Finite Group Databasebeta

← #86780

Subgroups — #86780

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×#550C211yesmaximal, characteristic, complemented
2#5833C212yesmaximal, complemented
3#5824C212yesmaximal, complemented
4#5719C212yesmaximal, complemented
5#5718C212yesmaximal, complemented
6#5718C211yesmaximal, characteristic, complemented
7#5716C211yesmaximal, characteristic, complemented
8#5621C214yesmaximal, complemented
9#5611C212yesmaximal, complemented
10#5608C212yesmaximal, complemented
11#5597C214yesmaximal, complemented
12#5579C211yesmaximal, characteristic, complemented
13#5578C211yesmaximal, characteristic, complemented
14#5562C212yesmaximal, complemented
15#5548C211yesmaximal, characteristic, complemented
16#5543C211yesmaximal, characteristic, complemented
17#5539C212yesmaximal, complemented
18#3548—31nomaximal, Sylow
19C2×C3×#41C2212yes—
20C2×#237C2212yes—
21C2×#237C2212yes—
22C2×#237C2214yes—
23C2×#235C2212yes—
24C3×#127C2211yescharacteristic
25C3×#127C2212yes—
26C3×#121C2214yes—
27C3×#121C2212yes—
28C3×C4×Q8C2211yescharacteristic
29C3×C4×D4C2212yes—
30C3×C4×D4C2211yescharacteristic
31#946C2212yes—
32#946C2211yescharacteristic
33#944C2211yescharacteristic
34#940C2211yescharacteristic
35#940C2212yes—
36#939C2214yes—
37#939C2214yes—
38#939C2214yes—
39#939C2214yes—
40#939C2214yes—
41#938C2212yes—
42#938C2212yes—
43#938C2212yes—
44#938C2214yes—
45#938C2211yescharacteristic
46#936C2212yes—
47#935C2214yes—
48#935C2214yes—
49D4×Dic3C2212yes—
50D4×Dic3C2212yes—
51D4×Dic3C2212yes—
52D4×Dic3C2212yes—
53#930C2214yes—
54#929C2214yes—
55C4×#67C2212yes—
56C2×C4⋊Dic3C2212yes—
57#924C2214yes—
58#924C2212yes—
59C2×Dic3⋊C4C2214yes—
60#897C2211yescharacteristic
61#896C2212yes—
62#896C2214yes—
63#895C2211yescharacteristic
64D3×C4⋊C4C2212yes—
65#890C2212yes—
66#889C2212yes—
67#888C2211yescharacteristic
68#888C2211yescharacteristic
69#885C2212yes—
70#884C2212yes—
71#884C2214yes—
72#883C2214yes—
73#878C2214yes—
74#878C2214yes—
75#878C2214yes—
76C4×D12C2211yescharacteristic
77C4×D12C2212yes—
78#872C2212yes—
79C42×D3C2211yescharacteristic
80#870C2212yes—
81#869C2211yescharacteristic
82C4×Dic6C2212yes—
83C4×Dic6C2211yescharacteristic
84#562—34no—
85#554—31no—
86#554—32no—
87#554—31no—
88#550—32no—
89#550—31no—
90#550D311yescharacteristic, complemented
91#547—32no—
92#547—32no—
93#540—32no—
94#540—32no—
95#537—32no—
96#537—34no—
97#537—32no—
98#533—31no—
99#532—31no—
100#532—31no—
101C3×#41—44no—
102C2×C3×Q8C2311yescharacteristic
103C2×C3×D4C2312yes—
104C2×C3×D4C2311yescharacteristic
105C2×C3×D4C2312yes—
106C2×C3×D4C2314yes—
107C2×C3×D4C2314yes—
108C2×C3×D4C2312yes—
109C22×C3×C4C2312yes—
110C22×C3×C4C2314yes—
111C2×#67C2312yes—
112C2×#67C2314yes—
113C2×#67C2314yes—
114C2×#67C2314yes—
115C2×#67C2312yes—
116C2×#67C2312yes—
117C22×Dic3C2314yes—
118C22×Dic3C2314yes—
119C22×Dic3C2314yes—
120#237—44no—
121#237—48no—
122#237—44no—
123#235—44no—
124C2×D12C2311yescharacteristic
125C2×D12C2312yes—
126C2×C4×D3C2311yescharacteristic
127C2×C4×D3C2311yescharacteristic
128C2×C4×D3C2311yescharacteristic
129C2×C4×D3C2312yes—
130C2×C4×D3C2314yes—
131C2×Dic6C2312yes—
132C2×Dic6C2311yescharacteristic
133C2×Dic6C2312yes—
