Finite Group Databasebeta

← #82871

Subgroups — #82871

Order 384.

568 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#123C211yesmaximal, characteristic
2C2×#935C211yesmaximal, characteristic
3C2×#924C211yesmaximal, characteristic, complemented
4C2×#878C211yesmaximal, characteristic, complemented
5#5222C211yesmaximal, characteristic
6#5210C211yesmaximal, characteristic, complemented
7#4983C211yesmaximal, characteristic, complemented
8#4982C211yesmaximal, characteristic, complemented
9#4979C211yesmaximal, characteristic, complemented
10#4975C211yesmaximal, characteristic, complemented
11#4953C211yesmaximal, characteristic, complemented
12#4952C211yesmaximal, characteristic, complemented
13#4951C211yesmaximal, characteristic, complemented
14#4946C211yesmaximal, characteristic, complemented
15#4945C211yesmaximal, characteristic, complemented
16#2680—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C23×C3×C4C2211yescharacteristic
19C23×Dic3C2211yescharacteristic
20C22×Dic6C2211yescharacteristic
21C3×#123—24no—
22C2×C3×C4⋊C4C2211yescharacteristic
23C2×C3×C4⋊C4C2211yescharacteristic
24C2×C3×C22⋊C4C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2211yescharacteristic
27C2×#217C2211yescharacteristic
28C2×#217C2211yescharacteristic
29C2×#217C2211yescharacteristic
30C2×#217C2211yescharacteristic
31C2×#217C2211yescharacteristic
32C2×#217C2211yescharacteristic
33C2×#217C2211yescharacteristic
34#935—24no—
35C2×C4⋊Dic3C2211yescharacteristic
36C2×C4⋊Dic3C2211yescharacteristic
37#924—24no—
38C2×Dic3⋊C4C2211yescharacteristic
39C2×Dic3⋊C4C2211yescharacteristic
40C2×Dic3⋊C4C2211yescharacteristic
41C2×Dic3⋊C4C2211yescharacteristic
42C2×Dic3⋊C4C2211yescharacteristic
43C2×Dic3⋊C4C2211yescharacteristic
44C2×Dic3⋊C4C2211yescharacteristic
45C2×Dic3⋊C4C2211yescharacteristic
46C2×C4×Dic3C2211yescharacteristic
47#878—24no—
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C3⋊#95C2211yescharacteristic
55C3⋊#95C2211yescharacteristic
56C2×#123—31no—
57C2×#123D311yescharacteristic
58C2×C22⋊Q8—31no—
59C2×C22⋊Q8—31no—
60#398—31no—
61#398—31no—
62#397—31no—
63#397—31no—
64#395—31no—
65#394—31no—
66#388—31no—
67#388—31no—
68#385—31no—
69#385—31no—
70#384—31no—
71C24×C3C2311yescharacteristic
72C24×C3C2311yescharacteristic
73C2×C3×D4—22no—
74C2×C3×D4—22no—
75C2×C3×D4—41no—
76C22×C3×C4—22no—
77C22×C3×C4—22no—
78C22×C3×C4—41no—
79C22×C3×C4C2311yescharacteristic
80C22×C3×C4C2311yescharacteristic
81C22×C3×C4C2311yescharacteristic
82C22×C3×C4C2311yescharacteristic
83C22×C3×C4C2311yescharacteristic
84C22×Dic3—22no—
85C22×Dic3—22no—
86C22×Dic3—41no—
87C22×Dic3C2311yescharacteristic
88C22×Dic3C2311yescharacteristic
89C22×Dic3C2311yescharacteristic
90C22×Dic3C2311yescharacteristic
91C22×Dic3C2311yescharacteristic
92C22×Dic3C2311yescharacteristic
93C22×Dic3C2311yescharacteristic
94C22×Dic3C2311yescharacteristic
95C2×Dic6—22no—
96C2×Dic6—22no—
97C2×Dic6—41no—
98C3×C4⋊C4—22no—
99C3×C4⋊C4—22no—
100C3×C22⋊C4—22no—
101C3×C22⋊C4—22no—
102C3×C22⋊C4D414yes—
103#217—22no—
104#217—22no—
105#217—22no—
106#217—22no—
107#217—22no—
108#217—22no—
109#217D414yes—
110C4⋊Dic3—22no—
111C4⋊Dic3—22no—
112Dic3⋊C4—22no—
113Dic3⋊C4—22no—
114Dic3⋊C4—22no—
115Dic3⋊C4—22no—
116Dic3⋊C4—22no—
117Dic3⋊C4—22no—
118Dic3⋊C4—41no—
119Dic3⋊C4—41no—
120C4×Dic3—22no—
121C22×Q8—31no—
122C22×D4C2×D311yescharacteristic
123C23×C4—31no—
124C23×C4C2×D311yescharacteristic
125#123—24no—
126#123—64no—
127C22⋊Q8—64no—
128C22⋊Q8—64no—
129C2×C4⋊C4—31no—
130C2×C4⋊C4—31no—
131C2×C4⋊C4—31no—
132C2×C4⋊C4—31no—
133C2×C4⋊C4—31no—
134C2×C4⋊C4—31no—
135C2×C4⋊C4—31no—
136C2×C4⋊C4—31no—
137C2×C4⋊C4—31no—
138C2×C4⋊C4—31no—
139C2×C4⋊C4C2×D311yescharacteristic
140C2×C4⋊C4C2×D311yescharacteristic
141C2×C22⋊C4—31no—
142C2×C22⋊C4—31no—
143C2×C22⋊C4—31no—
144C2×C22⋊C4—31no—
145C2×C22⋊C4—31no—
146C2×C22⋊C4—31no—
147C2×C22⋊C4—31no—
148C2×C22⋊C4C2×D311yescharacteristic
149C2×C22⋊C4C2×D311yescharacteristic
150C2×C22⋊C4C2×D311yescharacteristic
151C2×C42—31no—
152#95—31no—
153#95—31no—
154#95—31no—
155#95—31no—
156#95—31no—
