Finite Group Databasebeta

← #86995

Subgroups — #86995

Order 384.

1002 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×#556C211yesmaximal, characteristic, complemented
2#5732C211yesmaximal, characteristic, complemented
3#5728C211yesmaximal, characteristic, complemented
4#5712C214yesmaximal, complemented
5#5710C214yesmaximal, complemented
6#5682C212yesmaximal, complemented
7#5681C212yesmaximal, complemented
8#5632C212yesmaximal, complemented
9#5628C212yesmaximal, complemented
10#5582C212yesmaximal, complemented
11#5578C211yesmaximal, characteristic, complemented
12Q8×D12C211yesmaximal, characteristic, complemented
13Q8×Dic6C212yesmaximal, complemented
14#5548C211yesmaximal, characteristic
15D4×Dic6C211yesmaximal, characteristic, complemented
16#5533C212yesmaximal, complemented
17#5526C212yesmaximal, complemented
18#3550—31nomaximal, Sylow
19C2×D3×Q8C2212yes—
20C2×#235C2212yes—
21C22×Dic6C2212yes—
22C3×C4⋊Q8C2211yescharacteristic
23C3×#126C2214yes—
24C3×#124C2212yes—
25C3×C22⋊Q8C2212yes—
26C3×C4×Q8C2212yes—
27C3×C4×Q8C2211yescharacteristic
28C3×C4×D4C2211yescharacteristic
29C3×#117C2212yes—
30#947C2212yes—
31#946C2212yes—
32#946C2211yescharacteristic
33Q8×Dic3C2212yes—
34Q8×Dic3C2212yes—
35#944C2211yescharacteristic
36#940C2211yescharacteristic
37#938C2211yescharacteristic
38D4×Dic3C2212yes—
39C4×#67C2212yes—
40#924C2212yes—
41#898C2214yes—
42#897C2212yes—
43#897C2214yes—
44#897C2212yes—
45#897C2211yescharacteristic
46#897C2212yes—
47#896C2214yes—
48#896C2212yes—
49#895C2211yescharacteristic
50#894C2214yes—
51#892C2212yes—
52#892C2212yes—
53#890C2212yes—
54#889C2212yes—
55#888C2214yes—
56#888C2214yes—
57#888C2211yescharacteristic
58#888C2212yes—
59#888C2211yescharacteristic
60#887C2212yes—
61#886C2214yes—
62#886C2214yes—
63#885C2214yes—
64#885C2214yes—
65#885C2212yes—
66#884C2212yes—
67#883C2214yes—
68#67⋊C4C2214yes—
69#879C2214yes—
70#878C2214yes—
71#878C2214yes—
72#876C2214yes—
73#875C2214yes—
74C4×D12C2212yes—
75C4×D12C2211yescharacteristic
76#872C2212yes—
77C42×D3C2211yescharacteristic
78#869C2212yes—
79#869C2214yes—
80#869C2211yescharacteristic
81C4×Dic6C2212yes—
82C4×Dic6C2214yes—
83C4×Dic6C2211yescharacteristic
84#563—34no—
85#559—32no—
86Q82—32no—
87#556—31no—
88#556—32no—
89#556D311yescharacteristic, complemented
90D4×Q8—32no—
91D4×Q8—31no—
92D4×Q8—31no—
93#542—34no—
94#542—32no—
95#542—32no—
96#536—32no—
97#536—32no—
98#533—31no—
99#533—31no—
100#532—31no—
101C2×C3×Q8C2311yescharacteristic
