Finite Group Databasebeta

← #74422

Subgroups — #74422

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
1C22×C4⋊Dic3C211yesmaximal, characteristic, complemented
2C2×#897C211yesmaximal, characteristic, complemented
3C2×#67⋊C4C211yesmaximal, characteristic
4C2×#878C211yesmaximal, characteristic, complemented
5C3×#385C211yesmaximal, characteristic, complemented
6#5213C211yesmaximal, characteristic
7#4994C211yesmaximal, characteristic, complemented
8#4984C211yesmaximal, characteristic, complemented
9#4981C211yesmaximal, characteristic, complemented
10#4958C211yesmaximal, characteristic
11#4950C211yesmaximal, characteristic, complemented
12#4677C211yesmaximal, characteristic, complemented
13#4668C211yesmaximal, characteristic, complemented
14#4656C211yesmaximal, characteristic, complemented
15#4653C211yesmaximal, characteristic, complemented
16#2449—31nomaximal, Sylow
17C23×C3×C4C2211yescharacteristic
18C22×#67C2211yescharacteristic
19C23×Dic3C2211yescharacteristic
20C23×Dic3C2211yescharacteristic
21C22×C4×D3C2211yescharacteristic
22C22×Dic6C2211yescharacteristic
23C2×C3×C4⋊C4C2211yescharacteristic
24C2×C3×C4⋊C4C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2211yescharacteristic
27C2×#217C2211yescharacteristic
28C2×(C2×D3)⋊C4C2211yescharacteristic
29C2×(C2×D3)⋊C4C2211yescharacteristic
30C2×(C2×D3)⋊C4C2211yescharacteristic
31C2×(C2×D3)⋊C4C2211yescharacteristic
32C2×(C2×D3)⋊C4C2211yescharacteristic
33C2×C4⋊Dic3—24no—
34C2×C4⋊Dic3C2211yescharacteristic
35C2×C4⋊Dic3C2211yescharacteristic
36C2×C4⋊Dic3C2211yescharacteristic
37C2×C4⋊Dic3C2211yescharacteristic
38C2×C4⋊Dic3C2211yescharacteristic
39C2×C4⋊Dic3C2211yescharacteristic
40C2×C4⋊Dic3C2211yescharacteristic
41C2×Dic3⋊C4C2211yescharacteristic
42C2×Dic3⋊C4C2211yescharacteristic
43C2×Dic3⋊C4C2211yescharacteristic
44C2×C4×Dic3C2211yescharacteristic
45#897—24no—
46#67⋊C4—24no—
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#386—31no—
68#385—31no—
69#385—31no—
70#385D311yescharacteristic, complemented
71C24×C3C2311yescharacteristic
72C23×D3C2311yescharacteristic
73C22×C3×C4—22no—
74C22×C3×C4—22no—
75C22×C3×C4—22no—
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C22×C3×C4C2311yescharacteristic
81C22×C3×C4C2311yescharacteristic
82C2×#67—22no—
83C2×#67—22no—
84C2×#67Q814yescomplemented
85C22×Dic3—22no—
86C22×Dic3—22no—
87C22×Dic3—22no—
88C22×Dic3—22no—
89C22×Dic3—41no—
90C22×Dic3—41no—
91C22×Dic3C2311yescharacteristic
92C22×Dic3C2311yescharacteristic
93C22×Dic3C2311yescharacteristic
94C22×Dic3C2311yescharacteristic
95C22×Dic3C2311yescharacteristic
96C22×Dic3C2311yescharacteristic
97C22×Dic3C2311yescharacteristic
98C2×C4×D3—22no—
99C2×C4×D3—22no—
100C2×C4×D3—41no—
101C2×Dic6—22no—
102C2×Dic6—22no—
103C2×Dic6—41no—
104C3×C4⋊C4—22no—
105C3×C4⋊C4—22no—
106C3×C22⋊C4—22no—
107C3×C22⋊C4D414yes—
108#217—22no—
109(C2×D3)⋊C4—22no—
110(C2×D3)⋊C4—22no—
111(C2×D3)⋊C4—22no—
112(C2×D3)⋊C4—22no—
113(C2×D3)⋊C4—41no—
114C4⋊Dic3—22no—
115C4⋊Dic3—22no—
116C4⋊Dic3—22no—
117C4⋊Dic3—22no—
118C4⋊Dic3—22no—
119C4⋊Dic3—41no—
120C4⋊Dic3D414yescomplemented
121Dic3⋊C4—22no—
122Dic3⋊C4—22no—
123Dic3⋊C4—22no—
124C4×Dic3—22no—
125C22×Q8—31no—
126C22×D4—31no—
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—64no—
145C2×C4⋊C4C2×D311yescharacteristic
146C2×C4⋊C4C2×D311yescharacteristic
147C2×C22⋊C4—31no—
148C2×C22⋊C4—31no—
149C2×C22⋊C4—31no—
150C2×C22⋊C4—31no—
151C2×C22⋊C4—31no—
152C2×C22⋊C4—31no—
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×C3C2×D412yes—
169C23×C3C2×Q812yes—
170C23×C3#4112yes—
171C23×C3C2411yescharacteristic
172C22×D3—21no—
173C22×D3—21no—
174C22×D3—21no—
175C22×D3—21no—
176C22×D3—21no—
177C22×D3—21no—
178C22×D3C2×Q812yes—
179C2×C3×C4—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—21no—
194C2×C3×C4C2×D412yes—
195C2×C3×C4C2×D412yes—
196C2×C3×C4C2×D412yes—
197C2×C3×C4#4112yes—
198#67—28no—
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—44no—
218C2×Dic3—44no—
219C2×Dic3C2×D412yes—
220C2×Dic3C2×D412yes—
221C2×Dic3C2×Q812yes—
222C4×D3—44no—
223Dic6—44no—
224C24—31no—
