Finite Group Databasebeta

← #86137

Subgroups — #86137

Order 384.

1022 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#529C211yesmaximal, characteristic, complemented
2#5707C214yesmaximal, complemented
3#5690C212yesmaximal, complemented
4#5681C212yesmaximal, complemented
5#5667C214yesmaximal, complemented
6#5643C212yesmaximal, complemented
7#5619C212yesmaximal, complemented
8#5585C211yesmaximal, characteristic, complemented
9#5570C211yesmaximal, characteristic, complemented
10#5553C211yesmaximal, characteristic, complemented
11#5552C212yesmaximal, complemented
12#5551C212yesmaximal, complemented
13D4×Dic6C211yesmaximal, characteristic, complemented
14#5534C212yesmaximal, complemented
15#5526C212yesmaximal, complemented
16#5477C211yesmaximal, characteristic, complemented
17#5470C211yesmaximal, characteristic, complemented
18#3554—31nomaximal, Sylow
19C2×#235C2212yes—
20C2×#235C2212yes—
21C22×Dic6C2212yes—
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#947C2212yes—
34Q8×Dic3C2212yes—
35#940C2212yes—
36#939C2214yes—
37#938C2214yes—
38D4×Dic3C2212yes—
39#930C2212yes—
40#930C2212yes—
41#930C2212yes—
42#926C2212yes—
43#924C2212yes—
44#924C2211yescharacteristic
45#924C2211yescharacteristic
46#924C2212yes—
47#898C2214yes—
48#897C2214yes—
49#897C2214yes—
50#896C2214yes—
51#896C2214yes—
52#894C2214yes—
53#893C2212yes—
54#893C2212yes—
55D3×C4⋊C4C2214yes—
56#890C2214yes—
57#890C2212yes—
58#888C2212yes—
59#886C2214yes—
60#886C2214yes—
61#885C2214yes—
62#885C2214yes—
63#884C2214yes—
64#883C2214yes—
65#883C2214yes—
66#883C2214yes—
67#67⋊C4C2214yes—
68#878C2214yes—
69#875C2212yes—
70#875C2212yes—
71#874C2211yescharacteristic
72C4×D12C2212yes—
73C4×D12C2212yes—
74C4×D12C2212yes—
75#870C2212yes—
76#869C2211yescharacteristic
77#869C2211yescharacteristic
78#869C2211yescharacteristic
79#869C2212yes—
80C4×Dic6C2212yes—
81C4×Dic6C2211yescharacteristic
82C4×Dic6C2211yescharacteristic
83C4×Dic6C2212yes—
84#562—34no—
85#559—32no—
86#559—31no—
87#557—31no—
88#554—32no—
89#550—31no—
90D4×Q8—31no—
91#548—32no—
92#547—32no—
93#542—32no—
94#542—32no—
95#540—32no—
96#539—34no—
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×#67C2314yes—
110C2×#67C2314yes—
111C22×Dic3C2314yes—
112#235—44no—
113#235—44no—
114C2×D12C2312yes—
115C2×D12C2312yes—
116C2×D12C2312yes—
117C2×C4×D3C2314yes—
118C2×C4×D3C2314yes—
119C2×C4×D3C2314yes—
120C2×Dic6—44no—
121C2×Dic6C2311yescharacteristic
122C2×Dic6C2312yes—
123C2×Dic6C2311yescharacteristic
124C2×Dic6C2312yes—
125C2×Dic6C2314yes—
126C3×C4⋊C4C2311yescharacteristic
127C3×C4⋊C4C2311yescharacteristic
128C3×C4⋊C4C2312yes—
129C3×C4⋊C4C2312yes—
130C3×C4⋊C4C2312yes—
131C3×C4⋊C4C2312yes—
132C3×C22⋊C4C2312yes—
133C3×C22⋊C4C2312yes—
134C3×C22⋊C4C2312yes—
135C3×C22⋊C4C2312yes—
136C3×C22⋊C4C2312yes—
137C3×C42C2311yescharacteristic
138C3×C42C2311yescharacteristic
139C3×C42C2312yes—
140C3×C42C2311yescharacteristic
141C3×C42C2311yescharacteristic
142#217C2314yes—
143#217C2312yes—
144#217C2312yes—
145(C2×D3)⋊C4C2314yes—
146(C2×D3)⋊C4C2314yes—
147(C2×D3)⋊C4C2314yes—
148(C2×D3)⋊C4C2314yes—
