Finite Group Databasebeta

← #87035

Subgroups — #87035

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×#557C211yesmaximal, characteristic, complemented
2#5727C211yesmaximal, characteristic, complemented
3#5726C212yesmaximal, complemented
4#5725C211yesmaximal, characteristic, complemented
5#5724C212yesmaximal, complemented
6#5700C214yesmaximal, complemented
7#5697C212yesmaximal, complemented
8#5692C214yesmaximal, complemented
9#5690C212yesmaximal, complemented
10#5585C212yesmaximal, complemented
11#5585C211yesmaximal, characteristic, complemented
12#5582C212yesmaximal, complemented
13#5582C212yesmaximal, complemented
14#5580C211yesmaximal, characteristic, complemented
15#5578C211yesmaximal, characteristic, complemented
16#5577C211yesmaximal, characteristic, complemented
17#5576C212yesmaximal, complemented
18#3554—31nomaximal, Sylow
19C2×#239C2212yes—
20C2×#239C2212yes—
21C2×D3×Q8C2212yes—
22C3×C4⋊Q8C2212yes—
23C3×C4⋊Q8C2211yescharacteristic
24C3×#125C2212yes—
25C3×#125C2214yes—
26C3×C4×Q8C2212yes—
27C3×C4×Q8C2211yescharacteristic
28C3×C4×Q8C2211yescharacteristic
29C3×C4×Q8C2212yes—
30#947C2212yes—
31#947C2211yescharacteristic
32#947C2212yes—
33#946C2211yescharacteristic
34#946C2212yes—
35#946C2211yescharacteristic
36#944C2211yescharacteristic
37#898C2212yes—
38#897C2214yes—
39#897C2211yescharacteristic
40#896C2214yes—
41#896C2214yes—
42#896C2214yes—
43#896C2214yes—
44#896C2212yes—
45#895C2214yes—
46#895C2211yescharacteristic
47#895C2214yes—
48#895C2214yes—
49#895C2212yes—
50#895C2212yes—
51#895C2212yes—
52#894C2214yes—
53#894C2214yes—
54#894C2214yes—
55#894C2214yes—
56#894C2212yes—
57#894C2214yes—
58#894C2214yes—
59#893C2212yes—
60#893C2212yes—
61#893C2212yes—
62#893C2212yes—
63D3×C4⋊C4C2214yes—
64#890C2211yescharacteristic
65#889C2212yes—
66#888C2212yes—
67#888C2211yescharacteristic
68#876C2214yes—
69#875C2214yes—
70#875C2214yes—
71#875C2212yes—
72#874C2212yes—
73#874C2211yescharacteristic
74C4×D12C2212yes—
75C4×D12C2214yes—
76C4×D12C2212yes—
77C4×D12C2212yes—
78C4×D12C2211yescharacteristic
79#872C2212yes—
80#872C2212yes—
81C42×D3C2211yescharacteristic
82C4×Dic6C2212yes—
83C4×Dic6C2211yescharacteristic
84#562—34no—
85#559—32no—
86#559—31no—
87#557D311yescharacteristic, complemented
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#529—31no—
101C2×C3×Q8C2311yescharacteristic
102C2×C3×Q8C2311yescharacteristic
103C2×C3×Q8C2312yes—
104#239—44no—
105#239—44no—
106D3×Q8—44no—
107C2×D12C2312yes—
108C2×D12C2311yescharacteristic
109C2×D12C2312yes—
110C2×D12C2314yes—
111C2×D12C2312yes—
112C2×D12C2312yes—
113C2×D12C2314yes—
114C2×D12C2314yes—
115C2×C4×D3C2311yescharacteristic
116C2×C4×D3C2311yescharacteristic
117C2×C4×D3C2312yes—
118C2×C4×D3C2311yescharacteristic
119C2×C4×D3C2314yes—
120C2×C4×D3C2314yes—
121C2×C4×D3C2314yes—
122C2×C4×D3C2314yes—
123C2×Dic6C2311yescharacteristic
124C2×Dic6C2312yes—
125C2×Dic6C2314yes—
126C3×C4⋊C4C2312yes—
127C3×C4⋊C4C2311yescharacteristic
128C3×C4⋊C4C2312yes—
129C3×C4⋊C4C2311yescharacteristic
130C3×C4⋊C4C2314yes—
131C3×C4⋊C4C2312yes—
132C3×C4⋊C4C2314yes—
133C3×C4⋊C4C2312yes—