225C24C22×D311yescharacteristic
226C2×Q8—62no—
227C2×Q8—62no—
228C2×Q8—121no—
229C2×D4—34no—
230C2×D4—62no—
231C2×D4—62no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—31no—
236C22×C4—31no—
237C22×C4—31no—
238C22×C4—31no—
239C22×C4—31no—
240C22×C4—31no—
241C22×C4—31no—
242C22×C4—62no—
243C22×C4—62no—
244C22×C4—62no—
245C22×C4—62no—
246C22×C4—62no—
247C22×C4—62no—
248C22×C4—121no—
249C22×C4—121no—
250C22×C4—121no—
251C22×C4C22×D311yescharacteristic
252C22×C4C22×D311yescharacteristic
253C22×C4C22×D311yescharacteristic
254C22×C4C22×D311yescharacteristic
255C22×C4C22×D311yescharacteristic
256C22×C4C22×D311yescharacteristic
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—34no—
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—62no—
271C22⋊C4—62no—
272C22⋊C4—62no—
273C22⋊C4—62no—
274C22⋊C4—62no—
275C22⋊C4—121no—
276C22⋊C4D1214yes—
277C42—62no—
278C22×C3—22no—
279C22×C3—22no—
280C22×C3—22no—
281C22×C3—22no—
282C22×C3—22no—
283C22×C3—22no—
284C22×C3C22⋊Q814yescomplemented
285C22×C3C22×D411yescharacteristic
286C22×C3C22×D411yescharacteristic
287C22×C3C22×Q811yescharacteristic
288C22×C3C2×#4111yescharacteristic
289C22×C3C2×#4111yescharacteristic
290C22×C3H(5,𝔽2)11yescharacteristic
291C22×C32−1+411yescharacteristic
292C2×D3—22no—
293C2×D3—22no—
294C2×D3—22no—
295C2×D3—41no—
296C2×D3—41no—
297C2×D3—41no—
298C2×D3—41no—
299C3×C4—22no—
300C3×C4—22no—
301C3×C4—22no—
302C3×C4—22no—
303C3×C4—41no—
304C3×C4—41no—
305Dic3—22no—
306Dic3—22no—
307Dic3—22no—
308Dic3—41no—
309Dic3—41no—
310Dic3—41no—
311Dic3—41no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—32no—
317C23—61no—
318C23—61no—
319C23—61no—
320C23—61no—
321C23—61no—
322C23—61no—
323C23C2×D1212yes—
324C23#23712yes—
325C23D3×Q812yes—
326C23C23×D311yescharacteristic
327Q8—124no—
328D4—68no—
329C2×C4—28no—
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—21no—
343C2×C4—21no—
344C2×C4—32no—
345C2×C4—32no—
346C2×C4—32no—
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—61no—
358C2×C4—61no—
359C2×C4—61no—
360C2×C4—61no—
361C2×C4—61no—
362C2×C4—61no—
363C2×C4—61no—
364C2×C4—61no—
365C2×C4—124no—
366C2×C4—124no—
367C2×C4—124no—
368C2×C4C2×D1212yes—
369C2×C4C2×D1212yes—
370C2×C4D3×D412yes—
371C2×C4#23712yes—
372C2×C3—24no—
373C2×C3C2×C22⋊Q811yescharacteristic
374C2×C3#52511yescharacteristic
375C2×C3#53911yescharacteristic
376C2×C3D4211yescharacteristic
377C2×C3#54711yescharacteristic
378C2×C3D4×Q811yescharacteristic
379C2×C3#55411yescharacteristic
380D3—42no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—62no—
388C22—62no—
389C22—62no—
390C22—121no—
391C22—121no—
392C22—121no—
393C22—121no—
394C22#89714yescomplemented
395C22C22×D1211yescharacteristic
396C22C2×D3×D411yescharacteristic
397C22C2×#23711yescharacteristic
398C22C2×#23711yescharacteristic
399C22C2×D3×Q811yescharacteristic
400C22#100911yescharacteristic
401C22#101011yescharacteristic
402C4—22no—
403C4—22no—
404C4—22no—
405C4—22no—
406C4—41no—
407C4—41no—
408C4—62no—
409C4—62no—
410C4—62no—
411C4—121no—
412C4—121no—
413C4—121no—
414C4—121no—
415C3#244911yescharacteristic, complemented, Sylow
416C2—24no—
417C2—122no—
418C2C2×#89711yescharacteristic
419C2D4×D1211yescharacteristic
420C2#555511yescharacteristic
421C2#558611yescharacteristic
422C2#563211yescharacteristic
423C2#563511yescharacteristic
424C2#566511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#385C2
prime radicalC22×#67C22
hypercenter#385D3
hypocenterC3#2449
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊Dic3C2semidirect
2. C2×#897C2semidirect
4. C2×#878C2semidirect
5. C3×#385C2semidirect
7. #4994C2semidirect
8. #4984C2semidirect
9. #4981C2semidirect
11. #4950C2semidirect
12. #4677C2semidirect
13. #4668C2semidirect
14. #4656C2semidirect
15. #4653C2semidirect
70. #385D3semidirect
84. C2×#67Q8semidirect
120. C4⋊Dic3D4semidirect
284. C22×C3C22⋊Q8semidirect
394. C22#897semidirect
415. C3#2449semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Dic32
C2×#8972
C2×#67⋊C42
C2×#8782
C3×#3852
#52132
#49942
#49842
#49812
#49582
#49502
#46772
#46682
#46562
#46532
#24493