Finite Group Databasebeta

← #86693

Subgroups — #86693

Order 384.

1114 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#547C211yesmaximal, characteristic, complemented
2#5834C212yesmaximal, complemented
3#5833C212yesmaximal, complemented
4#5728C211yesmaximal, characteristic, complemented
5#5726C211yesmaximal, characteristic, complemented
6#5667C214yesmaximal, complemented
7#5663C214yesmaximal, complemented
8#5635C212yesmaximal, complemented
9#5629C212yesmaximal, complemented
10#5608C212yesmaximal, complemented
11#5599C212yesmaximal, complemented
12#5555C211yesmaximal, characteristic, complemented
13#5551C211yesmaximal, characteristic, complemented
14#5548C211yesmaximal, characteristic, complemented
15#5548C211yesmaximal, characteristic, complemented
16#5511C212yesmaximal, complemented
17#5510C212yesmaximal, complemented
18#3548—31nomaximal, Sylow
19C2×C3×#41C2212yes—
20C2×#239C2212yes—
21C2×#237C2212yes—
22C2×#235C2212yes—
23C2×#235C2212yes—
24C2×#235C2212yes—
25C3×C4⋊Q8C2211yescharacteristic
26C3×#123C2214yes—
27C3×C22⋊Q8C2212yes—
28C3×#121C2212yes—
29C3×C4×D4C2211yescharacteristic
30C3×C4×D4C2211yescharacteristic
31C2×C3×C4⋊C4C2212yes—
32#946C2212yes—
33#946C2212yes—
34Q8×Dic3C2212yes—
35#940C2214yes—
36#940C2211yescharacteristic
37#940C2211yescharacteristic
38#939C2214yes—
39#939C2214yes—
40#938C2214yes—
41#938C2211yescharacteristic
42#938C2211yescharacteristic
43#938C2212yes—
44#930C2214yes—
45#930C2214yes—
46#930C2212yes—
47C4×#67C2212yes—
48C4×#67C2214yes—
49C4×#67C2212yes—
50C2×C4⋊Dic3C2212yes—
51#924C2214yes—
52#924C2212yes—
53#897C2214yes—
54#897C2212yes—
55#897C2212yes—
56#896C2214yes—
57#895C2214yes—
58#895C2212yes—
59#895C2211yescharacteristic
60#895C2211yescharacteristic
61#894C2214yes—
62#892C2212yes—
63D3×C4⋊C4C2212yes—
64D3×C4⋊C4C2212yes—
65#890C2212yes—
66#889C2214yes—
67#888C2211yescharacteristic
68#888C2211yescharacteristic
69#887C2212yes—
70#886C2214yes—
71#885C2212yes—
72#885C2212yes—
73#884C2212yes—
74#884C2212yes—
75#884C2214yes—
76#884C2214yes—
77#883C2214yes—
78#67⋊C4C2214yes—
79#878C2214yes—
80C4×D12C2211yescharacteristic
81C4×D12C2211yescharacteristic
82C42×D3C2211yescharacteristic
83#869C2211yescharacteristic
84#562—34no—
85#554—32no—
86#554—32no—
87#550—32no—
88#550—32no—
89#547—31no—
90#547—31no—
91#547—31no—
92#547D311yescharacteristic, complemented
93#540—34no—
94#537—32no—
95#537—32no—
96#537—32no—
97#537—32no—
98#533—31no—
99#532—31no—
100#532—31no—
101C3×#41—44no—
102C2×C3×Q8C2312yes—
