Finite Group Databasebeta

← #86939

Subgroups — #86939

Order 384.

1114 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#554C211yesmaximal, characteristic, complemented
2#5730C211yesmaximal, characteristic, complemented
3#5725C211yesmaximal, characteristic, complemented
4#5697C212yesmaximal, complemented
5#5690C212yesmaximal, complemented
6#5647C214yesmaximal, complemented
7#5631C212yesmaximal, complemented
8#5629C212yesmaximal, complemented
9#5629C214yesmaximal, complemented
10#5578C211yesmaximal, characteristic, complemented
11#5577C211yesmaximal, characteristic, complemented
12#5562C212yesmaximal, complemented
13#5553C212yesmaximal, complemented
14#5553C211yesmaximal, characteristic, complemented
15#5548C211yesmaximal, characteristic, complemented
16#5510C212yesmaximal, complemented
17#5507C212yesmaximal, complemented
18#3548—31nomaximal, Sylow
19C2×#239C2212yes—
20C2×#239C2212yes—
21C2×#235C2214yes—
22C2×#235C2212yes—
23C2×#235C2212yes—
24C3×C4⋊Q8C2211yescharacteristic
25C3×#125C2212yes—
26C3×C22⋊Q8C2214yes—
27C3×C22⋊Q8C2212yes—
28C3×C4×Q8C2211yescharacteristic
29C3×C4×D4C2211yescharacteristic
30C3×C4×D4C2212yes—
31C2×C3×C4⋊C4C2212yes—
32#947C2212yes—
33#946C2214yes—
34#946C2211yescharacteristic
35#944C2211yescharacteristic
36#940C2211yescharacteristic
37#940C2212yes—
38#938C2211yescharacteristic
39#930C2214yes—
40#930C2212yes—
41#930C2214yes—
42#929C2214yes—
43C4×#67C2212yes—
44#897C2214yes—
45#897C2212yes—
46#897C2211yescharacteristic
47#896C2214yes—
48#896C2214yes—
49#896C2212yes—
50#895C2214yes—
51#895C2211yescharacteristic
52#895C2212yes—
53#895C2214yes—
54#895C2214yes—
55#895C2212yes—
56#895C2212yes—
57#894C2214yes—
58#894C2214yes—
59#893C2212yes—
60#893C2212yes—
61#893C2212yes—
62#893C2212yes—
63D3×C4⋊C4C2214yes—
64D3×C4⋊C4C2212yes—
65#890C2212yes—
66#889C2212yes—
67#888C2211yescharacteristic
68#888C2211yescharacteristic
69#885C2212yes—
70#884C2214yes—
71#884C2214yes—
72#884C2212yes—
73#884C2214yes—
74#884C2214yes—
75#883C2214yes—
76#874C2212yes—
77#874C2211yescharacteristic
78C4×D12C2212yes—
79C4×D12C2211yescharacteristic
80#872C2212yes—
81C42×D3C2211yescharacteristic
82C4×Dic6C2212yes—
83C4×Dic6C2211yescharacteristic
84#562—34no—
85#554—31no—
86#554—32no—
87#554D311yescharacteristic, complemented
88#550—31no—
89#550—32no—
90#550—31no—
91#547—32no—
92#547—32no—
93#540—32no—
94#540—32no—
95#537—34no—
96#537—32no—
97#537—32no—
98#533—31no—
99#532—31no—
100#532—31no—
101C2×C3×Q8C2311yescharacteristic
