Finite Group Databasebeta

← #82361

Subgroups — #82361

Order 384.

584 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×C4×D4C211yesmaximal, characteristic
2C2×#878C211yesmaximal, characteristic, complemented
3#5220C211yesmaximal, characteristic
4#5220C211yesmaximal, characteristic
5#5212C211yesmaximal, characteristic, complemented
6#5210C211yesmaximal, characteristic, complemented
7#5210C211yesmaximal, characteristic, complemented
8C2×C3⋊#95C211yesmaximal, characteristic, complemented
9#4986C211yesmaximal, characteristic, complemented
10#4982C211yesmaximal, characteristic, complemented
11#4950C211yesmaximal, characteristic, complemented
12#4948C211yesmaximal, characteristic, complemented
13#4947C211yesmaximal, characteristic, complemented
14#4933C211yesmaximal, characteristic, complemented
15#4929C211yesmaximal, characteristic, complemented
16#2450—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C23×C3×C4C2211yescharacteristic
19C23×C3×C4C2211yescharacteristic
20C23×Dic3C2211yescharacteristic
21C23×Dic3C2211yescharacteristic
22C22×Dic6C2211yescharacteristic
23C3×C4×D4—24no—
24C2×C3×C4⋊C4C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2211yescharacteristic
27C2×C3×C42C2211yescharacteristic
28C2×#217C2211yescharacteristic
29C2×#217C2211yescharacteristic
30C2×#217C2211yescharacteristic
31C2×#217C2211yescharacteristic
32C2×#217C2211yescharacteristic
33C2×#217C2211yescharacteristic
34C2×C4⋊Dic3C2211yescharacteristic
35C2×C4⋊Dic3C2211yescharacteristic
36C2×C4⋊Dic3C2211yescharacteristic
37C2×Dic3⋊C4C2211yescharacteristic
38C2×Dic3⋊C4C2211yescharacteristic
39C2×Dic3⋊C4C2211yescharacteristic
40C2×Dic3⋊C4C2211yescharacteristic
41#878—24no—
42C3⋊#95C2211yescharacteristic
43C3⋊#95C2211yescharacteristic
44C3⋊#95C2211yescharacteristic
45C3⋊#95C2211yescharacteristic
46C3⋊#95C2211yescharacteristic
47C3⋊#95C2211yescharacteristic
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C2×C22⋊Q8—31no—
55C2×C4×D4D311yescharacteristic
56#400—31no—
57#398—31no—
58#396—31no—
59#393—31no—
60#391—31no—
61#387—31no—
62#386—31no—
63#386—31no—
64#385—31no—
65#385—31no—
66#385—31no—
67#380—31no—
68C2×#95—31no—
69C24×C3C2311yescharacteristic
70C24×C3C2311yescharacteristic
71C2×C3×D4—22no—
72C2×C3×D4—22no—
73C2×C3×D4Q814yescomplemented
74C22×C3×C4—22no—
75C22×C3×C4—22no—
76C22×C3×C4—22no—
77C22×C3×C4—22no—
78C22×C3×C4—41no—
79C22×C3×C4—41no—
80C22×C3×C4C2311yescharacteristic
81C22×C3×C4C2311yescharacteristic
82C22×C3×C4C2311yescharacteristic
83C22×C3×C4C2311yescharacteristic
84C22×C3×C4C2311yescharacteristic
85C22×Dic3—22no—
86C22×Dic3—22no—
87C22×Dic3—22no—
88C22×Dic3—41no—
89C22×Dic3—41no—
90C22×Dic3—41no—
91C22×Dic3C2311yescharacteristic
92C22×Dic3C2311yescharacteristic
93C22×Dic3C2311yescharacteristic
94C22×Dic3C2311yescharacteristic
95C22×Dic3C2311yescharacteristic
96C22×Dic3C2311yescharacteristic
97C22×Dic3C2311yescharacteristic
98C22×Dic3C2311yescharacteristic
99C2×Dic6—22no—
100C2×Dic6—22no—
101C2×Dic6—41no—
102C3×C4⋊C4—22no—
103C3×C22⋊C4—22no—
104C3×C22⋊C4D414yes—
105C3×C42—22no—
106#217—22no—
107#217—22no—
108#217—22no—
109#217—41no—
110#217—41no—
111#217—41no—
112C4⋊Dic3—22no—
113C4⋊Dic3—41no—
114C4⋊Dic3—41no—
115Dic3⋊C4—22no—
116Dic3⋊C4—22no—
117Dic3⋊C4—22no—
118Dic3⋊C4—41no—
119C22×Q8—31no—
120C22×D4C2×D311yescharacteristic
121C23×C4—31no—
122C23×C4—31no—
123C23×C4C2×D311yescharacteristic
124C23×C4C2×D311yescharacteristic
125C22⋊Q8—64no—
126C4×D4—24no—
127C2×C4⋊C4—31no—
128C2×C4⋊C4—31no—
129C2×C4⋊C4—31no—
130C2×C4⋊C4—31no—
131C2×C4⋊C4—31no—
132C2×C4⋊C4—31no—
133C2×C4⋊C4—31no—
134C2×C4⋊C4C2×D311yescharacteristic
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⋊C4C2×D311yescharacteristic
142C2×C22⋊C4C2×D311yescharacteristic
143C2×C42C2×D311yescharacteristic
144#95—31no—
145#95—31no—
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#95—31no—
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×C3C2×D412yes—
167C23×C3C2×Q812yes—
168C23×C3C2×Q812yes—
169C23×C3#4112yes—
170C23×C3C2411yescharacteristic
171C3×D4—28no—