103C2×C3×D4C2311yescharacteristic
104C2×C3×D4C2311yescharacteristic
105C2×C3×D4C2314yes—
106C22×C3×C4C2312yes—
107C22×C3×C4C2312yes—
108C22×C3×C4C2314yes—
109C2×#67C2312yes—
110C2×#67C2312yes—
111C2×#67C2314yes—
112C2×#67C2312yes—
113C2×#67C2312yes—
114C2×#67C2314yes—
115C2×#67C2312yes—
116C2×#67C2312yes—
117C22×Dic3C2314yes—
118#239—44no—
119#237—44no—
120#235—44no—
121#235—44no—
122#235—44no—
123C2×D12C2311yescharacteristic
124C2×D12C2311yescharacteristic
125C2×D12C2312yes—
126C2×D12C2314yes—
127C2×D12C2312yes—
128C2×C4×D3C2311yescharacteristic
129C2×C4×D3C2311yescharacteristic
130C2×C4×D3C2311yescharacteristic
131C2×C4×D3C2312yes—
132C2×C4×D3C2312yes—
133C2×C4×D3C2314yes—
134C2×C4×D3C2314yes—
135C2×Dic6C2312yes—
136C2×Dic6C2312yes—
137C2×Dic6C2312yes—
138C3×C4⋊C4C2312yes—
139C3×C4⋊C4C2311yescharacteristic
140C3×C4⋊C4C2311yescharacteristic
141C3×C4⋊C4C2314yes—
142C3×C4⋊C4C2312yes—
143C3×C22⋊C4C2314yes—
144C3×C22⋊C4C2312yes—
145C3×C22⋊C4C2312yes—
146C3×C42C2311yescharacteristic
147#217C2312yes—
148#217C2314yes—
149#217C2312yes—
150(C2×D3)⋊C4C2312yes—
151(C2×D3)⋊C4C2312yes—
152(C2×D3)⋊C4C2314yes—
153(C2×D3)⋊C4C2312yes—
154(C2×D3)⋊C4C2314yes—
155(C2×D3)⋊C4C2312yes—
156(C2×D3)⋊C4C2314yes—
157C4⋊Dic3C2312yes—
158C4⋊Dic3C2312yes—
159C4⋊Dic3C2311yescharacteristic
160C4⋊Dic3C2311yescharacteristic
161C4⋊Dic3C2314yes—
162C4⋊Dic3C2314yes—
163Dic3⋊C4C2314yes—
164Dic3⋊C4C2314yes—
165Dic3⋊C4C2312yes—
166Dic3⋊C4C2314yes—
167Dic3⋊C4C2312yes—
168C4×Dic3C2311yescharacteristic
169C4×Dic3C2311yescharacteristic
170C4×Dic3C2311yescharacteristic
171C4×Dic3C2314yes—
172C2×#41—32no—
173C2×#41—32no—
174C2×#41—32no—
175C2×#41—32no—
176C2×#41—32no—
177C2×#41C2×D312yes—
178C4⋊Q8—31no—
179C4⋊Q8—31no—
180C4⋊Q8—31no—
181C4⋊Q8C2×D311yescharacteristic
182#127—31no—
183#127—31no—
184#127—34no—
185#125—32no—
186#125—34no—
187#124—32no—
188#124—32no—
189#123—34no—
190#123—34no—
191#123—34no—
192#123C2×D314yes—
193C22⋊Q8—32no—
194C22⋊Q8—32no—
195C22⋊Q8—32no—
196C22⋊Q8—34no—
197C22⋊Q8—32no—
198C22⋊Q8—32no—
199C22⋊Q8—34no—
200C22⋊Q8—34no—
201C22⋊Q8—34no—
202C22⋊Q8C2×D312yes—
203#121—31no—
204#121—31no—
205#121—34no—
206#121—32no—
207#121—34no—
208#121—34no—
209#121—34no—
210#121—32no—
211#121—34no—
212#121—31no—
213#121—31no—
214#121—34no—
215#121—32no—
216#121—32no—
217#121—32no—
218#121—34no—
219#121—34no—
220#121C2×D312yes—
221C4×Q8—32no—
222C4×Q8—32no—
223C4×D4—32no—
224C4×D4—32no—
225C4×D4—34no—
226C4×D4—34no—
227C4×D4—31no—
228C4×D4—31no—
229C4×D4C2×D311yescharacteristic
