Finite Group Databasebeta

← #86175

Subgroups — #86175

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×#529C211yesmaximal, characteristic, complemented
2#5710C214yesmaximal, complemented
3#5694C212yesmaximal, complemented
4#5677C212yesmaximal, complemented
5#5667C214yesmaximal, complemented
6#5628C212yesmaximal, complemented
7#5608C212yesmaximal, complemented
8#5579C211yesmaximal, characteristic, complemented
9Q8×D12C211yesmaximal, characteristic, complemented
10#5565C212yesmaximal, complemented
11#5563C211yesmaximal, characteristic, complemented
12#5563C212yesmaximal, complemented
13#5555C211yesmaximal, characteristic, complemented
14#5535C212yesmaximal, complemented
15#5519C212yesmaximal, complemented
16#5477C211yesmaximal, characteristic, complemented
17#5470C211yesmaximal, characteristic, complemented
18#3547—31nomaximal, Sylow
19C2×D3×Q8C2212yes—
20C2×#237C2212yes—
21C2×#235C2212yes—
22C3×#126C2212yes—
23C3×#125C2211yescharacteristic
24C3×#124C2211yescharacteristic
25C3×#123C2212yes—
26C3×C22⋊Q8C2211yescharacteristic
27C3×#121C2211yescharacteristic
28C3×C4×Q8C2211yescharacteristic
29C3×C4×D4C2211yescharacteristic
30C3×C4×D4C2212yes—
31C3×#117C2212yes—
32C2×C3×C42C2211yescharacteristic
33#946C2212yes—
34#939C2214yes—
35#938C2214yes—
36#938C2212yes—
37#930C2212yes—
38#930C2211yescharacteristic
39#930C2211yescharacteristic
40#930C2212yes—
41#930C2212yes—
42#926C2212yes—
43#926C2212yes—
44C2×C4⋊Dic3C2212yes—
45#924C2212yes—
46#924C2212yes—
47#898C2214yes—
48#897C2214yes—
49#897C2212yes—
50#897C2212yes—
51#896C2214yes—
52#894C2214yes—
53#894C2214yes—
54#892C2214yes—
55#892C2212yes—
56D3×C4⋊C4C2212yes—
57#890C2214yes—
58#888C2214yes—
59#886C2214yes—
60#885C2214yes—
61#885C2214yes—
62#885C2214yes—
63#884C2214yes—
64#884C2214yes—
65#883C2214yes—
66#67⋊C4C2214yes—
67#878C2214yes—
68#878C2214yes—
69#875C2212yes—
70#874C2211yescharacteristic
71C4×D12C2212yes—
72C4×D12C2211yescharacteristic
73C4×D12C2211yescharacteristic
74C4×D12C2212yes—
75C4×D12C2212yes—
76#870C2212yes—
77#870C2212yes—
78#869C2211yescharacteristic
79#869C2211yescharacteristic
80#869C2211yescharacteristic
81#869C2212yes—
82C4×Dic6C2212yes—
83C4×Dic6C2212yes—
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#547—31no—
93#544—32no—
94#542—34no—
95#540—32no—
96#537—32no—
97#536—32no—
98#533—31no—
99#532—31no—
100#529D311yescharacteristic, complemented
101C2×C3×Q8C2311yescharacteristic
102C2×C3×D4C2311yescharacteristic
103C2×C3×D4C2312yes—
104C22×C3×C4C2311yescharacteristic
105C22×C3×C4C2312yes—
106C22×C3×C4C2311yescharacteristic
107C22×C3×C4C2311yescharacteristic
108C2×#67C2314yes—
109C2×#67C2312yes—
110C2×#67C2314yes—
111C2×#67C2312yes—
112C22×Dic3C2314yes—
113D3×Q8—44no—
114#237—44no—
115#235—44no—
116C2×D12C2311yescharacteristic
117C2×D12C2311yescharacteristic
118C2×D12C2312yes—
119C2×D12C2312yes—
120C2×C4×D3C2312yes—
121C2×C4×D3C2312yes—
122C2×C4×D3C2314yes—
123C2×C4×D3C2314yes—
124C2×Dic6C2312yes—
125C2×Dic6C2312yes—
126C2×Dic6C2312yes—
127C2×Dic6C2314yes—
128C3×C4⋊C4C2312yes—
129C3×C4⋊C4C2312yes—
130C3×C4⋊C4C2311yescharacteristic
131C3×C4⋊C4C2311yescharacteristic
132C3×C4⋊C4C2312yes—
133C3×C4⋊C4C2312yes—
134C3×C22⋊C4C2312yes—
135C3×C22⋊C4C2312yes—
136C3×C22⋊C4C2312yes—
137C3×C22⋊C4C2312yes—
138C3×C22⋊C4C2312yes—
