Finite Group Databasebeta

← #85823

Subgroups — #85823

Order 384.

1012 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#41C211yesmaximal, characteristic, complemented
2#5833C211yesmaximal, characteristic
3#5825C211yesmaximal, characteristic, complemented
4#5583C212yesmaximal, complemented
5#5578C212yesmaximal, complemented
6#5564C214yesmaximal, complemented
7#5561C212yesmaximal, complemented
8#5557C212yesmaximal, complemented
9#5538C214yesmaximal, complemented
10#5534C212yesmaximal, complemented
11#5526C211yesmaximal, characteristic, complemented
12#5519C211yesmaximal, characteristic, complemented
13#5518C212yesmaximal, complemented
14#5479C212yesmaximal, complemented
15C4×#235C211yesmaximal, characteristic, complemented
16#5470C212yesmaximal, complemented
17C2×C4×Dic6C211yesmaximal, characteristic, complemented
18#3461—31nomaximal, Sylow
19C2×C3×#41C2211yescharacteristic
20C2×#235C2211yescharacteristic
21C22×Dic6C2211yescharacteristic
22C3×C4×Q8C2211yescharacteristic
23C3×C4×Q8C2211yescharacteristic
24C3×C4×D4C2212yes—
25C3×C4×D4C2211yescharacteristic
26C3×C4×D4C2211yescharacteristic
27C3×C4×D4C2212yes—
28C3×#117C2212yes—
29C3×#117C2211yescharacteristic
30C2×C3×C42C2211yescharacteristic
31C2×C3×C42C2212yes—
32#946C2212yes—
33#944C2212yes—
34#939C2214yes—
35#938C2212yes—
36#936C2212yes—
37#935C2214yes—
38#930C2212yes—
39#929C2214yes—
40C4×#67C2212yes—
41C4×#67C2212yes—
42C4×#67C2214yes—
43#926C2214yes—
44#926C2211yescharacteristic
45C2×C4⋊Dic3C2211yescharacteristic
46#924C2212yes—
47#924C2212yes—
48#924C2214yes—
49#924C2212yes—
50C2×Dic3⋊C4C2212yes—
51C2×C4×Dic3C2212yes—
52#898C2214yes—
53#897C2212yes—
54#897C2212yes—
55#896C2214yes—
56#890C2212yes—
57#889C2214yes—
58#889C2214yes—
59#888C2212yes—
60#886C2214yes—
61#885C2214yes—
62#883C2214yes—
63#879C2214yes—
64#879C2214yes—
65#878C2214yes—
66#876C2214yes—
67#875C2212yes—
68C4×D12C2211yescharacteristic
69C4×D12C2211yescharacteristic
70#872C2212yes—
71#872C2214yes—
72C42×D3C2212yes—
73#870C2214yes—
74#870C2212yes—
75#869C2212yes—
76#869C2212yes—
77C4×Dic6C2212yes—
78C4×Dic6C2212yes—
79C4×Dic6C2212yes—
80C4×Dic6C2212yes—
81C4×Dic6C2211yescharacteristic
82C4×Dic6C2214yes—
83C4×Dic6C2211yescharacteristic
84#544—31no—
85#542—32no—
86#542—32no—
87#541—32no—
88#541—32no—
89#539—34no—
90#537—31no—
91#536—31no—
92#536—31no—
93#533—32no—
94#533—32no—
95#529—34no—
96#529—32no—
97#529—32no—
98C4×#41—31no—
99C4×#41D311yescharacteristic, complemented
100C2×C4×Q8—31no—
