Finite Group Databasebeta

← #86232

Subgroups — #86232

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×#532C211yesmaximal, characteristic, complemented
2#5831C212yesmaximal, complemented
3#5824C212yesmaximal, complemented
4#5727C211yesmaximal, characteristic, complemented
5#5721C211yesmaximal, characteristic, complemented
6#5718C211yesmaximal, characteristic, complemented
7#5713C211yesmaximal, characteristic, complemented
8#5682C212yesmaximal, complemented
9#5675C212yesmaximal, complemented
10#5621C214yesmaximal, complemented
11#5619C212yesmaximal, complemented
12#5603C212yesmaximal, complemented
13#5564C214yesmaximal, complemented
14#5559C212yesmaximal, complemented
15#5539C212yesmaximal, complemented
16#5479C211yesmaximal, characteristic, complemented
17#5470C211yesmaximal, characteristic, complemented
18#3554—31nomaximal, Sylow
19C2×C3×#41C2212yes—
20C2×D3×Q8C2212yes—
21C2×#237C2212yes—
22C3×C4⋊Q8C2211yescharacteristic
23C3×#127C2211yescharacteristic
24C3×#124C2212yes—
25C3×#121C2212yes—
26C3×#121C2212yes—
27C3×C4×D4C2212yes—
28C3×C4×D4C2212yes—
29C2×C3×C42C2211yescharacteristic
30#946C2212yes—
31Q8×Dic3C2212yes—
32#944C2212yes—
33#940C2212yes—
34#939C2214yes—
35#939C2214yes—
36#938C2212yes—
37#938C2214yes—
38#936C2212yes—
39#936C2212yes—
40#936C2214yes—
41#935C2214yes—
42#935C2214yes—
43D4×Dic3C2212yes—
44D4×Dic3C2212yes—
45D4×Dic3C2212yes—
46#930C2211yescharacteristic
47#929C2212yes—
48#929C2214yes—
49C4×#67C2212yes—
50C4×#67C2214yes—
51#926C2212yes—
52#924C2211yescharacteristic
53#924C2211yescharacteristic
54#924C2212yes—
55#924C2211yescharacteristic
56#924C2214yes—
57C2×Dic3⋊C4C2214yes—
58#896C2214yes—
59#893C2212yes—
60#890C2212yes—
61#889C2214yes—
62#888C2212yes—
63#887C2212yes—
64#886C2214yes—
65#885C2214yes—
66#883C2214yes—
67#883C2214yes—
68#879C2214yes—
69#878C2214yes—
70#878C2214yes—
71#878C2214yes—
72#876C2212yes—
73#875C2211yescharacteristic
74C4×D12C2211yescharacteristic
75#872C2212yes—
76#870C2212yes—
77#870C2211yescharacteristic
78#869C2211yescharacteristic
79#869C2211yescharacteristic
80C4×Dic6C2211yescharacteristic
81C4×Dic6C2211yescharacteristic
82C4×Dic6C2211yescharacteristic
83C4×Dic6C2212yes—
84#562—34no—
85#559—31no—
86#559—32no—
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#532D311yescharacteristic, complemented
100#529—31no—
101C3×#41—44no—
102C2×C3×Q8C2312yes—
103C2×C3×D4C2312yes—
104C2×C3×D4C2312yes—
105C2×C3×D4C2312yes—
106C2×C3×D4C2314yes—
107C22×C3×C4C2311yescharacteristic
108C22×C3×C4C2311yescharacteristic
109C22×C3×C4C2314yes—
110C22×C3×C4C2311yescharacteristic
111C2×#67C2314yes—
112C2×#67C2314yes—
113C2×#67C2312yes—
114C22×Dic3C2314yes—
115C22×Dic3C2314yes—
116D3×Q8—44no—
117#237—44no—
118C2×D12C2311yescharacteristic
119C2×C4×D3C2312yes—
120C2×C4×D3C2314yes—
121C2×Dic6C2311yescharacteristic
122C2×Dic6C2311yescharacteristic
123C2×Dic6C2311yescharacteristic
124C2×Dic6C2312yes—
125C2×Dic6C2314yes—
126C3×C4⋊C4C2312yes—
127C3×C4⋊C4C2312yes—
128C3×C22⋊C4C2314yes—
129C3×C22⋊C4C2314yes—
130C3×C42C2311yescharacteristic
131C3×C42C2311yescharacteristic
132C3×C42C2311yescharacteristic
133C3×C42C2311yescharacteristic
134#217C2314yes—
135#217C2314yes—