102C2×C3×Q8C2312yes—
103C2×C3×D4C2311yescharacteristic
104C22×C3×C4C2312yes—
105C22×C3×C4C2314yes—
106C2×#67C2312yes—
107C2×#67C2314yes—
108C2×#67C2314yes—
109C2×#67C2312yes—
110C2×#67C2312yes—
111#239—44no—
112#239—44no—
113#235—48no—
114#235—44no—
115#235—44no—
116C2×D12C2312yes—
117C2×D12C2311yescharacteristic
118C2×D12C2312yes—
119C2×D12C2314yes—
120C2×D12C2312yes—
121C2×D12C2314yes—
122C2×D12C2312yes—
123C2×D12C2314yes—
124C2×C4×D3C2311yescharacteristic
125C2×C4×D3C2311yescharacteristic
126C2×C4×D3C2311yescharacteristic
127C2×C4×D3C2312yes—
128C2×C4×D3C2314yes—
129C2×C4×D3C2314yes—
130C2×C4×D3C2314yes—
131C2×C4×D3C2314yes—
132C2×Dic6C2312yes—
133C2×Dic6C2311yescharacteristic
134C2×Dic6C2312yes—
135C3×C4⋊C4C2311yescharacteristic
136C3×C4⋊C4C2311yescharacteristic
137C3×C4⋊C4C2312yes—
138C3×C4⋊C4C2312yes—
139C3×C4⋊C4C2312yes—
140C3×C4⋊C4C2314yes—
141C3×C4⋊C4C2314yes—
142C3×C22⋊C4C2314yes—
143C3×C22⋊C4C2312yes—
144C3×C42C2311yescharacteristic
145C3×C42C2312yes—
146#217C2312yes—
147(C2×D3)⋊C4C2312yes—
148(C2×D3)⋊C4C2312yes—
149(C2×D3)⋊C4C2312yes—
150(C2×D3)⋊C4C2314yes—
151(C2×D3)⋊C4C2314yes—
152(C2×D3)⋊C4C2314yes—
153(C2×D3)⋊C4C2314yes—
154(C2×D3)⋊C4C2314yes—
155(C2×D3)⋊C4C2312yes—
156C4⋊Dic3C2311yescharacteristic
157C4⋊Dic3C2312yes—
158C4⋊Dic3C2311yescharacteristic
159C4⋊Dic3C2314yes—
160Dic3⋊C4C2312yes—
161Dic3⋊C4C2312yes—
162Dic3⋊C4C2312yes—
163Dic3⋊C4C2312yes—
164Dic3⋊C4C2312yes—
165Dic3⋊C4C2312yes—
166Dic3⋊C4C2314yes—
167C4×Dic3C2311yescharacteristic
168C4×Dic3C2312yes—
169C4×Dic3C2311yescharacteristic
170C4×Dic3C2311yescharacteristic
171C2×#41—32no—
172C2×#41—32no—
173C2×#41—32no—
174C2×#41—34no—
175C2×#41—32no—
176C4⋊Q8—31no—
177C4⋊Q8—31no—
178C4⋊Q8—31no—
179C4⋊Q8C2×D311yescharacteristic
180#127—31no—
181#127—32no—
182#127—31no—
183#127—32no—
184#125—32no—
185#125—32no—
186#125C2×D312yes—
187#124—32no—
188#124—32no—
189#123—34no—
190#123—34no—
191#123—34no—
192#123—34no—
193C22⋊Q8—31no—
194C22⋊Q8—34no—
195C22⋊Q8—32no—
196C22⋊Q8—31no—
197C22⋊Q8—34no—
198C22⋊Q8—32no—
199C22⋊Q8—34no—
200C22⋊Q8—34no—
201C22⋊Q8C2×D312yes—
202C22⋊Q8C2×D314yes—
203#121—31no—
204#121—34no—
205#121—32no—
206#121—34no—
207#121—32no—
208#121—32no—
209#121—31no—
210#121—34no—
211#121—34no—
212#121—32no—
213#121—34no—
214#121—32no—
215#121—34no—
216#121—34no—
217#121—34no—
218#121—34no—
219C4×Q8—32no—
220C4×Q8—31no—
221C4×Q8C2×D311yescharacteristic
222C4×D4—32no—
223C4×D4—32no—
224C4×D4—32no—
225C4×D4—32no—
226C4×D4—32no—
227C4×D4—31no—
