Finite Group Databasebeta

← #86708

Subgroups — #86708

Order 384.

1022 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#547C211yesmaximal, characteristic, complemented
2#5833C212yesmaximal, complemented
3#5831C212yesmaximal, complemented
4#5728C211yesmaximal, characteristic, complemented
5#5727C211yesmaximal, characteristic, complemented
6#5667C214yesmaximal, complemented
7#5662C214yesmaximal, complemented
8#5642C212yesmaximal, complemented
9#5628C212yesmaximal, complemented
10#5619C212yesmaximal, complemented
11#5616C212yesmaximal, complemented
12#5563C211yesmaximal, characteristic, complemented
13#5557C211yesmaximal, characteristic, complemented
14#5548C211yesmaximal, characteristic, complemented
15#5543C211yesmaximal, characteristic, complemented
16#5514C212yesmaximal, complemented
17#5500C212yesmaximal, complemented
18#3547—31nomaximal, Sylow
19C2×C3×#41C2212yes—
20C2×D3×Q8C2212yes—
21C2×#235C2212yes—
22C3×C4⋊Q8C2211yescharacteristic
23C3×#123C2214yes—
24C3×C22⋊Q8C2212yes—
25C3×#121C2212yes—
26C3×C4×D4C2211yescharacteristic
27C3×C4×D4C2211yescharacteristic
28C2×C3×C4⋊C4C2212yes—
29#946C2212yes—
30#946C2212yes—
31Q8×Dic3C2212yes—
32#940C2211yescharacteristic
33#939C2214yes—
34#939C2214yes—
35#938C2214yes—
36#938C2211yescharacteristic
37#938C2211yescharacteristic
38#938C2212yes—
39#936C2214yes—
40#936C2211yescharacteristic
41D4×Dic3C2212yes—
42#930C2214yes—
43#930C2212yes—
44C4×#67C2212yes—
45C4×#67C2214yes—
46C4×#67C2212yes—
47#926C2212yes—
48#926C2212yes—
49C2×C4⋊Dic3C2212yes—
50#924C2214yes—
51#924C2214yes—
52#924C2212yes—
53#898C2214yes—
54#897C2212yes—
55#897C2211yescharacteristic
56#896C2214yes—
57#895C2211yescharacteristic
58#894C2214yes—
59#893C2212yes—
60#892C2212yes—
61#892C2212yes—
62#890C2211yescharacteristic
63#890C2214yes—
64#890C2212yes—
65#890C2212yes—
66#888C2214yes—
67#888C2212yes—
68#888C2211yescharacteristic
69#887C2212yes—
70#886C2214yes—
71#885C2212yes—
72#885C2214yes—
73#885C2214yes—
74#884C2212yes—
75#883C2214yes—
76#883C2212yes—
77#879C2212yes—
78#878C2214yes—
79#877C2214yes—
80C4×D12C2211yescharacteristic
81C42×D3C2211yescharacteristic
82#869C2211yescharacteristic
83C4×Dic6C2211yescharacteristic
84#562—34no—
85#555—32no—
86#555—31no—
87#554—31no—
88#553—32no—
89#552—32no—
90D4×Q8—31no—
91#548—32no—
92#547D311yescharacteristic, complemented
93#544—32no—
94#542—34no—
95#540—32no—
96#537—32no—
97#536—32no—
98#533—31no—
99#532—31no—
100#529—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×#67C2312yes—
112C2×#67C2312yes—
113C2×#67C2314yes—
114C22×Dic3C2314yes—
115D3×Q8—44no—
116#235—44no—
117C2×D12C2311yescharacteristic
118C2×D12C2312yes—
119C2×C4×D3C2311yescharacteristic
120C2×C4×D3C2311yescharacteristic
121C2×C4×D3C2312yes—
122C2×C4×D3C2311yescharacteristic
123C2×C4×D3C2314yes—
124C2×Dic6C2311yescharacteristic
125C2×Dic6C2312yes—
126C2×Dic6C2312yes—
127C2×Dic6C2314yes—
128C3×C4⋊C4C2312yes—
129C3×C4⋊C4C2311yescharacteristic
130C3×C4⋊C4C2312yes—
131C3×C4⋊C4C2314yes—
132C3×C4⋊C4C2311yescharacteristic
133C3×C22⋊C4C2314yes—
134C3×C22⋊C4C2312yes—
135C3×C22⋊C4C2312yes—
136C3×C42C2311yescharacteristic
137#217C2312yes—