136#217C2314yes—
137#217C2314yes—
138#217C2312yes—
139#217C2312yes—
140#217C2312yes—
141(C2×D3)⋊C4C2312yes—
142(C2×D3)⋊C4C2312yes—
143(C2×D3)⋊C4C2312yes—
144(C2×D3)⋊C4C2312yes—
145(C2×D3)⋊C4C2314yes—
146C4⋊Dic3C2312yes—
147C4⋊Dic3C2311yescharacteristic
148C4⋊Dic3C2311yescharacteristic
149C4⋊Dic3C2311yescharacteristic
150C4⋊Dic3C2312yes—
151C4⋊Dic3C2312yes—
152C4⋊Dic3C2311yescharacteristic
153C4⋊Dic3C2314yes—
154Dic3⋊C4C2312yes—
155Dic3⋊C4C2312yes—
156Dic3⋊C4C2312yes—
157Dic3⋊C4C2312yes—
158Dic3⋊C4C2312yes—
159Dic3⋊C4C2312yes—
160Dic3⋊C4C2312yes—
161Dic3⋊C4C2312yes—
162Dic3⋊C4C2314yes—
163Dic3⋊C4C2314yes—
164Dic3⋊C4C2314yes—
165C4×Dic3C2312yes—
166C4×Dic3C2312yes—
167C4×Dic3C2312yes—
168C4×Dic3C2314yes—
169C2×#41—32no—
170C2×#41C2×D312yes—
171C22×Q8—32no—
172C4⋊Q8—32no—
173C4⋊Q8—32no—
174C4⋊Q8—31no—
175C4⋊Q8—31no—
176C4⋊Q8C2×D311yescharacteristic
177#127—32no—
178#127C2×D311yescharacteristic
179#126—34no—
180#126—32no—
181#125—34no—
182#125—32no—
183#125—31no—
184#125—32no—
185#124—32no—
186#124—34no—
187#124—32no—
188#124—34no—
189#124—31no—
190#124C2×D312yes—
191#123—34no—
192#123—34no—
193#123—32no—
194#123—34no—
195#123—34no—
196#123—34no—
197#123—34no—
198C22⋊Q8—32no—
199C22⋊Q8—31no—
200C22⋊Q8—31no—
201C22⋊Q8—31no—
202C22⋊Q8—32no—
203C22⋊Q8—34no—
204C22⋊Q8—34no—
205C22⋊Q8—34no—
206C22⋊Q8—34no—
207C22⋊Q8—34no—
208#121—34no—
209#121—34no—
210#121—34no—
211#121—32no—
212#121—31no—
213#121C2×D312yes—
214#121C2×D312yes—
215C4×Q8—32no—
216C4×Q8—32no—
217C4×Q8—31no—
218C4×Q8—31no—
219C4×Q8—31no—
220C4×Q8—32no—
221C4×D4—32no—
222C4×D4—32no—
223C4×D4—32no—
224C4×D4—32no—
225C4×D4—34no—
226C4×D4—32no—
227C4×D4—31no—
228C4×D4C2×D312yes—
229C4×D4C2×D312yes—
230#117—32no—
231#117—32no—
232C2×C4⋊C4—34no—
233C2×C42C2×D311yescharacteristic
234C23×C3C2411yescharacteristic
235C23×C3C2414yes—
236C22×D3C2412yes—
237C3×Q8—24no—
238C3×D4—24no—
239C3×D4—24no—
240C3×D4—42no—
241C3×D4—44no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—41no—
245C2×C3×C4—44no—
246C2×C3×C4C2414yes—
247C2×C3×C4C2411yescharacteristic
248C2×C3×C4C2411yescharacteristic
249C2×C3×C4C2411yescharacteristic
250C2×C3×C4C2411yescharacteristic
251C2×C3×C4C2411yescharacteristic
252C2×C3×C4C2411yescharacteristic
253#67—44no—
254#67—44no—
255#67—42no—
256C2×Dic3—44no—
257C2×Dic3—44no—
258C2×Dic3C2412yes—
259C2×Dic3C2412yes—
260C2×Dic3C2412yes—
261C2×Dic3C2412yes—
262C2×Dic3C2412yes—
263C2×Dic3C2412yes—
264C2×Dic3C2412yes—
265D12—22no—
266C4×D3—24no—
267C4×D3—44no—
268Dic6—22no—
269Dic6—22no—
270Dic6—22no—
271Dic6—42no—
272Dic6—44no—
273#41—44no—
274#41—124no—
275C2×Q8—31no—
276C2×Q8—31no—
277C2×Q8—31no—
278C2×Q8—32no—
279C2×Q8—34no—
280C2×Q8—124no—
281C2×Q8C22×D312yes—
282C2×D4—32no—
283C2×D4—34no—
284C2×D4—34no—
285C2×D4—31no—
286C2×D4C22×D312yes—
287C2×D4C22×D312yes—
288C2×D4C22×D312yes—
289C2×D4C22×D314yes—
290C22×C4—34no—
291C22×C4—34no—
292C22×C4—32no—
293C22×C4—34no—
294C22×C4C22×D311yescharacteristic
295C22×C4C22×D311yescharacteristic
296C22×C4C22×D311yescharacteristic
297C22×C4C22×D314yes—
298C4⋊C4—32no—
299C4⋊C4—32no—
300C4⋊C4—34no—
301C4⋊C4—34no—
302C4⋊C4—32no—
303C4⋊C4—32no—
304C4⋊C4—32no—
305C4⋊C4—32no—
306C4⋊C4—31no—
