Finite Group Databasebeta

← #86342

Subgroups — #86342

Order 384.

1012 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#536C211yesmaximal, characteristic, complemented
2#5832C211yesmaximal, characteristic
3#5827C211yesmaximal, characteristic, complemented
4#5816C211yesmaximal, characteristic, complemented
5C2×Q8×Dic3C211yesmaximal, characteristic, complemented
6#5732C212yesmaximal, complemented
7#5721C212yesmaximal, complemented
8#5681C212yesmaximal, complemented
9#5671C212yesmaximal, complemented
10#5665C214yesmaximal, complemented
11#5649C214yesmaximal, complemented
12#5642C212yesmaximal, complemented
13#5641C212yesmaximal, complemented
14#5639C212yesmaximal, complemented
15#5635C212yesmaximal, complemented
16#5523C211yesmaximal, characteristic, complemented
17#5517C211yesmaximal, characteristic, complemented
18#3507—31nomaximal, Sylow
19C2×C3×#41C2211yescharacteristic
20C22×C3×Q8C2211yescharacteristic
21C2×#235C2211yescharacteristic
22C3×C4⋊Q8C2212yes—
23C3×#124C2212yes—
24C3×#123C2214yes—
25C3×C22⋊Q8C2212yes—
26C3×C22⋊Q8C2212yes—
27C3×#117C2211yescharacteristic
28#947C2212yes—
29#947C2211yescharacteristic
30#946C2212yes—
31#946C2212yes—
32Q8×Dic3C2212yes—
33Q8×Dic3C2212yes—
34Q8×Dic3C2212yes—
35Q8×Dic3C2212yes—
36Q8×Dic3C2212yes—
37#944C2212yes—
38#944C2211yescharacteristic
39#940C2211yescharacteristic
40#939C2212yes—
41#939C2212yes—
42#936C2211yescharacteristic
43D4×Dic3C2214yes—
44D4×Dic3C2212yes—
45#929C2214yes—
46C4×#67C2212yes—
47C4×#67C2212yes—
48#926C2212yes—
49#926C2211yescharacteristic
50C2×C4⋊Dic3C2212yes—
51C2×C4⋊Dic3C2212yes—
52C2×C4⋊Dic3C2211yescharacteristic
53C2×C4×Dic3C2212yes—
54C2×C4×Dic3C2211yescharacteristic
55#898C2214yes—
56#898C2214yes—
57#897C2214yes—
58#892C2214yes—
59#892C2212yes—
60D3×C4⋊C4C2212yes—
61#890C2211yescharacteristic
62#890C2211yescharacteristic
63#890C2214yes—
64#889C2214yes—
65#889C2214yes—
66#888C2211yescharacteristic
67#888C2211yescharacteristic
68#887C2214yes—
69#886C2214yes—
70#885C2214yes—
71#883C2214yes—
72#67⋊C4C2214yes—
73#67⋊C4C2214yes—
74#879C2214yes—
75#879C2214yes—
76#878C2214yes—
77#878C2212yes—
78#878C2212yes—
79#877C2214yes—
80C4×D12C2212yes—
81C4×Dic6C2212yes—
82C4×Dic6C2212yes—
83C4×Dic6C2212yes—
84#559—32no—
85#557—32no—
86#556—32no—
87#556—32no—
88#556—32no—
89#554—32no—
90#552—32no—
91#552—32no—
92#539—34no—
93#536—31no—
94#536D311yescharacteristic, complemented
95#533—31no—
96#532—31no—
97#529—34no—
98#520—31no—
99#520—31no—
100C2×C4×Q8—31no—
101C3×#41—24no—
102C2×C3×Q8—24no—
103C2×C3×Q8C2312yes—
104C2×C3×Q8C2311yescharacteristic
105C2×C3×Q8C2312yes—
106C2×C3×D4C2311yescharacteristic
107C2×C3×D4C2312yes—
108C22×C3×C4C2312yes—
109C22×C3×C4C2312yes—
110C22×C3×C4C2311yescharacteristic
111C2×#67C2312yes—
112C2×#67C2312yes—
113C2×#67C2312yes—
114C22×Dic3C2311yescharacteristic
115C22×Dic3C2314yes—
116C22×Dic3C2311yescharacteristic
117C22×Dic3C2312yes—
118#235—24no—
119C2×D12C2311yescharacteristic
120C2×C4×D3C2312yes—
121C2×C4×D3C2314yes—
122C2×Dic6C2311yescharacteristic
123C2×Dic6C2312yes—
124C2×Dic6C2312yes—
125C3×C4⋊C4C2314yes—
126C3×C4⋊C4C2311yescharacteristic
127C3×C4⋊C4C2311yescharacteristic
128C3×C4⋊C4C2314yes—
129C3×C22⋊C4C2314yes—