138#217C2314yes—
139#217C2312yes—
140#217C2312yes—
141#217C2312yes—
142#217C2314yes—
143(C2×D3)⋊C4C2312yes—
144(C2×D3)⋊C4C2312yes—
145(C2×D3)⋊C4C2314yes—
146(C2×D3)⋊C4C2314yes—
147(C2×D3)⋊C4C2312yes—
148(C2×D3)⋊C4C2314yes—
149C4⋊Dic3C2311yescharacteristic
150C4⋊Dic3C2312yes—
151C4⋊Dic3C2312yes—
152C4⋊Dic3C2311yescharacteristic
153C4⋊Dic3C2314yes—
154C4⋊Dic3C2312yes—
155C4⋊Dic3C2312yes—
156C4⋊Dic3C2314yes—
157Dic3⋊C4C2312yes—
158Dic3⋊C4C2312yes—
159Dic3⋊C4C2314yes—
160Dic3⋊C4C2314yes—
161Dic3⋊C4C2314yes—
162Dic3⋊C4C2312yes—
163C4×Dic3C2312yes—
164C4×Dic3C2311yescharacteristic
165C4×Dic3C2311yescharacteristic
166C4×Dic3C2311yescharacteristic
167C4×Dic3C2314yes—
168C4×Dic3C2314yes—
169C2×#41—32no—
170C2×#41C2×D312yes—
171C22×Q8—32no—
172C4⋊Q8—31no—
173C4⋊Q8—34no—
174C4⋊Q8—32no—
175C4⋊Q8—31no—
176C4⋊Q8C2×D311yescharacteristic
177#127—31no—
178#126—34no—
179#126—32no—
180#125—31no—
181#125—32no—
182#125—34no—
183#125—32no—
184#124—31no—
185#124—34no—
186#124—32no—
187#124—34no—
188#124—34no—
189#123—34no—
190#123—32no—
191#123—34no—
192#123—34no—
193#123C2×D314yes—
194C22⋊Q8—32no—
195C22⋊Q8—32no—
196C22⋊Q8—32no—
197C22⋊Q8—34no—
198C22⋊Q8—34no—
199C22⋊Q8—31no—
200C22⋊Q8—32no—
201C22⋊Q8—34no—
202C22⋊Q8—34no—
203C22⋊Q8C2×D312yes—
204#121—31no—
205#121—31no—
206#121—34no—
207#121—32no—
208#121—34no—
209#121—34no—
210#121—34no—
211#121—32no—
212#121—31no—
213#121—32no—
214#121C2×D312yes—
215C4×Q8—32no—
216C4×Q8—32no—
217C4×Q8—31no—
218C4×D4—32no—
219C4×D4—32no—
220C4×D4—32no—
221C4×D4—34no—
222C4×D4—32no—
223C4×D4—31no—
224C4×D4C2×D311yescharacteristic
225C4×D4C2×D311yescharacteristic
226#117—32no—
227#117—32no—
228#117—32no—
229#117—32no—
230#117—34no—
231C2×C4⋊C4—32no—
232C2×C4⋊C4C2×D312yes—
233C2×C42—31no—
234C23×C3C2412yes—
235C23×C3C2412yes—
236C22×D3C2412yes—
237C22×D3C2411yescharacteristic
238C3×Q8—24no—
239C3×D4—22no—
240C3×D4—22no—
241C3×D4—44no—
242C2×C3×C4—24no—
243C2×C3×C4—24no—
244C2×C3×C4—44no—
245C2×C3×C4C2412yes—
246C2×C3×C4C2414yes—
247C2×C3×C4C2411yescharacteristic
248C2×C3×C4C2412yes—
249C2×C3×C4C2411yescharacteristic
250C2×C3×C4C2411yescharacteristic
251#67—42no—
252#67—42no—
253#67—44no—
254#67—42no—
255#67—42no—
256C2×Dic3—44no—
257C2×Dic3C2411yescharacteristic
258C2×Dic3C2411yescharacteristic
259C2×Dic3C2412yes—
260C2×Dic3C2411yescharacteristic
261C2×Dic3C2414yes—
262C2×Dic3C2412yes—
263C2×Dic3C2412yes—
264D12—22no—
265D12—42no—
266C4×D3—22no—
267C4×D3—24no—
268C4×D3—22no—
269C4×D3—41no—
270C4×D3—44no—
271Dic6—22no—
272Dic6—42no—
273Dic6—42no—
274Dic6—44no—
275#41—44no—
276#41—124no—
277C2×Q8—31no—
278C2×Q8—32no—
279C2×Q8—34no—
280C2×Q8—32no—
281C2×Q8—124no—
282C2×Q8C22×D312yes—
283C2×D4—32no—
284C2×D4—34no—
285C2×D4—32no—
286C2×D4—32no—
287C2×D4—32no—
288C2×D4—31no—
289C2×D4—32no—
290C2×D4C22×D311yescharacteristic
291C2×D4C22×D311yescharacteristic
292C2×D4C22×D314yes—
293C22×C4—34no—
294C22×C4—31no—
295C22×C4—31no—
296C22×C4—32no—
297C22×C4—31no—
298C22×C4—34no—
299C22×C4C22×D312yes—
300C22×C4C22×D312yes—
301C22×C4C22×D314yes—
302C4⋊C4—31no—
303C4⋊C4—32no—
304C4⋊C4—34no—
305C4⋊C4—32no—
306C4⋊C4—31no—
307C4⋊C4—32no—
308C4⋊C4—34no—
