Finite Group Databasebeta

← #74709

Subgroups — #74709

Order 384.

648 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
1C2×D4×Dic3C211yesmaximal, characteristic
2C2×#924C211yesmaximal, characteristic, complemented
3C22×Dic3⋊C4C211yesmaximal, characteristic, complemented
4C2×#878C211yesmaximal, characteristic, complemented
5C3×#386C211yesmaximal, characteristic
6#5220C211yesmaximal, characteristic
7#4950C211yesmaximal, characteristic, complemented
8#4949C211yesmaximal, characteristic, complemented
9#4948C211yesmaximal, characteristic, complemented
10#4946C211yesmaximal, characteristic, complemented
11#4946C211yesmaximal, characteristic, complemented
12#4657C211yesmaximal, characteristic, complemented
13#4656C211yesmaximal, characteristic, complemented
14#4655C211yesmaximal, characteristic, complemented
15#4647C211yesmaximal, characteristic, complemented
16#2449—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C23×C3×C4C2211yescharacteristic
19C23×Dic3C2211yescharacteristic
20C23×Dic3C2211yescharacteristic
21C23×Dic3C2211yescharacteristic
22C22×Dic6C2211yescharacteristic
23C2×C3×C22⋊C4C2211yescharacteristic
24C2×C3×C22⋊C4C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×#217C2211yescharacteristic
27C2×#217C2211yescharacteristic
28C2×#217C2211yescharacteristic
29C2×#217C2211yescharacteristic
30C2×#217C2211yescharacteristic
31D4×Dic3—24no—
32C2×C4⋊Dic3C2211yescharacteristic
33C2×C4⋊Dic3C2211yescharacteristic
34C2×C4⋊Dic3C2211yescharacteristic
35C2×C4⋊Dic3C2211yescharacteristic
36#924—24no—
37C2×Dic3⋊C4—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×C22⋊Q8—31no—
57C2×C22⋊Q8—31no—
58C2×C4×D4—31no—
59C22×C4⋊C4—31no—
60#399—31no—
61#398—31no—
62#396—31no—
63#395—31no—
64#389—31no—
65#387—31no—
66#386—31no—
67#386D311yescharacteristic
68#385—31no—
69#385—31no—
70#385—31no—
71C24×C3C2311yescharacteristic
72C24×C3C2311yescharacteristic
73C2×C3×D4—22no—
74C2×C3×D4—22no—
75C2×C3×D4Q814yescomplemented
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—22no—
87C22×Dic3—22no—
88C22×Dic3—22no—
89C22×Dic3—22no—
90C22×Dic3—22no—
91C22×Dic3—41no—
92C22×Dic3—41no—
93C22×Dic3C2311yescharacteristic
94C22×Dic3C2311yescharacteristic
95C22×Dic3C2311yescharacteristic
96C22×Dic3C2311yescharacteristic
97C22×Dic3C2311yescharacteristic
98C22×Dic3C2311yescharacteristic
99C22×Dic3C2311yescharacteristic
100C22×Dic3C2311yescharacteristic
101C2×Dic6—22no—
102C2×Dic6—22no—
103C2×Dic6—41no—
104C3×C22⋊C4—22no—
105C3×C22⋊C4—22no—
106C3×C22⋊C4—41no—
107#217—22no—
108#217—22no—
109#217—22no—
110#217—22no—
111#217D414yes—
112C4⋊Dic3—22no—
113C4⋊Dic3—22no—
114C4⋊Dic3—22no—
115C4⋊Dic3—41no—
116Dic3⋊C4—22no—
117Dic3⋊C4—22no—
118Dic3⋊C4—22no—
119Dic3⋊C4—22no—
120Dic3⋊C4—22no—
121Dic3⋊C4—22no—
122Dic3⋊C4—22no—
123Dic3⋊C4D414yescomplemented
124C4×Dic3—22no—
125C22×Q8—31no—
126C22×D4C2×D311yescharacteristic
127C23×C4—31no—
128C23×C4—31no—
129C23×C4—31no—
130C23×C4C2×D311yescharacteristic
131C22⋊Q8—64no—
132C22⋊Q8—64no—
133C4×D4—64no—
134C2×C4⋊C4—31no—
135C2×C4⋊C4—31no—
136C2×C4⋊C4—31no—
137C2×C4⋊C4—31no—
138C2×C4⋊C4—31no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4—31no—
141C2×C4⋊C4—31no—
142C2×C4⋊C4—31no—
143C2×C4⋊C4—31no—
144C2×C4⋊C4—31no—
145C2×C4⋊C4—31no—
146C2×C4⋊C4—64no—
147C2×C22⋊C4—31no—
148C2×C22⋊C4—31no—
149C2×C22⋊C4—31no—
150C2×C22⋊C4—31no—
151C2×C22⋊C4—31no—
152C2×C22⋊C4C2×D311yescharacteristic
153C2×C22⋊C4C2×D311yescharacteristic
154C2×C22⋊C4C2×D311yescharacteristic
155C2×C42—31no—
156#95—31no—
157#95—31no—
158#95—31no—
159#95—31no—
160#95—31no—
161#95—31no—
162#95C2×D311yescharacteristic
163#95C2×D311yescharacteristic
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×C3—21no—
173C23×C3—21no—
174C23×C3C2×D412yes—
175C23×C3C2×Q812yes—
176C23×C3C2×Q812yes—
177C23×C3#4112yes—
178C23×C3C2411yescharacteristic
179C3×D4—28no—
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—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—44no—
194C2×C3×C4C2×D412yes—
195C2×C3×C4C2×Q812yes—
196C2×Dic3—28no—
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—21no—
214C2×Dic3—21no—
215C2×Dic3—21no—
216C2×Dic3—44no—
217C2×Dic3—44no—