134C2×Dic6C2312yes—
135C3×C4⋊C4C2311yescharacteristic
136C3×C4⋊C4C2311yescharacteristic
137C3×C4⋊C4C2312yes—
138C3×C22⋊C4C2314yes—
139C3×C22⋊C4C2312yes—
140C3×C42C2311yescharacteristic
141C3×C42C2312yes—
142#217C2314yes—
143#217C2314yes—
144#217C2314yes—
145#217C2314yes—
146#217C2312yes—
147(C2×D3)⋊C4C2312yes—
148(C2×D3)⋊C4C2312yes—
149(C2×D3)⋊C4C2312yes—
150(C2×D3)⋊C4C2314yes—
151(C2×D3)⋊C4C2312yes—
152C4⋊Dic3C2312yes—
153C4⋊Dic3C2311yescharacteristic
154C4⋊Dic3C2312yes—
155C4⋊Dic3C2312yes—
156C4⋊Dic3C2311yescharacteristic
157C4⋊Dic3C2314yes—
158Dic3⋊C4C2312yes—
159Dic3⋊C4C2312yes—
160Dic3⋊C4C2312yes—
161Dic3⋊C4C2314yes—
162Dic3⋊C4C2314yes—
163Dic3⋊C4C2312yes—
164Dic3⋊C4C2312yes—
165Dic3⋊C4C2312yes—
166Dic3⋊C4C2314yes—
167C4×Dic3C2311yescharacteristic
168C4×Dic3C2312yes—
169C4×Dic3C2311yescharacteristic
170C4×Dic3C2311yescharacteristic
171C2×#41—32no—
172C2×#41—34no—
173C2×#41—32no—
174C2×#41—32no—
175C2×#41C2×D312yes—
176C4⋊Q8—31no—
177C4⋊Q8—31no—
178C4⋊Q8—31no—
179C4⋊Q8—31no—
180#127—31no—
181#127—32no—
182#127C2×D311yescharacteristic
183#127C2×D312yes—
184#125—32no—
185#125—32no—
186#125—32no—
187#124—32no—
188#124—32no—
189#123—34no—
190#123—34no—
191#123—34no—
192#123—34no—
193C22⋊Q8—31no—
194C22⋊Q8—32no—
195C22⋊Q8—32no—
196C22⋊Q8—34no—
197C22⋊Q8—32no—
198C22⋊Q8—31no—
199C22⋊Q8—34no—
200C22⋊Q8—34no—
201C22⋊Q8—34no—
202C22⋊Q8—34no—
203#121—32no—
204#121—32no—
205#121—31no—
206#121—34no—
207#121—34no—
208#121—32no—
209#121—34no—
210#121—34no—
211#121—34no—
212#121—34no—
213#121—34no—
214#121—31no—
215#121—32no—
216#121—34no—
217#121C2×D312yes—
218#121C2×D314yes—
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⋊C4—32no—
235C2×C42—31no—
236C23×C3C2414yes—
237C23×C3C2412yes—
238C22×D3C2411yescharacteristic
239C22×D3C2412yes—
240C3×Q8—22no—
241C3×D4—24no—
242C3×D4—24no—
243C3×D4—22no—
244C3×D4—44no—
245C3×D4—42no—
246C3×D4—44no—
247C2×C3×C4—24no—
248C2×C3×C4—44no—
249C2×C3×C4C2411yescharacteristic
250C2×C3×C4C2412yes—
251C2×C3×C4C2412yes—
252C2×C3×C4C2412yes—
253C2×C3×C4C2411yescharacteristic
254C2×C3×C4C2411yescharacteristic
255#67—24no—
256#67—42no—
257#67—42no—
258#67—44no—
259#67—44no—
260#67—44no—
261C2×Dic3—28no—
262C2×Dic3—44no—
263C2×Dic3—44no—
264C2×Dic3C2411yescharacteristic
265C2×Dic3C2411yescharacteristic
266C2×Dic3C2412yes—
267C2×Dic3C2412yes—
268C2×Dic3C2412yes—
269C2×Dic3C2414yes—
270C2×Dic3C2411yescharacteristic
271D12—22no—
272D12—42no—
273C4×D3—22no—
274C4×D3—24no—
275C4×D3—22no—
276C4×D3—22no—
277C4×D3—44no—
278Dic6—22no—
279Dic6—24no—
280Dic6—42no—
281Dic6—42no—
282#41—44no—
283#41—124no—
284#41—124no—
285#41—124no—
286#41—128no—
287C2×Q8—32no—
288C2×Q8—31no—
289C2×Q8—32no—
290C2×Q8—32no—
291C2×Q8C22×D311yescharacteristic
292C2×D4—32no—
293C2×D4—34no—
294C2×D4—32no—
295C2×D4—34no—
296C2×D4—34no—
297C2×D4—32no—
298C2×D4—31no—
299C2×D4—32no—
300C2×D4C22×D312yes—
301C2×D4C22×D311yescharacteristic
302C2×D4C22×D312yes—
303C2×D4C22×D314yes—
304C2×D4C22×D314yes—
305C2×D4C22×D312yes—
306C22×C4—34no—
307C22×C4—34no—
308C22×C4—34no—
309C22×C4—31no—
310C22×C4—31no—
311C22×C4—31no—
312C22×C4—32no—
313C22×C4—34no—
314C22×C4C22×D312yes—
315C22×C4C22×D314yes—
316C4⋊C4—32no—