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—44no—
184C2×C3×C4—44no—
185C2×C3×C4C2×D412yes—
186C2×C3×C4C2×D412yes—
187C2×C3×C4C2×Q812yes—
188C2×C3×C4#4112yes—
189C2×Dic3—21no—
190C2×Dic3—21no—
191C2×Dic3—21no—
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—44no—
213C2×Dic3—44no—
214C2×Dic3#4112yes—
215Dic6—44no—
216C24C22×D311yescharacteristic
217C24C22×D311yescharacteristic
218C2×Q8—62no—
219C2×Q8—62no—
220C2×Q8—121no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4Dic614yescomplemented
224C22×C4—22no—
225C22×C4—22no—
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—41no—
238C22×C4—62no—
239C22×C4—62no—
240C22×C4—62no—
241C22×C4—121no—
242C22×C4—121no—
243C22×C4—121no—
244C22×C4C22×D311yescharacteristic
245C22×C4C22×D311yescharacteristic
246C22×C4C22×D311yescharacteristic
247C22×C4C22×D311yescharacteristic
248C22×C4C22×D311yescharacteristic
249C4⋊C4—22no—
250C4⋊C4—62no—
251C4⋊C4—62no—
252C4⋊C4—62no—
253C4⋊C4—62no—
254C4⋊C4—121no—
255C4⋊C4—121no—
256C4⋊C4—121no—
257C22⋊C4—22no—
258C22⋊C4—62no—
259C22⋊C4—62no—
260C22⋊C4—62no—
261C22⋊C4—121no—
262C22⋊C4—121no—
263C22⋊C4—121no—
264C22⋊C4#6714yes—
265C42—22no—
266C22×C3—22no—
267C22×C3—22no—
268C22×C3—22no—
269C22×C3—22no—
270C22×C3—22no—
271C22×C3—22no—
272C22×C3—22no—
273C22×C3—22no—
274C22×C3—22no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3C22⋊Q814yes—
280C22×C3C22×D411yescharacteristic
281C22×C3C22×Q811yescharacteristic
282C22×C3C2×#4111yescharacteristic
283C22×C3C2×#4111yescharacteristic
284C22×C3C2×#4111yescharacteristic
285C22×C3H(5,𝔽2)11yescharacteristic
286C22×C3H(5,𝔽2)11yescharacteristic
287C3×C4—22no—
288C3×C4—22no—
289C3×C4—22no—
290C3×C4—22no—
291C3×C4—41no—
292Dic3—22no—
293Dic3—41no—
294Dic3—41no—
295Dic3—41no—
296Dic3—41no—
297Dic3—41no—
298Dic3—41no—
299Dic3—41no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23C2×Dic612yes—
311C23C2×Dic612yes—
312C23#23512yes—
313C23C2×#6712yes—
314C23C23×D311yescharacteristic
315Q8—124no—
316D4—28no—
317C2×C4—21no—
318C2×C4—21no—
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—32no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—61no—
332C2×C4—61no—
333C2×C4—61no—
334C2×C4—61no—
335C2×C4—61no—
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—124no—
355C2×C4—124no—
356C2×C4C2×Dic612yes—
357C2×C4#23712yes—
358C2×C4C2×#6712yes—
359C2×C4C2×#6712yes—
360C2×C3—24no—
361C2×C3—42no—
362C2×C3C2×C22⋊Q811yescharacteristic
363C2×C3#52911yescharacteristic
364C2×C3#53811yescharacteristic
365C2×C3#54611yescharacteristic
366C2×C3#54611yescharacteristic
367C2×C3#55411yescharacteristic
368C2×C3#55411yescharacteristic
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22#92414yes—
383C22C22×Dic611yescharacteristic
384C22C2×#23511yescharacteristic
385C22C2×#23711yescharacteristic
386C22#100411yescharacteristic
387C22D3×#4111yescharacteristic
388C22#100911yescharacteristic
389C22C22×#6711yescharacteristic
390C4—22no—
391C4—22no—
392C4—22no—
393C4—22no—
394C4—41no—
395C4—62no—
396C4—121no—
397C4—121no—
398C4—121no—
399C4—121no—
400C4—121no—
401C4—121no—
402C4—121no—
403C3#245011yescharacteristic, complemented, Sylow
404C2—24no—
405C2—42no—
406C2#553811yescharacteristic
407C2#553911yescharacteristic
408C2#554311yescharacteristic
409C2#556011yescharacteristic
410C2C2×#92411yescharacteristic
411C2#580611yescharacteristic
412C2#582911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C4×D4C2
prime radicalC22×C3×D4C22
hypercenterC2×C4×D4D3
hypocenterC3#2450
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#878C2semidirect
5. #5212C2semidirect
6. #5210C2semidirect
7. #5210C2semidirect
8. C2×C3⋊#95C2semidirect
9. #4986C2semidirect
10. #4982C2semidirect
11. #4950C2semidirect
12. #4948C2semidirect
13. #4947C2semidirect
14. #4933C2semidirect
15. #4929C2semidirect
73. C2×C3×D4Q8semidirect
223. C2×D4Dic6semidirect
403. C3#2450semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D42
C2×#8782
#52202
#52122
#52102
C2×C3⋊#952
#49862
#49822
#49502
#49482
#49472
#49332
#49292
#24503