130C3×C22⋊C4C2312yes—
131C3×C22⋊C4C2314yes—
132C3×C42C2312yes—
133#217C2312yes—
134#217C2312yes—
135#217C2312yes—
136#217C2314yes—
137(C2×D3)⋊C4C2314yes—
138(C2×D3)⋊C4C2314yes—
139(C2×D3)⋊C4C2314yes—
140(C2×D3)⋊C4C2312yes—
141(C2×D3)⋊C4C2312yes—
142C4⋊Dic3C2312yes—
143C4⋊Dic3C2312yes—
144C4⋊Dic3C2312yes—
145C4⋊Dic3C2314yes—
146C4⋊Dic3C2314yes—
147C4⋊Dic3C2311yescharacteristic
148C4⋊Dic3C2311yescharacteristic
149C4⋊Dic3C2314yes—
150C4⋊Dic3C2312yes—
151Dic3⋊C4C2312yes—
152Dic3⋊C4C2312yes—
153Dic3⋊C4C2314yes—
154Dic3⋊C4C2312yes—
155Dic3⋊C4C2312yes—
156Dic3⋊C4C2314yes—
157Dic3⋊C4C2314yes—
158Dic3⋊C4C2312yes—
159Dic3⋊C4C2312yes—
160C4×Dic3C2311yescharacteristic
161C4×Dic3C2311yescharacteristic
162C4×Dic3C2312yes—
163C4×Dic3C2314yes—
164C4×Dic3C2314yes—
165C4×Dic3C2314yes—
166C4×Dic3C2314yes—
167C4×Dic3C2311yescharacteristic
168C4×Dic3C2311yescharacteristic
169C2×#41—31no—
170C2×#41C2×D311yescharacteristic
171C22×Q8C2×D311yescharacteristic
172C4⋊Q8—31no—
173C4⋊Q8—32no—
174C4⋊Q8—31no—
175C4⋊Q8—31no—
176C4⋊Q8C2×D312yes—
177#127—31no—
178#126—34no—
179#126—34no—
180#126—34no—
181#126—34no—
182#125—34no—
183#125—31no—
184#125—31no—
185#125—34no—
186#125—34no—
187#124—31no—
188#124—32no—
189#124—31no—
190#124—34no—
191#124C2×D312yes—
192#123—34no—
193#123—34no—
194#123—34no—
195#123C2×D314yes—
196C22⋊Q8—32no—
197C22⋊Q8—32no—
198C22⋊Q8—34no—
199C22⋊Q8—32no—
200C22⋊Q8—32no—
201C22⋊Q8—34no—
202C22⋊Q8C2×D312yes—
203C22⋊Q8C2×D312yes—
204#121—32no—
205#121—32no—
206C4×Q8—32no—
207C4×Q8—32no—
208C4×Q8—32no—
209C4×Q8—32no—
210C4×Q8—32no—
211C4×Q8—34no—
212C4×Q8—32no—
213C4×Q8—32no—
214C4×Q8—32no—
215C4×D4—34no—
216C4×D4—32no—
217C4×D4—32no—
218C4×D4—32no—
219C4×D4—34no—
220C4×D4—34no—
221C4×D4—32no—
222#117—31no—
223#117—32no—
224#117—32no—
225#117—34no—
226#117—34no—
227#117C2×D311yescharacteristic
228C2×C4⋊C4—31no—
229C2×C4⋊C4—32no—
230C2×C4⋊C4—32no—
231C2×C4⋊C4—32no—
232C2×C42—32no—
233C2×C42—31no—
234C23×C3C2412yes—
235C23×C3C2411yescharacteristic
236C22×D3C2412yes—
237C3×Q8—24no—
238C3×Q8—24no—
239C3×Q8—22no—
240C3×D4—24no—
241C3×D4—22no—
242C2×C3×C4—24no—
243C2×C3×C4—24no—
244C2×C3×C4#4114yes—
245C2×C3×C4C2414yes—
246C2×C3×C4C2412yes—
247C2×C3×C4C2412yes—
248C2×C3×C4C2412yes—
249C2×C3×C4C2411yescharacteristic
250C2×C3×C4C2411yescharacteristic
251#67—24no—
252#67—24no—
253#67—42no—
254C2×Dic3—22no—
255C2×Dic3—41no—
256C2×Dic3—44no—
257C2×Dic3—42no—
258C2×Dic3C2411yescharacteristic
259C2×Dic3C2414yes—
260C2×Dic3C2412yes—
261C2×Dic3C2412yes—
262C2×Dic3C2411yescharacteristic
263C2×Dic3C2411yescharacteristic
264C2×Dic3C2412yes—
265C2×Dic3C2411yescharacteristic
266D12—22no—
267C4×D3—24no—
268C4×D3—44no—
269Dic6—22no—
270Dic6—24no—
271Dic6—42no—
272#41—24no—
273#41—64no—
274C2×Q8—24no—
275C2×Q8—32no—
276C2×Q8—31no—
277C2×Q8—32no—
278C2×Q8C22×D312yes—
279C2×Q8C22×D312yes—
280C2×Q8C22×D311yescharacteristic
281C2×D4—32no—
282C2×D4—32no—
283C2×D4—32no—
284C2×D4—31no—
285C2×D4C22×D312yes—
286C2×D4C22×D311yescharacteristic
287C22×C4—31no—
288C22×C4—32no—
289C22×C4—31no—
290C22×C4—34no—
291C22×C4—32no—
292C22×C4—34no—
293C22×C4C22×D312yes—
294C22×C4C22×D312yes—
295C22×C4C22×D311yescharacteristic
296C4⋊C4—32no—
297C4⋊C4—31no—
298C4⋊C4—32no—