134C3×C4⋊C4C2314yes—
135C3×C42C2311yescharacteristic
136C3×C42C2312yes—
137C3×C42C2312yes—
138C3×C42C2314yes—
139(C2×D3)⋊C4C2312yes—
140(C2×D3)⋊C4C2312yes—
141(C2×D3)⋊C4C2312yes—
142(C2×D3)⋊C4C2312yes—
143(C2×D3)⋊C4C2312yes—
144(C2×D3)⋊C4C2312yes—
145(C2×D3)⋊C4C2314yes—
146(C2×D3)⋊C4C2314yes—
147(C2×D3)⋊C4C2314yes—
148(C2×D3)⋊C4C2312yes—
149(C2×D3)⋊C4C2314yes—
150(C2×D3)⋊C4C2314yes—
151(C2×D3)⋊C4C2314yes—
152(C2×D3)⋊C4C2314yes—
153C4⋊Dic3C2311yescharacteristic
154C4⋊Dic3C2312yes—
155C4⋊Dic3C2311yescharacteristic
156C4⋊Dic3C2312yes—
157Dic3⋊C4C2312yes—
158Dic3⋊C4C2312yes—
159Dic3⋊C4C2312yes—
160Dic3⋊C4C2312yes—
161Dic3⋊C4C2312yes—
162Dic3⋊C4C2312yes—
163Dic3⋊C4C2312yes—
164Dic3⋊C4C2314yes—
165C4×Dic3C2311yescharacteristic
166C4×Dic3C2312yes—
167C4×Dic3C2311yescharacteristic
168C4×Dic3C2311yescharacteristic
169C2×#41—32no—
170C2×#41—32no—
171C22×Q8—32no—
172C4⋊Q8—31no—
173C4⋊Q8—31no—
174C4⋊Q8—32no—
175C4⋊Q8C2×D311yescharacteristic
176C4⋊Q8C2×D312yes—
177#127—31no—
178#127—32no—
179#126—32no—
180#126—34no—
181#125—31no—
182#125—32no—
183#125C2×D314yes—
184#125C2×D312yes—
185#124—31no—
186#124—32no—
187#124—32no—
188#124—32no—
189#124—34no—
190#124—34no—
191#123—32no—
192#123—34no—
193#123—34no—
194#123—34no—
195#123—34no—
196#123—34no—
197#123—34no—
198C22⋊Q8—31no—
199C22⋊Q8—31no—
200C22⋊Q8—32no—
201C22⋊Q8—31no—
202C22⋊Q8—32no—
203C22⋊Q8—34no—
204C22⋊Q8—34no—
205C22⋊Q8—34no—
206C22⋊Q8—34no—
207C22⋊Q8—34no—
208#121—31no—
209#121—32no—
210#121—32no—
211#121—34no—
212#121—34no—
213#121—34no—
214#121—32no—
215C4×Q8—31no—
216C4×Q8—32no—
217C4×Q8C2×D311yescharacteristic
218C4×Q8C2×D311yescharacteristic
219C4×Q8C2×D312yes—
220C4×Q8C2×D312yes—
221C4×D4—32no—
222C4×D4—32no—
223C4×D4—32no—
224C4×D4—32no—
225C4×D4—32no—
226C4×D4—31no—
227C4×D4—34no—
228C4×D4—32no—
229C4×D4—32no—
230#117—32no—
231#117—32no—
232C2×C4⋊C4—34no—
233C2×C42—31no—
234C22×D3C2412yes—
235C22×D3C2411yescharacteristic
236C22×D3C2414yes—
237C3×Q8—24no—
238C3×Q8—22no—
239C3×Q8—22no—
240C2×C3×C4C2411yescharacteristic
241C2×C3×C4C2412yes—
242C2×C3×C4C2412yes—
243C2×C3×C4C2412yes—
244C2×C3×C4C2412yes—
245C2×C3×C4C2414yes—
246C2×C3×C4C2411yescharacteristic
247C2×C3×C4C2411yescharacteristic
248C2×Dic3C2411yescharacteristic
249C2×Dic3C2411yescharacteristic
250C2×Dic3C2411yescharacteristic
251C2×Dic3C2412yes—
252C2×Dic3C2412yes—
253C2×Dic3C2412yes—
254D12—24no—
255D12—24no—
256D12—22no—
257D12—44no—
258D12—42no—
259D12—42no—
260D12—44no—
261D12—44no—
262C4×D3—24no—
263C4×D3—22no—
264C4×D3—22no—
265C4×D3—41no—
266C4×D3—44no—
267C4×D3—44no—
268C4×D3—44no—
269C4×D3—44no—
270Dic6—22no—
271Dic6—42no—
272Dic6—44no—
273#41—124no—
274#41—124no—
275C2×Q8—31no—
276C2×Q8—32no—
277C2×Q8—34no—
278C2×Q8—124no—
279C2×Q8C22×D311yescharacteristic
280C2×Q8C22×D312yes—
281C2×Q8C22×D311yescharacteristic
282C2×D4—32no—
283C2×D4—31no—
284C2×D4—32no—
285C2×D4—32no—
286C2×D4—32no—
287C2×D4—34no—
288C2×D4—34no—
289C2×D4—34no—
290C22×C4—31no—
291C22×C4—31no—
292C22×C4—32no—
293C22×C4—31no—
294C22×C4—34no—
295C22×C4—34no—
296C22×C4—34no—
297C22×C4—34no—
298C4⋊C4—32no—
299C4⋊C4—32no—