230C4×D4C2×D311yescharacteristic
231#117—32no—
232C2×C4⋊C4—32no—
233C2×C4⋊C4—32no—
234C2×C4⋊C4—32no—
235C2×C4⋊C4C2×D312yes—
236C2×C42—31no—
237C23×C3C2412yes—
238C23×C3C2412yes—
239C22×D3C2412yes—
240C22×D3C2411yescharacteristic
241C22×D3C2412yes—
242C3×Q8—24no—
243C3×D4—22no—
244C3×D4—22no—
245C3×D4—44no—
246C2×C3×C4—24no—
247C2×C3×C4—24no—
248C2×C3×C4—44no—
249C2×C3×C4C2414yes—
250C2×C3×C4C2411yescharacteristic
251C2×C3×C4C2412yes—
252C2×C3×C4C2412yes—
253C2×C3×C4C2411yescharacteristic
254C2×C3×C4C2411yescharacteristic
255#67—24no—
256#67—24no—
257#67—44no—
258#67—42no—
259#67—42no—
260#67—42no—
261#67—42no—
262#67—44no—
263C2×Dic3—44no—
264C2×Dic3C2411yescharacteristic
265C2×Dic3C2411yescharacteristic
266C2×Dic3C2414yes—
267C2×Dic3C2412yes—
268C2×Dic3C2412yes—
269C2×Dic3C2411yescharacteristic
270D12—22no—
271D12—22no—
272D12—42no—
273D12—42no—
274D12—44no—
275C4×D3—22no—
276C4×D3—22no—
277C4×D3—24no—
278C4×D3—24no—
279C4×D3—22no—
280C4×D3—44no—
281C4×D3—44no—
282Dic6—24no—
283Dic6—42no—
284Dic6—42no—
285#41—44no—
286#41—124no—
287#41—124no—
288#41—124no—
289#41—124no—
290#41—124no—
291C2×Q8—32no—
292C2×Q8—32no—
293C2×Q8—32no—
294C2×Q8C22×D312yes—
295C2×D4—32no—
296C2×D4—32no—
297C2×D4—32no—
298C2×D4—32no—
299C2×D4—34no—
300C2×D4—34no—
301C2×D4—32no—
302C2×D4—32no—
303C2×D4—31no—
304C2×D4—31no—
305C2×D4—32no—
306C2×D4—32no—
307C2×D4—34no—
308C2×D4C22×D311yescharacteristic
309C2×D4C22×D311yescharacteristic
310C2×D4C22×D314yes—
311C22×C4—34no—
312C22×C4—31no—
313C22×C4—31no—
314C22×C4—31no—
315C22×C4—32no—
316C22×C4—32no—
317C22×C4—34no—
318C22×C4—34no—
319C22×C4C22×D312yes—
320C22×C4C22×D312yes—
321C22×C4C22×D314yes—
322C4⋊C4—31no—
323C4⋊C4—31no—
324C4⋊C4—32no—
325C4⋊C4—34no—
326C4⋊C4—34no—
327C4⋊C4—34no—
328C4⋊C4—32no—
329C4⋊C4—32no—
330C4⋊C4—32no—
331C4⋊C4—34no—
332C4⋊C4—34no—
333C4⋊C4C22×D311yescharacteristic
334C4⋊C4C22×D312yes—
335C4⋊C4C22×D311yescharacteristic
336C4⋊C4C22×D314yes—
337C4⋊C4C22×D312yes—
338C22⋊C4—34no—
339C22⋊C4—32no—
340C22⋊C4—32no—
341C22⋊C4—34no—
342C22⋊C4—34no—
343C22⋊C4—34no—
344C22⋊C4—32no—
345C22⋊C4—32no—
346C22⋊C4—32no—
347C22⋊C4—32no—
348C22⋊C4C22×D314yes—
349C22⋊C4C22×D312yes—
350C22⋊C4C22×D312yes—
351C42—31no—
352C42—31no—
353C42—31no—
354C42—34no—
355C42C22×D311yescharacteristic
356C22×C3—22no—
357C22×C3—22no—
358C22×C3—22no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—22no—
362C22×C3C2511yescharacteristic
363C2×D3—22no—
364C2×D3—21no—
365C2×D3—21no—
366C2×D3—22no—
367C2×D3—22no—
368C2×D3—22no—