139C3×C42C2311yescharacteristic
140C3×C42C2311yescharacteristic
141C3×C42C2311yescharacteristic
142C3×C42C2311yescharacteristic
143C3×C42C2312yes—
144#217C2314yes—
145#217C2314yes—
146(C2×D3)⋊C4C2312yes—
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—
154C4⋊Dic3C2312yes—
155C4⋊Dic3C2312yes—
156C4⋊Dic3C2311yescharacteristic
157C4⋊Dic3C2312yes—
158C4⋊Dic3C2311yescharacteristic
159C4⋊Dic3C2312yes—
160C4⋊Dic3C2312yes—
161C4⋊Dic3C2314yes—
162C4⋊Dic3C2314yes—
163Dic3⋊C4C2314yes—
164Dic3⋊C4C2314yes—
165Dic3⋊C4C2314yes—
166Dic3⋊C4C2314yes—
167C4×Dic3C2314yes—
168C4×Dic3C2314yes—
169C2×#41—32no—
170C2×#41—32no—
171C22×Q8—32no—
172C4⋊Q8—34no—
173C4⋊Q8—31no—
174C4⋊Q8—31no—
175C4⋊Q8—31no—
176C4⋊Q8—32no—
177#127—31no—
178#126—34no—
179#126C2×D312yes—
180#125—34no—
181#125—32no—
182#125—32no—
183#125C2×D311yescharacteristic
184#124—34no—
185#124—34no—
186#124—34no—
187#124—32no—
188#124C2×D311yescharacteristic
189#123—34no—
190#123—34no—
191#123—34no—
192#123—34no—
193#123C2×D312yes—
194C22⋊Q8—32no—
195C22⋊Q8—32no—
196C22⋊Q8—32no—
197C22⋊Q8—32no—
198C22⋊Q8—32no—
199C22⋊Q8—34no—
200C22⋊Q8—34no—
201C22⋊Q8—34no—
202C22⋊Q8—34no—
203C22⋊Q8C2×D311yescharacteristic
204#121—32no—
205#121—34no—
206#121—34no—
207#121—31no—
208#121—31no—
209#121—32no—
210#121—32no—
211#121—32no—
212#121—34no—
213#121—34no—
214#121C2×D311yescharacteristic
215C4×Q8—32no—
216C4×Q8—32no—
217C4×Q8C2×D311yescharacteristic
218C4×D4—34no—
219C4×D4—31no—
220C4×D4—32no—
221C4×D4—31no—
222C4×D4—32no—
223C4×D4—32no—
224C4×D4C2×D311yescharacteristic
225C4×D4C2×D312yes—
226#117—32no—
227#117—32no—
228#117—32no—
229#117—34no—
230#117C2×D312yes—
231C2×C4⋊C4—32no—
232C2×C4⋊C4—32no—
233C2×C42C2×D311yescharacteristic
234C23×C3C2411yescharacteristic
235C23×C3C2412yes—
236C22×D3C2412yes—
237C22×D3C2412yes—
238C3×Q8—22no—
239C3×D4—22no—
240C3×D4—42no—
241C2×C3×C4—22no—
242C2×C3×C4—24no—
243C2×C3×C4—22no—
244C2×C3×C4—41no—
245C2×C3×C4C2411yescharacteristic
246C2×C3×C4C2412yes—
247C2×C3×C4C2411yescharacteristic
248C2×C3×C4C2411yescharacteristic
249C2×C3×C4C2412yes—
250C2×C3×C4C2412yes—
251C2×C3×C4C2411yescharacteristic
252C2×C3×C4C2411yescharacteristic
253C2×C3×C4C2411yescharacteristic
254#67—42no—
255#67—42no—
256#67—44no—
257#67—44no—
258C2×Dic3—44no—
259C2×Dic3C2414yes—
260C2×Dic3C2412yes—
261C2×Dic3C2414yes—
262C2×Dic3C2412yes—
263D12—22no—
264D12—22no—
265D12—42no—
266D12—42no—
267C4×D3—24no—
268C4×D3—24no—
269C4×D3—44no—
270C4×D3—44no—
271Dic6—24no—
272Dic6—42no—
273Dic6—42no—
274Dic6—44no—
275#41—124no—
276#41—124no—
277C2×Q8—32no—
278C2×Q8—32no—
279C2×Q8—34no—
280C2×Q8—32no—
281C2×Q8—124no—
282C2×Q8C22×D311yescharacteristic
283C2×D4—34no—
284C2×D4—32no—
285C2×D4—34no—
286C2×D4—32no—
287C2×D4—31no—
288C2×D4—31no—
289C2×D4—32no—
290C2×D4—32no—
291C2×D4C22×D311yescharacteristic
292C2×D4C22×D312yes—
293C22×C4—34no—
294C22×C4—32no—
295C22×C4—32no—
296C22×C4—34no—
297C22×C4—34no—
298C22×C4C22×D311yescharacteristic
299C22×C4C22×D311yescharacteristic
300C22×C4C22×D312yes—
301C22×C4C22×D311yescharacteristic
302C4⋊C4—31no—
303C4⋊C4—32no—
304C4⋊C4—32no—
305C4⋊C4—34no—
306C4⋊C4—34no—
307C4⋊C4—32no—
308C4⋊C4—31no—
309C4⋊C4—34no—
310C4⋊C4—32no—
311C4⋊C4—34no—
312C4⋊C4—34no—
313C4⋊C4—34no—
314C4⋊C4—32no—
315C4⋊C4C22×D311yescharacteristic
316C4⋊C4C22×D311yescharacteristic
