#55311yescharacteristic
384C2×C3#55911yescharacteristic
385D3—42no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—62no—
401C22—62no—
402C22—62no—
403C22—121no—
404C22—121no—
405C22—121no—
406C22—121no—
407C22#100411yescharacteristic
408C22#100411yescharacteristic
409C22D3×#4111yescharacteristic
410C22D3×#4111yescharacteristic
411C22D3×#4111yescharacteristic
412C22D3×#4111yescharacteristic
413C22C22×#6711yescharacteristic
414C4—22no—
415C4—22no—
416C4—41no—
417C4—41no—
418C4—41no—
419C4—62no—
420C4—62no—
421C4—62no—
422C4—121no—
423C4—121no—
424C4—121no—
425C4—121no—
426C3#260211yescharacteristic, complemented, Sylow
427C2—42no—
428C2—81no—
429C2—122no—
430C2#565911yescharacteristic
431C2#566111yescharacteristic
432C2#566111yescharacteristic
433C2#566411yescharacteristic
434C2#580411yescharacteristic
435C2#582811yescharacteristic
436C2#583211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#123C2
prime radical#5223C2
hypercenterC2×#123D3
hypocenterC3#2602
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×C3×#123C2semidirect
2. C2×C4×#67C2semidirect
4. #5222C2semidirect
5. C4×#217C2semidirect
6. #4990C2semidirect
7. #4989C2semidirect
8. #4978C2semidirect
9. #4976C2semidirect
10. #4959C2semidirect
11. #4957C2semidirect
12. #4956C2semidirect
13. #4952C2semidirect
14. #4945C2semidirect
15. #4944C2semidirect
54. C2×#123D3semidirect
77. C22×C3×C4D4semidirect
245. C22×C4#67semidirect
426. C3#2602semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1232
C2×C4×#672
#52232
#52222
C4×#2172
#49902
#49892
#49782
#49762
#49592
#49572
#49562
#49522
#49452
#49442
#26023