228C4×D4—32no—
229C4×D4C2×D311yescharacteristic
230C4×D4C2×D312yes—
231#117—32no—
232C2×C4⋊C4—34no—
233C2×C4⋊C4—32no—
234C2×C4⋊C4C2×D312yes—
235C2×C42—31no—
236C23×C3C2412yes—
237C22×D3C2411yescharacteristic
238C22×D3C2414yes—
239C22×D3C2412yes—
240C3×Q8—22no—
241C3×Q8—42no—
242C3×D4—22no—
243C2×C3×C4—24no—
244C2×C3×C4—44no—
245C2×C3×C4C2411yescharacteristic
246C2×C3×C4C2412yes—
247C2×C3×C4C2412yes—
248C2×C3×C4C2414yes—
249C2×C3×C4C2412yes—
250C2×C3×C4C2411yescharacteristic
251C2×C3×C4C2411yescharacteristic
252#67—24no—
253#67—44no—
254#67—44no—
255#67—42no—
256#67—42no—
257C2×Dic3C2411yescharacteristic
258C2×Dic3C2411yescharacteristic
259C2×Dic3C2412yes—
260C2×Dic3C2412yes—
261C2×Dic3C2412yes—
262C2×Dic3C2411yescharacteristic
263D12—24no—
264D12—24no—
265D12—22no—
266D12—44no—
267D12—42no—
268D12—44no—
269D12—42no—
270D12—44no—
271C4×D3—22no—
272C4×D3—24no—
273C4×D3—22no—
274C4×D3—28no—
275C4×D3—22no—
276C4×D3—44no—
277C4×D3—44no—
278C4×D3—44no—
279Dic6—22no—
280Dic6—24no—
281Dic6—42no—
282#41—124no—
283#41—124no—
284#41—128no—
285#41—124no—
286#41—124no—
287C2×Q8—32no—
288C2×Q8—31no—
289C2×Q8—32no—
290C2×Q8C22×D311yescharacteristic
291C2×Q8C22×D312yes—
292C2×D4—32no—
293C2×D4—32no—
294C2×D4—34no—
295C2×D4—34no—
296C2×D4—32no—
297C2×D4—32no—
298C2×D4—31no—
299C2×D4—32no—
300C2×D4—34no—
301C2×D4—32no—
302C2×D4—34no—
303C2×D4—34no—
304C2×D4—32no—
305C2×D4C22×D311yescharacteristic
306C22×C4—31no—
307C22×C4—31no—
308C22×C4—31no—
309C22×C4—34no—
310C22×C4—32no—
311C22×C4—34no—
312C22×C4—34no—
313C22×C4—34no—
314C22×C4C22×D312yes—
315C22×C4C22×D314yes—
316C4⋊C4—32no—
317C4⋊C4—32no—
318C4⋊C4—31no—
319C4⋊C4—34no—
320C4⋊C4—34no—
321C4⋊C4—32no—
322C4⋊C4—32no—
323C4⋊C4—31no—
324C4⋊C4—32no—
325C4⋊C4—32no—
326C4⋊C4—32no—
327C4⋊C4C22×D312yes—
328C4⋊C4C22×D314yes—
329C4⋊C4C22×D311yescharacteristic
330C4⋊C4C22×D314yes—
331C4⋊C4C22×D312yes—
332C4⋊C4C22×D311yescharacteristic
333C4⋊C4C22×D312yes—
334C22⋊C4—32no—
335C22⋊C4—32no—
336C22⋊C4—32no—
337C22⋊C4—34no—
338C22⋊C4—34no—
339C22⋊C4—34no—
340C22⋊C4—34no—
341C22⋊C4—32no—
342C22⋊C4—34no—
343C22⋊C4—32no—
344C22⋊C4C22×D314yes—
345C22⋊C4C22×D312yes—
346C42—31no—
347C42—31no—
348C42—31no—
349C42—32no—
350C42C22×D311yescharacteristic
351C42C22×D312yes—
352C22×C3—22no—
353C22×C3—22no—
354C22×C3—22no—
355C22×C3C2511yescharacteristic
356C2×D3—21no—
357C2×D3—22no—
358C2×D3—24no—
359C2×D3—21no—
360C2×D3—24no—
361C2×D3—22no—
362C2×D3—21no—
363C2×D3—24no—
364C2×D3—22no—
365C3×C4—22no—
366C3×C4—22no—