299C4⋊C4—34no—
300C4⋊C4—34no—
301C4⋊C4—34no—
302C4⋊C4—34no—
303C4⋊C4—31no—
304C4⋊C4—32no—
305C4⋊C4—32no—
306C4⋊C4—34no—
307C4⋊C4—32no—
308C4⋊C4—32no—
309C4⋊C4—32no—
310C4⋊C4—32no—
311C4⋊C4—32no—
312C4⋊C4—34no—
313C4⋊C4—32no—
314C4⋊C4C22×D311yescharacteristic
315C4⋊C4C22×D311yescharacteristic
316C4⋊C4C22×D314yes—
317C4⋊C4C22×D314yes—
318C22⋊C4—32no—
319C22⋊C4—32no—
320C22⋊C4—34no—
321C22⋊C4—32no—
322C22⋊C4—32no—
323C22⋊C4—34no—
324C22⋊C4—34no—
325C22⋊C4—32no—
326C22⋊C4—34no—
327C22⋊C4C22×D314yes—
328C22⋊C4C22×D312yes—
329C22⋊C4C22×D314yes—
330C42—34no—
331C42—31no—
332C42—31no—
333C42—34no—
334C42—31no—
335C42—31no—
336C42—34no—
337C42—34no—
338C42—32no—
339C42C22×D312yes—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—22no—
343C22×C3—22no—
344C22×C3—22no—
345C22×C3C2×#4112yes—
346C22×C3C2511yescharacteristic
347C2×D3—22no—
348C2×D3—22no—
349C2×D3—22no—
350C3×C4—22no—
351C3×C4—24no—
352C3×C4—22no—
353C3×C4—22no—
354C3×C4C2×#4112yes—
355C3×C4C2×#4112yes—
356Dic3—22no—
357Dic3—22no—
358Dic3—21no—
359Dic3—21no—
360Dic3—22no—
361Dic3—24no—
362Dic3—21no—
363Dic32−1+412yes—
364C23—32no—
365C23C23×D312yes—
366C23C23×D311yescharacteristic
367Q8—24no—
368Q8—24no—
369Q8—22no—
370Q8—64no—
371Q8—62no—
372Q8—122no—
373D4—24no—
374D4—22no—
375D4—64no—
376D4—64no—
377D4—62no—
378D4—122no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—31no—
382C2×C4—31no—
383C2×C4—34no—
384C2×C4—32no—
385C2×C4—31no—
386C2×C4—32no—
387C2×C4—31no—
388C2×C4—32no—
389C2×C4—62no—
390C2×C4—64no—
391C2×C4—124no—
392C2×C4—122no—
393C2×C4—124no—
394C2×C4—121no—
395C2×C4#23714yes—
396C2×C4C23×D312yes—
397C2×C4C23×D314yes—
398C2×C4C23×D312yes—
399C2×C4C23×D312yes—
400C2×C4C23×D311yescharacteristic
401C2×C4C23×D311yescharacteristic
402C2×C3—22no—
403C2×C3—42no—
404C2×C3C22×#4111yescharacteristic
405C2×C3C2×2−1+411yescharacteristic
406C2×C3#58511yescharacteristic
407D3—42no—
408C22—21no—
409C22—21no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—62no—
414C22—62no—
415C22—62no—
416C22C2×#23712yes—
417C22C24×D311yescharacteristic
418C4—22no—
419C4—24no—
420C4—22no—
421C4—22no—
422C4—32no—
423C4—62no—
424C4—61no—
425C4—62no—
426C4—64no—
427C4—62no—
428C4—61no—
429C4—61no—
430C4C2×#23712yes—
431C4C2×#23712yes—
432C3#350711yescharacteristic, complemented, Sylow
433C2—22no—
434C2—42no—
435C2—122no—
436C2C22×#23711yescharacteristic
437C2D3×2−1+411yescharacteristic
438C2#596811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#536C2
prime radical#5832C2
hypercenter#536D3
hypocenterC3#3507
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C3×#536C2semidirect
3. #5827C2semidirect
4. #5816C2semidirect
5. C2×Q8×Dic3C2semidirect
6. #5732C2semidirect
7. #5721C2semidirect
8. #5681C2semidirect
9. #5671C2semidirect
10. #5665C2semidirect
11. #5649C2semidirect
12. #5642C2semidirect
13. #5641C2semidirect
14. #5639C2semidirect
15. #5635C2semidirect
16. #5523C2semidirect
17. #5517C2semidirect
94. #536D3semidirect
432. C3#3507semidirect

Maximal subgroups

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

subgroupindex
C3×#5362
#58322
#58272
#58162
C2×Q8×Dic32
#57322
#57212
#56812
#56712
#56652
#56492
#56422
#56412
#56392
#56352
#55232
#55172
#35073