149(C2×D3)⋊C4C2314yes—
150(C2×D3)⋊C4C2314yes—
151C4⋊Dic3C2311yescharacteristic
152C4⋊Dic3C2311yescharacteristic
153C4⋊Dic3C2312yes—
154C4⋊Dic3C2312yes—
155C4⋊Dic3C2312yes—
156C4⋊Dic3C2312yes—
157C4⋊Dic3C2312yes—
158C4⋊Dic3C2314yes—
159C4⋊Dic3C2314yes—
160Dic3⋊C4C2312yes—
161Dic3⋊C4C2312yes—
162Dic3⋊C4C2312yes—
163Dic3⋊C4C2312yes—
164Dic3⋊C4C2314yes—
165Dic3⋊C4C2314yes—
166C4×Dic3C2312yes—
167C4×Dic3C2312yes—
168C4×Dic3C2314yes—
169C2×#41—32no—
170C2×#41—32no—
171C22×Q8—32no—
172C4⋊Q8—32no—
173C4⋊Q8—31no—
174C4⋊Q8—31no—
175C4⋊Q8—31no—
176C4⋊Q8—32no—
177#127—32no—
178#127—31no—
179#126—34no—
180#126C2×D312yes—
181#125—34no—
182#125—32no—
183#125—32no—
184#125C2×D311yescharacteristic
185#124—32no—
186#124—34no—
187#124—34no—
188#124—32no—
189#124—32no—
190#124C2×D311yescharacteristic
191#123—34no—
192#123—34no—
193#123—34no—
194#123—34no—
195#123—34no—
196#123—34no—
197#123C2×D312yes—
198C22⋊Q8—31no—
199C22⋊Q8—31no—
200C22⋊Q8—32no—
201C22⋊Q8—32no—
202C22⋊Q8—34no—
203C22⋊Q8—34no—
204C22⋊Q8—34no—
205C22⋊Q8—34no—
206C22⋊Q8—34no—
207C22⋊Q8C2×D311yescharacteristic
208#121—34no—
209#121—34no—
210#121—32no—
211#121—32no—
212#121—32no—
213#121—34no—
214#121C2×D311yescharacteristic
215C4×Q8—32no—
216C4×Q8—31no—
217C4×Q8—31no—
218C4×Q8—32no—
219C4×Q8—32no—
220C4×Q8C2×D311yescharacteristic
221C4×D4—32no—
222C4×D4—32no—
223C4×D4—32no—
224C4×D4—34no—
225C4×D4—32no—
226C4×D4—32no—
227C4×D4—32no—
228C4×D4C2×D312yes—
229C4×D4C2×D311yescharacteristic
230#117—32no—
231#117C2×D312yes—
232C2×C4⋊C4—34no—
233C2×C42C2×D311yescharacteristic
234C23×C3C2411yescharacteristic
235C23×C3C2412yes—
236C22×D3C2414yes—
237C3×Q8—22no—
238C3×D4—22no—
239C3×D4—42no—
240C2×C3×C4—24no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—41no—
244C2×C3×C4C2412yes—
245C2×C3×C4C2411yescharacteristic
246C2×C3×C4C2411yescharacteristic
247C2×C3×C4C2412yes—
248C2×C3×C4C2412yes—
249C2×C3×C4C2411yescharacteristic
250C2×C3×C4C2411yescharacteristic
251C2×C3×C4C2411yescharacteristic
252C2×C3×C4C2411yescharacteristic
253#67—44no—
254#67—44no—
255#67—44no—
256C2×Dic3—44no—
257C2×Dic3C2412yes—
258C2×Dic3C2412yes—
259C2×Dic3C2412yes—
260C2×Dic3C2414yes—
261C2×Dic3C2412yes—
262D12—24no—
263D12—24no—
264D12—42no—
265C4×D3—44no—
266C4×D3—44no—
267C4×D3—44no—
268Dic6—22no—
269Dic6—24no—
270Dic6—22no—
271Dic6—44no—
272Dic6—42no—
273#41—124no—
274#41—124no—
275C2×Q8—31no—
276C2×Q8—32no—
277C2×Q8—31no—
278C2×Q8—32no—
279C2×Q8—34no—
280C2×Q8—124no—
281C2×Q8C22×D311yescharacteristic
282C2×D4—34no—
283C2×D4—34no—
284C2×D4—34no—
285C2×D4—32no—
286C2×D4—32no—
287C2×D4—32no—
288C2×D4C22×D311yescharacteristic
289C2×D4C22×D312yes—
290C22×C4—34no—
291C22×C4—34no—
292C22×C4—34no—
293C22×C4—34no—
294C22×C4C22×D311yescharacteristic
295C22×C4C22×D311yescharacteristic
296C22×C4C22×D312yes—
297C22×C4C22×D311yescharacteristic
298C4⋊C4—32no—
299C4⋊C4—34no—
300C4⋊C4—34no—
301C4⋊C4—32no—
302C4⋊C4—32no—
303C4⋊C4—32no—
304C4⋊C4—32no—
305C4⋊C4—31no—
306C4⋊C4—32no—
307C4⋊C4—31no—
308C4⋊C4—32no—
309C4⋊C4—34no—
310C4⋊C4—32no—
311C4⋊C4—32no—
312C4⋊C4—34no—
313C4⋊C4C22×D312yes—
314C4⋊C4C22×D312yes—