101C3×#41—24no—
102C2×C3×Q8C2311yescharacteristic
103C2×C3×D4C2312yes—
104C2×C3×D4C2311yescharacteristic
105C22×C3×C4C2311yescharacteristic
106C22×C3×C4C2312yes—
107C22×C3×C4C2312yes—
108C22×C3×C4C2311yescharacteristic
109C22×C3×C4C2312yes—
110C22×C3×C4C2311yescharacteristic
111C2×#67C2312yes—
112C2×#67C2314yes—
113C22×Dic3C2312yes—
114C22×Dic3C2312yes—
115#235—24no—
116C2×D12C2311yescharacteristic
117C2×C4×D3C2312yes—
118C2×C4×D3C2312yes—
119C2×C4×D3C2312yes—
120C2×Dic6—24no—
121C2×Dic6C2311yescharacteristic
122C2×Dic6C2312yes—
123C2×Dic6C2312yes—
124C2×Dic6C2314yes—
125C3×C4⋊C4C2311yescharacteristic
126C3×C4⋊C4C2312yes—
127C3×C4⋊C4C2312yes—
128C3×C4⋊C4C2311yescharacteristic
129C3×C22⋊C4C2312yes—
130C3×C22⋊C4C2312yes—
131C3×C22⋊C4C2312yes—
132C3×C42C2311yescharacteristic
133C3×C42C2311yescharacteristic
134C3×C42C2311yescharacteristic
135C3×C42C2311yescharacteristic
136C3×C42C2312yes—
137C3×C42C2312yes—
138C3×C42C2312yes—
139#217C2312yes—
140#217C2314yes—
141#217C2314yes—
142#217C2314yes—
143(C2×D3)⋊C4C2312yes—
144(C2×D3)⋊C4C2312yes—
145(C2×D3)⋊C4C2314yes—
146(C2×D3)⋊C4C2314yes—
147(C2×D3)⋊C4C2314yes—
148C4⋊Dic3C2312yes—
149C4⋊Dic3C2312yes—
150C4⋊Dic3C2311yescharacteristic
151C4⋊Dic3C2314yes—
152C4⋊Dic3C2311yescharacteristic
153C4⋊Dic3C2314yes—
154C4⋊Dic3C2314yes—
155Dic3⋊C4C2314yes—
156Dic3⋊C4C2314yes—
157Dic3⋊C4C2312yes—
158Dic3⋊C4C2314yes—
159Dic3⋊C4C2314yes—
160Dic3⋊C4C2312yes—
161Dic3⋊C4C2314yes—
162Dic3⋊C4C2314yes—
163Dic3⋊C4C2314yes—
164C4×Dic3C2312yes—
165C4×Dic3C2314yes—
166C4×Dic3C2314yes—
167C4×Dic3C2312yes—
168C4×Dic3C2312yes—
169C2×#41—31no—
170C2×#41C2×D311yescharacteristic
171C22×Q8—31no—
172C4⋊Q8—32no—
173C4⋊Q8—32no—
174C4⋊Q8—32no—
175C4⋊Q8—32no—
176#126—34no—
177#126—34no—
178#126—34no—
179#126—34no—
180#125—34no—
181#125—32no—
182#125—34no—
183#125—32no—
184#125—34no—
185#124—32no—
186#124—34no—
187#124—32no—
188#123—34no—
189#123—34no—
190#123—34no—
191#123—34no—
192C22⋊Q8—32no—
193C22⋊Q8—32no—
194C22⋊Q8—32no—
195C22⋊Q8—32no—
196C22⋊Q8—34no—
197C22⋊Q8—32no—
198C22⋊Q8—32no—
199C22⋊Q8—34no—
200C22⋊Q8—34no—
201#121—34no—
202#121—32no—
203#121—32no—
204C4×Q8—32no—
205C4×Q8—32no—
206C4×Q8—32no—
207C4×Q8—32no—
208C4×Q8—31no—
209C4×Q8—31no—
210C4×Q8—34no—
211C4×Q8C2×D311yescharacteristic
212C4×Q8C2×D311yescharacteristic
213C4×D4—32no—
214C4×D4—32no—
215C4×D4—34no—
216C4×D4—31no—
217C4×D4—31no—
218C4×D4C2×D311yescharacteristic
219C4×D4C2×D312yes—
220C4×D4C2×D312yes—
221C4×D4C2×D311yescharacteristic