309C4⋊C4—32no—
310C4⋊C4—32no—
311C4⋊C4—32no—
312C4⋊C4—34no—
313C4⋊C4—34no—
314C4⋊C4—34no—
315C4⋊C4—32no—
316C4⋊C4C22×D311yescharacteristic
317C4⋊C4C22×D312yes—
318C4⋊C4C22×D314yes—
319C4⋊C4C22×D311yescharacteristic
320C4⋊C4C22×D312yes—
321C22⋊C4—32no—
322C22⋊C4—32no—
323C22⋊C4—32no—
324C22⋊C4—34no—
325C22⋊C4—32no—
326C22⋊C4—34no—
327C22⋊C4—34no—
328C22⋊C4—34no—
329C22⋊C4—32no—
330C22⋊C4—34no—
331C22⋊C4—32no—
332C22⋊C4—32no—
333C22⋊C4C22×D314yes—
334C22⋊C4C22×D312yes—
335C22⋊C4C22×D312yes—
336C42—31no—
337C42—31no—
338C42—31no—
339C42—34no—
340C42—34no—
341C42—32no—
342C42C22×D311yescharacteristic
343C22×C3—22no—
344C22×C3—22no—
345C22×C3—22no—
346C22×C3—22no—
347C22×C3—22no—
348C22×C3—22no—
349C22×C3C2511yescharacteristic
350C2×D3—22no—
351C2×D3—21no—
352C2×D3—21no—
353C2×D3—22no—
354C2×D3—21no—
355C2×D3—22no—
356C3×C4—24no—
357C3×C4—22no—
358C3×C4—22no—
359C3×C4—21no—
360C3×C4H(5,𝔽2)12yes—
361C3×C42−1+412yes—
362Dic3—22no—
363Dic3—22no—
364Dic3—21no—
365Dic3—21no—
366Dic3—24no—
367Dic3—21no—
368Dic3—22no—
369C23—32no—
370C23—31no—
371C23C23×D312yes—
372C23C23×D312yes—
373Q8—24no—
374Q8—62no—
375Q8—124no—
376Q8—122no—
377Q8—122no—
378D4—22no—
379D4—22no—
380D4—44no—
381D4—62no—
382D4—122no—
383D4—122no—
384D4—122no—
385D4—122no—
386D4—122no—
387D4—124no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—31no—
391C2×C4—34no—
392C2×C4—32no—
393C2×C4—31no—
394C2×C4—31no—
395C2×C4—32no—
396C2×C4—32no—
397C2×C4—44no—
398C2×C4—64no—
399C2×C4—62no—
400C2×C4—62no—
401C2×C4—124no—
402C2×C4—121no—
403C2×C4—124no—
404C2×C4C23×D312yes—
405C2×C4C23×D311yescharacteristic
406C2×C4C23×D314yes—
407C2×C4C23×D312yes—
408C2×C4C23×D311yescharacteristic
409C2×C4C23×D311yescharacteristic
410C2×C3—42no—
411C2×C3—42no—
412C2×C3C2×H(5,𝔽2)11yescharacteristic
413C2×C3C2×2−1+411yescharacteristic
414C2×C3#58511yescharacteristic
415D3—42no—
416D3—41no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—62no—
424C22—61no—
425C22—61no—
426C22—62no—
427C22—61no—
428C22—62no—
429C22C24×D311yescharacteristic
430C4—22no—
431C4—24no—
432C4—22no—
433C4—21no—
434C4—61no—
435C4—62no—
436C4—62no—
437C4—61no—
438C4—62no—
439C4—61no—
440C4—64no—
441C4#100412yes—
442C4#101012yes—
443C3#354711yescharacteristic, complemented, Sylow
444C2—42no—
445C2—42no—
446C2—122no—
447C2—121no—
448C2C2×#100411yescharacteristic
449C2C2×#101011yescharacteristic
450C2#596811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#547C2
hypercenter#547D3
hypocenterC3#3547
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. #5833C2semidirect
3. #5831C2semidirect
4. #5728C2semidirect
5. #5727C2semidirect
6. #5667C2semidirect
7. #5662C2semidirect
8. #5642C2semidirect
9. #5628C2semidirect
10. #5619C2semidirect
11. #5616C2semidirect
12. #5563C2semidirect
13. #5557C2semidirect
14. #5548C2semidirect
15. #5543C2semidirect
16. #5514C2semidirect
17. #5500C2semidirect
92. #547D3semidirect
443. C3#3547semidirect

Maximal subgroups

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

subgroupindex
C3×#5472
#58332
#58312
#57282
#57272
#56672
#56622
#56422
#56282
#56192
#56162
#55632
#55572
#55482
#55432
#55142
#55002
#35473