315C4⋊C4C22×D312yes—
316C4⋊C4C22×D311yescharacteristic
317C4⋊C4C22×D311yescharacteristic
318C4⋊C4C22×D312yes—
319C22⋊C4—32no—
320C22⋊C4—32no—
321C22⋊C4—34no—
322C22⋊C4—34no—
323C22⋊C4—34no—
324C22⋊C4—34no—
325C22⋊C4—34no—
326C22⋊C4—34no—
327C22⋊C4—34no—
328C22⋊C4C22×D312yes—
329C22⋊C4C22×D312yes—
330C22⋊C4C22×D312yes—
331C22⋊C4C22×D312yes—
332C22⋊C4C22×D312yes—
333C42—32no—
334C42—32no—
335C42—34no—
336C42C22×D311yescharacteristic
337C42C22×D311yescharacteristic
338C42C22×D311yescharacteristic
339C42C22×D311yescharacteristic
340C42C22×D312yes—
341C22×C3—22no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—22no—
345C22×C3—22no—
346C22×C3—21no—
347C22×C3C2511yescharacteristic
348C2×D3—24no—
349C2×D3—24no—
350C2×D3—24no—
351C3×C4—22no—
352C3×C4—21no—
353C3×C4—21no—
354C3×C4—22no—
355C3×C4—21no—
356C3×C4—22no—
357C3×C4—21no—
358C3×C4H(5,𝔽2)12yes—
359C3×C42−1+412yes—
360Dic3—22no—
361Dic3—24no—
362Dic3—22no—
363Dic3—22no—
364Dic3—22no—
365C23—34no—
366C23C23×D311yescharacteristic
367C23C23×D312yes—
368Q8—22no—
369Q8—62no—
370Q8—64no—
371Q8—62no—
372Q8—124no—
373Q8—122no—
374D4—22no—
375D4—42no—
376D4—64no—
377D4—64no—
378D4—122no—
379D4—124no—
380D4—124no—
381D4—124no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—32no—
386C2×C4—32no—
387C2×C4—32no—
388C2×C4—34no—
389C2×C4—32no—
390C2×C4—41no—
391C2×C4—124no—
392C2×C4—124no—
393C2×C4—124no—
394C2×C4—124no—
395C2×C4C23×D312yes—
396C2×C4C23×D311yescharacteristic
397C2×C4C23×D311yescharacteristic
398C2×C4C23×D311yescharacteristic
399C2×C4C23×D311yescharacteristic
400C2×C4C23×D312yes—
401C2×C4C23×D312yes—
402C2×C4C23×D311yescharacteristic
403C2×C4C23×D311yescharacteristic
404C2×C3—42no—
405C2×C3—41no—
406C2×C3C2×H(5,𝔽2)11yescharacteristic
407C2×C3C2×2−1+411yescharacteristic
408C2×C3#58511yescharacteristic
409D3—44no—
410C22—22no—
411C22—21no—
412C22—21no—
413C22—22no—
414C22—21no—
415C22—22no—
416C22—64no—
417C22—64no—
418C22—64no—
419C22C24×D311yescharacteristic
420C4—21no—
421C4—22no—
422C4—21no—
423C4—22no—
424C4—21no—
425C4—21no—
426C4—22no—
427C4—62no—
428C4—64no—
429C4—62no—
430C4—62no—
431C4—62no—
432C4#100912yes—
433C4#101012yes—
434C3#355411yescharacteristic, complemented, Sylow
435C2—42no—
436C2—41no—
437C2—124no—
438C2C2×#100911yescharacteristic
439C2C2×#101011yescharacteristic
440C2#596411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#529C2
hypercenter#529D3
hypocenterC3#3554
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. #5707C2semidirect
3. #5690C2semidirect
4. #5681C2semidirect
5. #5667C2semidirect
6. #5643C2semidirect
7. #5619C2semidirect
8. #5585C2semidirect
9. #5570C2semidirect
10. #5553C2semidirect
11. #5552C2semidirect
12. #5551C2semidirect
13. D4×Dic6C2semidirect
14. #5534C2semidirect
15. #5526C2semidirect
16. #5477C2semidirect
17. #5470C2semidirect
100. #529D3semidirect
434. C3#3554semidirect

Maximal subgroups

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

subgroupindex
C3×#5292
#57072
#56902
#56812
#56672
#56432
#56192
#55852
#55702
#55532
#55522
#55512
D4×Dic62
#55342
#55262
#54772
#54702
#35543