222#117—31no—
223#117—34no—
224#117—32no—
225#117—34no—
226#117C2×D311yescharacteristic
227#117C2×D312yes—
228C2×C4⋊C4—31no—
229C2×C4⋊C4—32no—
230C2×C42—32no—
231C2×C42—32no—
232C2×C42C2×D311yescharacteristic
233C2×C42C2×D312yes—
234C23×C3C2412yes—
235C23×C3C2411yescharacteristic
236C22×D3C2412yes—
237C3×Q8—22no—
238C3×D4—24no—
239C3×D4—22no—
240C2×C3×C4—24no—
241C2×C3×C4—22no—
242C2×C3×C4—24no—
243C2×C3×C4—42no—
244C2×C3×C4—41no—
245C2×C3×C4#4114yes—
246C2×C3×C4C2411yescharacteristic
247C2×C3×C4C2411yescharacteristic
248C2×C3×C4C2411yescharacteristic
249C2×C3×C4C2412yes—
250C2×C3×C4C2412yes—
251C2×C3×C4C2412yes—
252C2×C3×C4C2411yescharacteristic
253C2×C3×C4C2411yescharacteristic
254C2×C3×C4C2411yescharacteristic
255#67—24no—
256#67—44no—
257C2×Dic3—24no—
258C2×Dic3—42no—
259C2×Dic3C2412yes—
260C2×Dic3C2414yes—
261C2×Dic3C2412yes—
262C2×Dic3C2414yes—
263C2×Dic3C2412yes—
264D12—22no—
265C4×D3—24no—
266C4×D3—24no—
267C4×D3—42no—
268Dic6—24no—
269Dic6—24no—
270Dic6—22no—
271Dic6—44no—
272#41—24no—
273#41—64no—
274C2×Q8—32no—
275C2×Q8—32no—
276C2×Q8—31no—
277C2×Q8—34no—
278C2×Q8—64no—
279C2×Q8C22×D311yescharacteristic
280C2×D4—32no—
281C2×D4—34no—
282C2×D4—31no—
283C2×D4C22×D312yes—
284C2×D4C22×D311yescharacteristic
285C22×C4—32no—
286C22×C4—32no—
287C22×C4—32no—
288C22×C4—32no—
289C22×C4—32no—
290C22×C4C22×D312yes—
291C22×C4C22×D311yescharacteristic
292C22×C4C22×D312yes—
293C22×C4C22×D311yescharacteristic
294C22×C4C22×D312yes—
295C22×C4C22×D311yescharacteristic
296C4⋊C4—32no—
297C4⋊C4—32no—
298C4⋊C4—34no—
299C4⋊C4—34no—
300C4⋊C4—32no—
301C4⋊C4—31no—
302C4⋊C4—31no—
303C4⋊C4—32no—
304C4⋊C4—34no—
305C4⋊C4—34no—
306C4⋊C4—34no—
307C4⋊C4—34no—
308C4⋊C4—34no—
309C4⋊C4—34no—
310C4⋊C4—34no—
311C4⋊C4—34no—
312C4⋊C4C22×D312yes—
313C4⋊C4C22×D311yescharacteristic
314C4⋊C4C22×D311yescharacteristic
315C4⋊C4C22×D312yes—
316C22⋊C4—32no—
317C22⋊C4—34no—
318C22⋊C4—34no—
319C22⋊C4—34no—
320C22⋊C4—32no—
321C22⋊C4—32no—
322C22⋊C4—34no—
323C22⋊C4—34no—
324C22⋊C4—34no—
325C22⋊C4C22×D312yes—
326C22⋊C4C22×D312yes—
327C22⋊C4C22×D312yes—
328C42—32no—
329C42—32no—
330C42—34no—
331C42—34no—
332C42—32no—
333C42C22×D311yescharacteristic
334C42C22×D311yescharacteristic
335C42C22×D312yes—
336C42C22×D312yes—
337C42C22×D311yescharacteristic
338C42C22×D311yescharacteristic
339C42C22×D312yes—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—22no—
343C22×C3—22no—
344C22×C3—22no—
345C22×C3C2×#4112yes—
346C22×C3C2511yescharacteristic