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—32no—
310C4⋊C4C22×D314yes—
311C4⋊C4C22×D311yescharacteristic
312C4⋊C4C22×D312yes—
313C4⋊C4C22×D312yes—
314C4⋊C4C22×D311yescharacteristic
315C4⋊C4C22×D312yes—
316C4⋊C4C22×D314yes—
317C4⋊C4C22×D314yes—
318C4⋊C4C22×D312yes—
319C22⋊C4—32no—
320C22⋊C4—32no—
321C22⋊C4—32no—
322C22⋊C4—32no—
323C22⋊C4—32no—
324C22⋊C4—32no—
325C22⋊C4—34no—
326C22⋊C4—34no—
327C22⋊C4—34no—
328C22⋊C4—34no—
329C22⋊C4—34no—
330C22⋊C4—32no—
331C22⋊C4—34no—
332C22⋊C4—34no—
333C42—31no—
334C42—31no—
335C42—31no—
336C42—32no—
337C42C22×D311yescharacteristic
338C42C22×D312yes—
339C42C22×D312yes—
340C42C22×D314yes—
341C22×C3C2511yescharacteristic
342C2×D3—22no—
343C2×D3—21no—
344C2×D3—24no—
345C2×D3—21no—
346C2×D3—22no—
347C2×D3—24no—
348C2×D3—21no—
349C2×D3—24no—
350C2×D3—22no—
351C3×C4—24no—
352C3×C4—22no—
353C3×C4—22no—
354C3×C4—22no—
355C3×C4—22no—
356C3×C4—21no—
357C3×C4H(5,𝔽2)12yes—
358C3×C42−1+412yes—
359Dic3—22no—
360Dic3—22no—
361Dic3—21no—
362Dic3—21no—
363Dic3—21no—
364Dic3—22no—
365C23—31no—
366C23—32no—
367C23—34no—
368Q8—22no—
369Q8—24no—
370Q8—22no—
371Q8—62no—
372Q8—124no—
373Q8—122no—
374D4—64no—
375D4—62no—
376D4—64no—
377D4—124no—
378D4—124no—
379D4—122no—
380D4—122no—
381D4—124no—
382C2×C4—31no—
383C2×C4—31no—
384C2×C4—31no—
385C2×C4—32no—
386C2×C4—32no—
387C2×C4—32no—
388C2×C4—62no—
389C2×C4—64no—
390C2×C4—62no—
391C2×C4—124no—
392C2×C4—124no—
393C2×C4—124no—
394C2×C4—124no—
395C2×C4—121no—
396C2×C4C23×D311yescharacteristic
397C2×C4C23×D312yes—
398C2×C4C23×D312yes—
399C2×C4C23×D312yes—
400C2×C4C23×D314yes—
401C2×C4C23×D312yes—
402C2×C4C23×D311yescharacteristic
403C2×C4C23×D311yescharacteristic
404C2×C3C2×H(5,𝔽2)11yescharacteristic
405C2×C3C2×2−1+411yescharacteristic
406C2×C3#58511yescharacteristic
407D3—42no—
408D3—41no—
409D3—44no—
410C22—61no—
411C22—62no—
412C22—64no—
413C22—62no—
414C22—61no—
415C22—64no—
416C22—61no—
417C22—64no—
418C22—62no—
419C22C24×D311yescharacteristic
420C4—22no—
421C4—22no—
422C4—24no—
423C4—22no—
424C4—21no—
425C4—22no—
426C4—61no—
427C4—61no—
428C4—62no—
429C4—62no—
430C4—62no—
431C4—61no—
432C4#100712yes—
433C4#100912yes—
434C3#355411yescharacteristic, complemented, Sylow
435C2—121no—
436C2—122no—
437C2—124no—
438C2C2×#100711yescharacteristic
439C2C2×#100911yescharacteristic
440C2#596811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#557C2
hypercenter#557D3
hypocenterC3#3554
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. C3×#557C2semidirect
2. #5727C2semidirect
3. #5726C2semidirect
4. #5725C2semidirect
5. #5724C2semidirect
6. #5700C2semidirect
7. #5697C2semidirect
8. #5692C2semidirect
9. #5690C2semidirect
10. #5585C2semidirect
11. #5585C2semidirect
12. #5582C2semidirect
13. #5582C2semidirect
14. #5580C2semidirect
15. #5578C2semidirect
16. #5577C2semidirect
17. #5576C2semidirect
87. #557D3semidirect
434. C3#3554semidirect

Maximal subgroups

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

subgroupindex
C3×#5572
#57272
#57262
#57252
#57242
#57002
#56972
#56922
#56902
#55852
#55822
#55802
#55782
#55772
#55762
#35543