317C4⋊C4—32no—
318C4⋊C4—32no—
319C4⋊C4—31no—
320C4⋊C4—34no—
321C4⋊C4—34no—
322C4⋊C4—34no—
323C4⋊C4—34no—
324C4⋊C4—32no—
325C4⋊C4—32no—
326C4⋊C4—32no—
327C4⋊C4—31no—
328C4⋊C4—32no—
329C4⋊C4—32no—
330C4⋊C4—32no—
331C4⋊C4C22×D311yescharacteristic
332C4⋊C4C22×D311yescharacteristic
333C4⋊C4C22×D312yes—
334C22⋊C4—34no—
335C22⋊C4—34no—
336C22⋊C4—34no—
337C22⋊C4—34no—
338C22⋊C4—32no—
339C22⋊C4—32no—
340C22⋊C4—32no—
341C22⋊C4—34no—
342C22⋊C4—32no—
343C22⋊C4—32no—
344C22⋊C4C22×D314yes—
345C22⋊C4C22×D312yes—
346C42—31no—
347C42—31no—
348C42—31no—
349C42—32no—
350C42C22×D311yescharacteristic
351C42C22×D312yes—
352C22×C3—24no—
353C22×C3—22no—
354C22×C3—24no—
355C22×C3—22no—
356C22×C3—24no—
357C22×C3—22no—
358C22×C3C2511yescharacteristic
359C2×D3—21no—
360C2×D3—22no—
361C2×D3—21no—
362C2×D3—22no—
363C2×D3—21no—
364C2×D3—22no—
365C3×C4—22no—
366C3×C4—21no—
367C3×C4—22no—
368C3×C4—22no—
369C3×C4H(5,𝔽2)12yes—
370C3×C42−1+412yes—
371Dic3—22no—
372Dic3—22no—
373Dic3—21no—
374Dic3—22no—
375Dic3—24no—
376Dic3—21no—
377Dic3H(5,𝔽2)12yes—
378C23—31no—
379C23—32no—
380C23C23×D312yes—
381C23C23×D314yes—
382Q8—22no—
383Q8—64no—
384Q8—62no—
385Q8—122no—
386Q8—122no—
387D4—24no—
388D4—24no—
389D4—22no—
390D4—44no—
391D4—42no—
392D4—44no—
393D4—64no—
394D4—62no—
395D4—122no—
396D4—122no—
397D4—122no—
398D4—124no—
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—34no—
409C2×C4—44no—
410C2×C4—62no—
411C2×C4—68no—
412C2×C4—64no—
413C2×C4—62no—
414C2×C4—62no—
415C2×C4—124no—
416C2×C4—124no—
417C2×C4—124no—
418C2×C4C23×D311yescharacteristic
419C2×C4C23×D312yes—
420C2×C4C23×D312yes—
421C2×C4C23×D312yes—
422C2×C4C23×D311yescharacteristic
423C2×C4C23×D311yescharacteristic
424C2×C3—44no—
425C2×C3—42no—
426C2×C3C2×H(5,𝔽2)11yescharacteristic
427C2×C3C2×H(5,𝔽2)11yescharacteristic
428C2×C3C2×2−1+411yescharacteristic
429D3—41no—
430D3—42no—
431C22—22no—
432C22—24no—
433C22—22no—
434C22—24no—
435C22—24no—
436C22—22no—
437C22—61no—
438C22—62no—
439C22—61no—
440C22—62no—
441C22—61no—
442C22—62no—
443C22C24×D311yescharacteristic
444C4—22no—
445C4—22no—
446C4—21no—
447C4—22no—
448C4—32no—
449C4—62no—
450C4—61no—
451C4—62no—
452C4—61no—
453C4—64no—
454C4—62no—
455C4#100412yes—
456C4#101012yes—
457C3#354811yescharacteristic, complemented, Sylow
458C2—42no—
459C2—44no—
460C2—121no—
461C2—122no—
462C2C2×#100411yescharacteristic
463C2C2×#101011yescharacteristic
464C2D3×H(5,𝔽2)11yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#550C2
hypercenter#550D3
hypocenterC3#3548
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C3×#550C2semidirect
2. #5833C2semidirect
3. #5824C2semidirect
4. #5719C2semidirect
5. #5718C2semidirect
6. #5718C2semidirect
7. #5716C2semidirect
8. #5621C2semidirect
9. #5611C2semidirect
10. #5608C2semidirect
11. #5597C2semidirect
12. #5579C2semidirect
13. #5578C2semidirect
14. #5562C2semidirect
15. #5548C2semidirect
16. #5543C2semidirect
17. #5539C2semidirect
90. #550D3semidirect
457. C3#3548semidirect

Maximal subgroups

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

subgroupindex
C3×#5502
#58332
#58242
#57192
#57182
#57162
#56212
#56112
#56082
#55972
#55792
#55782
#55622
#55482
#55432
#55392
#35483