218C2×Dic3C2×D412yes—
219C2×Dic3C2×D412yes—
220C2×Dic3C2×D412yes—
221C2×Dic3C2×D412yes—
222C2×Dic3#4112yes—
223Dic6—44no—
224C24C22×D311yescharacteristic
225C24C22×D311yescharacteristic
226C2×Q8—62no—
227C2×Q8—62no—
228C2×Q8—121no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4Dic614yescomplemented
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—31no—
235C22×C4—31no—
236C22×C4—31no—
237C22×C4—31no—
238C22×C4—31no—
239C22×C4—31no—
240C22×C4—31no—
241C22×C4—31no—
242C22×C4—41no—
243C22×C4—62no—
244C22×C4—62no—
245C22×C4—62no—
246C22×C4—62no—
247C22×C4—62no—
248C22×C4—62no—
249C22×C4—62no—
250C22×C4—121no—
251C22×C4—121no—
252C22×C4C22×D311yescharacteristic
253C22×C4C22×D311yescharacteristic
254C22×C4C22×D311yescharacteristic
255C22×C4C22×D311yescharacteristic
256C22×C4C22×D311yescharacteristic
257C4⋊C4—34no—
258C4⋊C4—62no—
259C4⋊C4—62no—
260C4⋊C4—62no—
261C4⋊C4—62no—
262C4⋊C4—62no—
263C4⋊C4—62no—
264C4⋊C4—62no—
265C4⋊C4—62no—
266C4⋊C4—62no—
267C4⋊C4—62no—
268C4⋊C4—121no—
269C22⋊C4—22no—
270C22⋊C4—22no—
271C22⋊C4—34no—
272C22⋊C4—41no—
273C22⋊C4—62no—
274C22⋊C4—62no—
275C22⋊C4—62no—
276C22⋊C4—62no—
277C42—62no—
278C22×C3—22no—
279C22×C3—22no—
280C22×C3—22no—
281C22×C3—22no—
282C22×C3—22no—
283C22×C3—22no—
284C22×C3—22no—
285C22×C3—22no—
286C22×C3—22no—
287C22×C3—41no—
288C22×C3—41no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3C22⋊Q814yescomplemented
292C22×C3C22×D411yescharacteristic
293C22×C3C22×D411yescharacteristic
294C22×C3C22×Q811yescharacteristic
295C22×C3C2×#4111yescharacteristic
296C22×C3C2×#4111yescharacteristic
297C22×C3H(5,𝔽2)11yescharacteristic
298C22×C32−1+411yescharacteristic
299C3×C4—22no—
300C3×C4—22no—
301C3×C4—41no—
302C3×C4—41no—
303C3×C4—41no—
304Dic3—22no—
305Dic3—22no—
306Dic3—22no—
307Dic3—22no—
308Dic3—22no—
309Dic3—41no—
310Dic3—41no—
311Dic3—41no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23C2×Dic612yes—
323C23C2×Dic612yes—
324C23D3×D412yes—
325C23#23712yes—
326C23C23×D311yescharacteristic
327Q8—124no—
328D4—28no—
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—32no—
343C2×C4—32no—
344C2×C4—32no—
345C2×C4—32no—
346C2×C4—32no—
347C2×C4—44no—
348C2×C4—68no—
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—61no—
364C2×C4—61no—
365C2×C4—61no—
366C2×C4—61no—
367C2×C4—61no—
368C2×C4—124no—
369C2×C4—124no—
370C2×C4C2×Dic612yes—
371C2×C4D3×D412yes—
372C2×C3—24no—
373C2×C3—42no—
374C2×C3C2×C22⋊Q811yescharacteristic
375C2×C3#52511yescharacteristic
376C2×C3#53911yescharacteristic
377C2×C3D4211yescharacteristic
378C2×C3#54711yescharacteristic
379C2×C3D4×Q811yescharacteristic
380C2×C3#55411yescharacteristic
381C22—22no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22#87814yescomplemented
395C22C22×Dic611yescharacteristic
396C22C2×D3×D411yescharacteristic
397C22C2×D3×D411yescharacteristic
398C22C2×#23711yescharacteristic
399C22#100411yescharacteristic
400C22D3×#4111yescharacteristic
401C22#101011yescharacteristic
402C4—22no—
403C4—22no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—62no—
408C4—62no—
409C4—62no—
410C4—62no—
411C4—62no—
412C4—121no—
413C4—121no—
414C4—121no—
415C3#244911yescharacteristic, complemented, Sylow
416C2—24no—
417C2—42no—
418C2C2×#87811yescharacteristic
419C2D4×Dic611yescharacteristic
420C2#553911yescharacteristic
421C2#559011yescharacteristic
422C2#560211yescharacteristic
423C2#560611yescharacteristic
424C2#565511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#386C2
prime radicalC22×C3×D4C22
hypercenter#386D3
hypocenterC3#2449
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×#924C2semidirect
3. C22×Dic3⋊C4C2semidirect
4. C2×#878C2semidirect
7. #4950C2semidirect
8. #4949C2semidirect
9. #4948C2semidirect
10. #4946C2semidirect
11. #4946C2semidirect
12. #4657C2semidirect
13. #4656C2semidirect
14. #4655C2semidirect
15. #4647C2semidirect
75. C2×C3×D4Q8semidirect
123. Dic3⋊C4D4semidirect
231. C2×D4Dic6semidirect
291. C22×C3C22⋊Q8semidirect
394. C22#878semidirect
415. C3#2449semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic32
C2×#9242
C22×Dic3⋊C42
C2×#8782
C3×#3862
#52202
#49502
#49492
#49482
#49462
#46572
#46562
#46552
#46472
#24493