317C4⋊C4C22×D312yes—
318C4⋊C4C22×D312yes—
319C4⋊C4C22×D312yes—
320C4⋊C4C22×D312yes—
321C22⋊C4—34no—
322C22⋊C4—34no—
323C22⋊C4—32no—
324C22⋊C4—34no—
325C22⋊C4—34no—
326C22⋊C4—32no—
327C22⋊C4—34no—
328C22⋊C4—32no—
329C22⋊C4—34no—
330C22⋊C4—32no—
331C22⋊C4C22×D312yes—
332C22⋊C4C22×D312yes—
333C22⋊C4C22×D312yes—
334C22⋊C4C22×D312yes—
335C22⋊C4C22×D312yes—
336C42—34no—
337C42—34no—
338C42C22×D311yescharacteristic
339C42C22×D311yescharacteristic
340C42C22×D311yescharacteristic
341C42C22×D311yescharacteristic
342C42C22×D312yes—
343C22×C3—22no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—22no—
347C22×C3—22no—
348C22×C3—21no—
349C22×C3C2511yescharacteristic
350C2×D3—22no—
351C2×D3—22no—
352C2×D3—22no—
353C2×D3—22no—
354C2×D3—22no—
355C2×D3—22no—
356C3×C4—22no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—22no—
360C3×C4—21no—
361C3×C4—22no—
362C3×C4—21no—
363C3×C4H(5,𝔽2)12yes—
364C3×C42−1+412yes—
365Dic3—24no—
366Dic3—22no—
367Dic3—22no—
368Dic3—24no—
369C23—32no—
370C23—32no—
371C23C23×D311yescharacteristic
372C23C23×D312yes—
373Q8—22no—
374Q8—64no—
375Q8—124no—
376Q8—122no—
377Q8—122no—
378D4—22no—
379D4—42no—
380D4—62no—
381D4—62no—
382D4—122no—
383D4—122no—
384D4—122no—
385D4—122no—
386D4—124no—
387D4—124no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—34no—
392C2×C4—32no—
393C2×C4—34no—
394C2×C4—32no—
395C2×C4—41no—
396C2×C4—64no—
397C2×C4—64no—
398C2×C4—124no—
399C2×C4—124no—
400C2×C4—124no—
401C2×C4C23×D311yescharacteristic
402C2×C4C23×D312yes—
403C2×C4C23×D311yescharacteristic
404C2×C4C23×D311yescharacteristic
405C2×C4C23×D311yescharacteristic
406C2×C4C23×D312yes—
407C2×C4C23×D312yes—
408C2×C4C23×D311yescharacteristic
409C2×C4C23×D311yescharacteristic
410C2×C3—42no—
411C2×C3—41no—
412C2×C3C2×H(5,𝔽2)11yescharacteristic
413C2×C3C2×2−1+411yescharacteristic
414C2×C3#58511yescharacteristic
415D3—42no—
416D3—42no—
417C22—21no—
418C22—22no—
419C22—21no—
420C22—22no—
421C22—21no—
422C22—22no—
423C22—62no—
424C22—62no—
425C22—62no—
426C22—62no—
427C22—62no—
428C22—62no—
429C22C24×D311yescharacteristic
430C4—21no—
431C4—22no—
432C4—21no—
433C4—22no—
434C4—21no—
435C4—21no—
436C4—22no—
437C4—62no—
438C4—62no—
439C4—64no—
440C4—64no—
441C4#100912yes—
442C4#101012yes—
443C3#354711yescharacteristic, complemented, Sylow
444C2—41no—
445C2—42no—
446C2—122no—
447C2—122no—
448C2C2×#100911yescharacteristic
449C2C2×#101011yescharacteristic
450C2#596411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#529C2
hypercenter#529D3
hypocenterC3#3547
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #529

Complements

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

NH ≅ G/Ntype
1. C3×#529C2semidirect
2. #5710C2semidirect
3. #5694C2semidirect
4. #5677C2semidirect
5. #5667C2semidirect
6. #5628C2semidirect
7. #5608C2semidirect
8. #5579C2semidirect
9. Q8×D12C2semidirect
10. #5565C2semidirect
11. #5563C2semidirect
12. #5563C2semidirect
13. #5555C2semidirect
14. #5535C2semidirect
15. #5519C2semidirect
16. #5477C2semidirect
17. #5470C2semidirect
100. #529D3semidirect
443. C3#3547semidirect

Maximal subgroups

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

subgroupindex
C3×#5292
#57102
#56942
#56772
#56672
#56282
#56082
#55792
Q8×D122
#55652
#55632
#55552
#55352
#55192
#54772
#54702
#35473