347C2×D3—22no—
348C2×D3—22no—
349C2×D3—22no—
350C3×C4—22no—
351C3×C4—21no—
352C3×C4—22no—
353C3×C4—22no—
354C3×C4—21no—
355C3×C4—21no—
356C3×C4C2×#4112yes—
357C3×C4C2×#4112yes—
358C3×C42−1+412yes—
359Dic3—24no—
360Dic3—22no—
361Dic3—22no—
362Dic3—22no—
363Dic3—24no—
364C23—32no—
365C23C23×D312yes—
366C23C23×D311yescharacteristic
367Q8—22no—
368Q8—64no—
369Q8—64no—
370Q8—62no—
371Q8—124no—
372D4—24no—
373D4—22no—
374D4—64no—
375D4—62no—
376D4—124no—
377C2×C4—24no—
378C2×C4—22no—
379C2×C4—24no—
380C2×C4—32no—
381C2×C4—34no—
382C2×C4—32no—
383C2×C4—32no—
384C2×C4—34no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—64no—
388C2×C4—64no—
389C2×C4—64no—
390C2×C4—122no—
391C2×C4—122no—
392C2×C4#23514yes—
393C2×C4C23×D311yescharacteristic
394C2×C4C23×D311yescharacteristic
395C2×C4C23×D311yescharacteristic
396C2×C4C23×D312yes—
397C2×C4C23×D312yes—
398C2×C4C23×D312yes—
399C2×C4C23×D311yescharacteristic
400C2×C4C23×D311yescharacteristic
401C2×C4C23×D311yescharacteristic
402C2×C3—22no—
403C2×C3—42no—
404C2×C3C22×#4111yescharacteristic
405C2×C3C2×2−1+411yescharacteristic
406C2×C3#58511yescharacteristic
407D3—42no—
408C22—21no—
409C22—21no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—62no—
414C22—62no—
415C22—62no—
416C22C2×#23512yes—
417C22C24×D311yescharacteristic
418C4—22no—
419C4—21no—
420C4—22no—
421C4—21no—
422C4—21no—
423C4—22no—
424C4—62no—
425C4—64no—
426C4—62no—
427C4—64no—
428C4—62no—
429C4C2×#23512yes—
430C4C2×#23512yes—
431C4#101012yes—
432C3#346111yescharacteristic, complemented, Sylow
433C2—22no—
434C2—42no—
435C2—122no—
436C2C22×#23511yescharacteristic
437C2C2×#101011yescharacteristic
438C2#596411yescharacteristic
subgroupG/H
centerC2×C4C23×D3
commutator, socle, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC3×C4×#41C2
prime radical#5833C2
hypercenterC4×#41D3
hypocenterC3#3461
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
1. C3×C4×#41C2semidirect
3. #5825C2semidirect
4. #5583C2semidirect
5. #5578C2semidirect
6. #5564C2semidirect
7. #5561C2semidirect
8. #5557C2semidirect
9. #5538C2semidirect
10. #5534C2semidirect
11. #5526C2semidirect
12. #5519C2semidirect
13. #5518C2semidirect
14. #5479C2semidirect
15. C4×#235C2semidirect
16. #5470C2semidirect
17. C2×C4×Dic6C2semidirect
99. C4×#41D3semidirect
432. C3#3461semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#412
#58332
#58252
#55832
#55782
#55642
#55612
#55572
#55382
#55342
#55262
#55192
#55182
#54792
C4×#2352
#54702
C2×C4×Dic62
#34613