367C3×C4—24no—
368C3×C4—21no—
369C3×C4—22no—
370C3×C4H(5,𝔽2)12yes—
371C3×C42−1+412yes—
372Dic3—22no—
373Dic3—22no—
374Dic3—21no—
375Dic3—22no—
376Dic3—21no—
377Dic3H(5,𝔽2)12yes—
378C23—31no—
379C23—32no—
380C23—34no—
381C23C23×D312yes—
382Q8—22no—
383Q8—42no—
384Q8—64no—
385Q8—62no—
386Q8—122no—
387D4—22no—
388D4—64no—
389D4—64no—
390D4—62no—
391D4—64no—
392D4—122no—
393D4—124no—
394D4—124no—
395D4—124no—
396D4—122no—
397D4—122no—
398D4—122no—
399D4—124no—
400D4—124no—
401C2×C4—24no—
402C2×C4—31no—
403C2×C4—31no—
404C2×C4—31no—
405C2×C4—32no—
406C2×C4—32no—
407C2×C4—32no—
408C2×C4—44no—
409C2×C4—68no—
410C2×C4—62no—
411C2×C4—62no—
412C2×C4—64no—
413C2×C4—62no—
414C2×C4—124no—
415C2×C4—124no—
416C2×C4—124no—
417C2×C4C23×D311yescharacteristic
418C2×C4C23×D312yes—
419C2×C4C23×D312yes—
420C2×C4C23×D314yes—
421C2×C4C23×D312yes—
422C2×C4C23×D311yescharacteristic
423C2×C4C23×D311yescharacteristic
424C2×C3—42no—
425C2×C3C2×H(5,𝔽2)11yescharacteristic
426C2×C3C2×H(5,𝔽2)11yescharacteristic
427C2×C3C2×2−1+411yescharacteristic
428D3—41no—
429D3—44no—
430D3—42no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—61no—
435C22—62no—
436C22—64no—
437C22—62no—
438C22—61no—
439C22—64no—
440C22—61no—
441C22—64no—
442C22—62no—
443C22C24×D311yescharacteristic
444C4—22no—
445C4—22no—
446C4—22no—
447C4—24no—
448C4—21no—
449C4—32no—
450C4—61no—
451C4—62no—
452C4—61no—
453C4—62no—
454C4—62no—
455C4#100712yes—
456C4#100912yes—
457C3#354811yescharacteristic, complemented, Sylow
458C2—42no—
459C2—121no—
460C2—122no—
461C2—124no—
462C2C2×#100711yescharacteristic
463C2C2×#100911yescharacteristic
464C2D3×H(5,𝔽2)11yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#554C2
hypercenter#554D3
hypocenterC3#3548
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #554

Complements

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

NH ≅ G/Ntype
1. C3×#554C2semidirect
2. #5730C2semidirect
3. #5725C2semidirect
4. #5697C2semidirect
5. #5690C2semidirect
6. #5647C2semidirect
7. #5631C2semidirect
8. #5629C2semidirect
9. #5629C2semidirect
10. #5578C2semidirect
11. #5577C2semidirect
12. #5562C2semidirect
13. #5553C2semidirect
14. #5553C2semidirect
15. #5548C2semidirect
16. #5510C2semidirect
17. #5507C2semidirect
87. #554D3semidirect
457. C3#3548semidirect

Maximal subgroups

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

subgroupindex
C3×#5542
#57302
#57252
#56972
#56902
#56472
#56312
#56292
#55782
#55772
#55622
#55532
#55482
#55102
#55072
#35483