157#95—31no—
158#95—31no—
159#95—31no—
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×C3—21no—
172C23×C3C2×D412yes—
173C23×C3C2×D412yes—
174C23×C3C2411yescharacteristic
175C3×D4—44no—
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—44no—
190C2×C3×C4C2×D412yes—
191C2×C3×C4C2×D412yes—
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—
210C2×Dic3—21no—
211C2×Dic3—21no—
212C2×Dic3—21no—
213C2×Dic3—44no—
214C2×Dic3C2×D412yes—
215C2×Dic3C2×D412yes—
216C2×Dic3#4112yes—
217Dic6—44no—
218C24C22×D311yescharacteristic
219C24C22×D311yescharacteristic
220C2×Q8—62no—
221C2×Q8—62no—
222C2×Q8—121no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—41no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—31no—
229C22×C4—31no—
230C22×C4—31no—
231C22×C4—31no—
232C22×C4—31no—
233C22×C4—31no—
234C22×C4—31no—
235C22×C4—31no—
236C22×C4—41no—
237C22×C4—62no—
238C22×C4—62no—
239C22×C4—121no—
240C22×C4C22×D311yescharacteristic
241C22×C4C22×D311yescharacteristic
242C22×C4C22×D311yescharacteristic
243C22×C4C22×D311yescharacteristic
244C22×C4C22×D311yescharacteristic
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—62no—
248C4⋊C4—62no—
249C4⋊C4—62no—
250C4⋊C4—62no—
251C4⋊C4—62no—
252C4⋊C4—62no—
253C4⋊C4—62no—
254C4⋊C4—62no—
255C4⋊C4—121no—
256C4⋊C4—121no—
257C22⋊C4—22no—
258C22⋊C4—22no—
259C22⋊C4—34no—
260C22⋊C4—62no—
261C22⋊C4—62no—
262C22⋊C4—62no—
263C22⋊C4—62no—
264C22⋊C4—62no—
265C22⋊C4—62no—
266C22⋊C4#6714yes—
267C42—62no—
268C22×C3—22no—
269C22×C3—22no—
270C22×C3—22no—
271C22×C3—22no—
272C22×C3—22no—
273C22×C3—22no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3C22×D411yescharacteristic
283C22×C3C22×D411yescharacteristic
284C22×C3C2×#4111yescharacteristic
285C22×C3H(5,𝔽2)11yescharacteristic
286C22×C32−1+411yescharacteristic
287C22×C32−1+411yescharacteristic
288C22×C32−1+411yescharacteristic
289C3×C4—22no—
290C3×C4—22no—
291C3×C4—41no—
292C3×C4—41no—
293C3×C4—41no—
294Dic3—22no—
295Dic3—22no—
296Dic3—22no—
297Dic3—41no—
298Dic3—41no—
299Dic3—41no—
300Dic3—41no—
301Dic3—41no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23D3×D412yes—
315C23C2×#6712yes—
316C23C23×D311yescharacteristic
317Q8—124no—
318D4—44no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—32no—
333C2×C4—32no—
334C2×C4—32no—
335C2×C4—44no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—61no—
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—124no—
358C2×C4C2×#6712yes—
359C2×C4C2×#6712yes—
360C2×C3—42no—
361C2×C3—42no—
362C2×C3#53611yescharacteristic
363C2×C3#53611yescharacteristic
364C2×C3#53911yescharacteristic
365C2×C3D4211yescharacteristic
366C2×C3#54711yescharacteristic
367C2×C3#54711yescharacteristic
368C2×C3#56311yescharacteristic
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22C2×D3×D411yescharacteristic
384C22#100411yescharacteristic
385C22D3×#4111yescharacteristic
386C22#101011yescharacteristic
387C22#101011yescharacteristic
388C22#101011yescharacteristic
389C22C22×#6711yescharacteristic
390C4—22no—
391C4—22no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—62no—
396C4—62no—
397C4—62no—
398C4—121no—
399C4—121no—
400C4—121no—
401C4—121no—
402C4—121no—
403C3#268011yescharacteristic, complemented, Sylow
404C2—42no—
405C2—42no—
406C2#564811yescharacteristic
407C2#565111yescharacteristic
408C2#565511yescharacteristic
409C2#566011yescharacteristic
410C2D4×#6711yescharacteristic
411C2#582411yescharacteristic
412C2#582511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#123C2
prime radicalC22×C3×D4C22
hypercenterC2×#123D3
hypocenterC3#2680
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
3. C2×#924C2semidirect
4. C2×#878C2semidirect
6. #5210C2semidirect
7. #4983C2semidirect
8. #4982C2semidirect
9. #4979C2semidirect
10. #4975C2semidirect
11. #4953C2semidirect
12. #4952C2semidirect
13. #4951C2semidirect
14. #4946C2semidirect
15. #4945C2semidirect
403. C3#2680semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1232
C2×#9352
C2×#9242
C2×#8782
#52222
#52102
#49832
#49822
#49792
#49752
#49532
#49522
#49512
#49462
#49452
#26803