369C2×D3—21no—
370C2×D3—22no—
371C2×D3—22no—
372C3×C4—22no—
373C3×C4—22no—
374C3×C4—24no—
375C3×C4—21no—
376C3×C4H(5,𝔽2)12yes—
377C3×C4H(5,𝔽2)12yes—
378Dic3—24no—
379Dic3—21no—
380Dic3—21no—
381Dic3—22no—
382Dic3—22no—
383Dic32−1+412yes—
384C23—31no—
385C23—32no—
386C23—32no—
387C23C23×D312yes—
388C23C23×D312yes—
389Q8—24no—
390Q8—64no—
391Q8—122no—
392Q8—122no—
393D4—22no—
394D4—22no—
395D4—44no—
396D4—64no—
397D4—64no—
398D4—62no—
399D4—62no—
400D4—122no—
401D4—124no—
402D4—122no—
403D4—122no—
404D4—122no—
405D4—122no—
406D4—122no—
407D4—124no—
408D4—124no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—31no—
412C2×C4—31no—
413C2×C4—31no—
414C2×C4—34no—
415C2×C4—32no—
416C2×C4—32no—
417C2×C4—44no—
418C2×C4—62no—
419C2×C4—64no—
420C2×C4—62no—
421C2×C4—64no—
422C2×C4—62no—
423C2×C4—124no—
424C2×C4—124no—
425C2×C4—124no—
426C2×C4C23×D311yescharacteristic
427C2×C4C23×D314yes—
428C2×C4C23×D312yes—
429C2×C4C23×D312yes—
430C2×C4C23×D311yescharacteristic
431C2×C4C23×D311yescharacteristic
432C2×C3—42no—
433C2×C3—42no—
434C2×C3C2×H(5,𝔽2)11yescharacteristic
435C2×C3C2×H(5,𝔽2)11yescharacteristic
436C2×C3C2×2−1+411yescharacteristic
437D3—42no—
438D3—41no—
439D3—42no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—62no—
447C22—61no—
448C22—62no—
449C22—61no—
450C22—62no—
451C22—62no—
452C22—61no—
453C22—62no—
454C22—62no—
455C22C24×D311yescharacteristic
456C4—22no—
457C4—24no—
458C4—22no—
459C4—21no—
460C4—32no—
461C4—62no—
462C4—61no—
463C4—62no—
464C4—61no—
465C4—64no—
466C4#100412yes—
467C4#100912yes—
468C3#354811yescharacteristic, complemented, Sylow
469C2—42no—
470C2—42no—
471C2—121no—
472C2—122no—
473C2—122no—
474C2C2×#100411yescharacteristic
475C2C2×#100911yescharacteristic
476C2D3×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#547C2
hypercenter#547D3
hypocenterC3#3548
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C3×#547C2semidirect
2. #5834C2semidirect
3. #5833C2semidirect
4. #5728C2semidirect
5. #5726C2semidirect
6. #5667C2semidirect
7. #5663C2semidirect
8. #5635C2semidirect
9. #5629C2semidirect
10. #5608C2semidirect
11. #5599C2semidirect
12. #5555C2semidirect
13. #5551C2semidirect
14. #5548C2semidirect
15. #5548C2semidirect
16. #5511C2semidirect
17. #5510C2semidirect
92. #547D3semidirect
468. C3#3548semidirect

Maximal subgroups

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

subgroupindex
C3×#5472
#58342
#58332
#57282
#57262
#56672
#56632
#56352
#56292
#56082
#55992
#55552
#55512
#55482
#55112
#55102
#35483