307C4⋊C4—31no—
308C4⋊C4—32no—
309C4⋊C4—31no—
310C4⋊C4—31no—
311C4⋊C4—32no—
312C4⋊C4—34no—
313C4⋊C4—32no—
314C4⋊C4—32no—
315C4⋊C4—34no—
316C4⋊C4—32no—
317C4⋊C4C22×D312yes—
318C4⋊C4C22×D312yes—
319C22⋊C4—32no—
320C22⋊C4—32no—
321C22⋊C4—34no—
322C22⋊C4—34no—
323C22⋊C4—32no—
324C22⋊C4—34no—
325C22⋊C4—34no—
326C22⋊C4—32no—
327C22⋊C4—32no—
328C22⋊C4—32no—
329C22⋊C4—32no—
330C22⋊C4—34no—
331C22⋊C4C22×D314yes—
332C22⋊C4C22×D314yes—
333C42—32no—
334C42—32no—
335C42—32no—
336C42—34no—
337C42C22×D311yescharacteristic
338C42C22×D311yescharacteristic
339C42C22×D311yescharacteristic
340C42C22×D311yescharacteristic
341C22×C3—24no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—24no—
345C22×C3—24no—
346C22×C3—21no—
347C22×C3C2511yescharacteristic
348C2×D3—22no—
349C2×D3—22no—
350C2×D3—22no—
351C3×C4—24no—
352C3×C4—21no—
353C3×C4—21no—
354C3×C4—21no—
355C3×C4—21no—
356C3×C4H(5,𝔽2)12yes—
357C3×C42−1+412yes—
358Dic3—22no—
359Dic3—22no—
360Dic3—22no—
361Dic3—22no—
362Dic3—22no—
363Dic3—22no—
364Dic3—22no—
365C23—32no—
366C23C23×D311yescharacteristic
367C23C23×D314yes—
368Q8—24no—
369Q8—62no—
370Q8—62no—
371Q8—62no—
372Q8—124no—
373Q8—122no—
374D4—24no—
375D4—24no—
376D4—44no—
377D4—42no—
378D4—62no—
379D4—124no—
380D4—122no—
381D4—124no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—32no—
385C2×C4—32no—
386C2×C4—32no—
387C2×C4—32no—
388C2×C4—32no—
389C2×C4—32no—
390C2×C4—32no—
391C2×C4—44no—
392C2×C4—41no—
393C2×C4—64no—
394C2×C4—124no—
395C2×C4—124no—
396C2×C4—124no—
397C2×C4C23×D314yes—
398C2×C4C23×D311yescharacteristic
399C2×C4C23×D311yescharacteristic
400C2×C4C23×D311yescharacteristic
401C2×C4C23×D311yescharacteristic
402C2×C4C23×D311yescharacteristic
403C2×C4C23×D311yescharacteristic
404C2×C3—44no—
405C2×C3—41no—
406C2×C3C2×H(5,𝔽2)11yescharacteristic
407C2×C3C2×2−1+411yescharacteristic
408C2×C3#58511yescharacteristic
409D3—42no—
410C22—21no—
411C22—24no—
412C22—21no—
413C22—24no—
414C22—21no—
415C22—24no—
416C22—62no—
417C22—62no—
418C22—62no—
419C22C24×D311yescharacteristic
420C4—21no—
421C4—24no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—62no—
426C4—62no—
427C4—62no—
428C4—62no—
429C4—62no—
430C4—62no—
431C4—62no—
432C4#100412yes—
433C4#101012yes—
434C3#355411yescharacteristic, complemented, Sylow
435C2—41no—
436C2—44no—
437C2—122no—
438C2C2×#100411yescharacteristic
439C2C2×#101011yescharacteristic
440C2#596411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#532C2
hypercenter#532D3
hypocenterC3#3554
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C3×#532C2semidirect
2. #5831C2semidirect
3. #5824C2semidirect
4. #5727C2semidirect
5. #5721C2semidirect
6. #5718C2semidirect
7. #5713C2semidirect
8. #5682C2semidirect
9. #5675C2semidirect
10. #5621C2semidirect
11. #5619C2semidirect
12. #5603C2semidirect
13. #5564C2semidirect
14. #5559C2semidirect
15. #5539C2semidirect
16. #5479C2semidirect
17. #5470C2semidirect
99. #532D3semidirect
434. C3#3554semidirect

Maximal subgroups

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

subgroupindex
C3×#5322
#58312
#58242
#57272
#57212
#57182
#57132
#56822
#56752
#56212
#56192
#56032
#55642
#55592
#55392
#54792
#54702
#35543