102C2×C3×Q8C2312yes—
103C2×C3×D4C2311yescharacteristic
104C22×C3×C4C2312yes—
105C2×#67C2312yes—
106C2×#67C2312yes—
107C2×#67C2312yes—
108C22×Dic3C2314yes—
109D3×Q8—44no—
110#235—44no—
111C2×D12C2311yescharacteristic
112C2×D12C2312yes—
113C2×C4×D3C2311yescharacteristic
114C2×C4×D3C2311yescharacteristic
115C2×C4×D3C2311yescharacteristic
116C2×C4×D3C2312yes—
117C2×C4×D3C2314yes—
118C2×Dic6—44no—
119C2×Dic6C2312yes—
120C2×Dic6C2311yescharacteristic
121C2×Dic6C2312yes—
122C2×Dic6C2312yes—
123C2×Dic6C2314yes—
124C2×Dic6C2314yes—
125C3×C4⋊C4C2312yes—
126C3×C4⋊C4C2311yescharacteristic
127C3×C4⋊C4C2314yes—
128C3×C4⋊C4C2311yescharacteristic
129C3×C4⋊C4C2312yes—
130C3×C4⋊C4C2312yes—
131C3×C22⋊C4C2312yes—
132C3×C22⋊C4C2314yes—
133C3×C22⋊C4C2314yes—
134C3×C42C2311yescharacteristic
135C3×C42C2312yes—
136C3×C42C2314yes—
137#217C2312yes—
138#217C2314yes—
139(C2×D3)⋊C4C2312yes—
140(C2×D3)⋊C4C2312yes—
141(C2×D3)⋊C4C2314yes—
142(C2×D3)⋊C4C2312yes—
143(C2×D3)⋊C4C2314yes—
144(C2×D3)⋊C4C2314yes—
145(C2×D3)⋊C4C2312yes—
146(C2×D3)⋊C4C2314yes—
147C4⋊Dic3C2314yes—
148C4⋊Dic3C2312yes—
149C4⋊Dic3C2311yescharacteristic
150C4⋊Dic3C2312yes—
151C4⋊Dic3C2311yescharacteristic
152C4⋊Dic3C2312yes—
153C4⋊Dic3C2314yes—
154C4⋊Dic3C2314yes—
155Dic3⋊C4C2312yes—
156Dic3⋊C4C2314yes—
157Dic3⋊C4C2312yes—
158Dic3⋊C4C2312yes—
159Dic3⋊C4C2312yes—
160Dic3⋊C4C2312yes—
161Dic3⋊C4C2312yes—
162Dic3⋊C4C2314yes—
163C4×Dic3C2311yescharacteristic
164C4×Dic3C2312yes—
165C4×Dic3C2314yes—
166C4×Dic3C2314yes—
167C4×Dic3C2311yescharacteristic
168C4×Dic3C2311yescharacteristic
169C2×#41—32no—
170C22×Q8—32no—
171C22×Q8—32no—
172C4⋊Q8—31no—
173C4⋊Q8—31no—
174C4⋊Q8—31no—
175C4⋊Q8—34no—
176C4⋊Q8—34no—
177C4⋊Q8—32no—
178C4⋊Q8—31no—
179C4⋊Q8—34no—
180C4⋊Q8—32no—
181C4⋊Q8C2×D311yescharacteristic
182#127—31no—
183#126—34no—
184#126—34no—
185#126—34no—
186#126C2×D314yes—
187#125—32no—
188#125—32no—
189#124—32no—
190#124—32no—
191#124—34no—
192#124—34no—
193#124—34no—
194#124C2×D312yes—
195#123—34no—
196#123—34no—
197#123—34no—
198#123—34no—
199C22⋊Q8—31no—
200C22⋊Q8—32no—
201C22⋊Q8—32no—
202C22⋊Q8—32no—
203C22⋊Q8—31no—
204C22⋊Q8—32no—
205C22⋊Q8—32no—
206C22⋊Q8—32no—
207C22⋊Q8—34no—
208C22⋊Q8—34no—
209C22⋊Q8—34no—
210C22⋊Q8—34no—
211C22⋊Q8C2×D312yes—
212#121—31no—
213#121—31no—
214#121—32no—
215C4×Q8—32no—
216C4×Q8—32no—
217C4×Q8—32no—
218C4×Q8—31no—
219C4×Q8—32no—
220C4×Q8—34no—
221C4×Q8C2×D311yescharacteristic
222C4×Q8C2×D312yes—
223C4×D4—32no—
224C4×D4—32no—
225C4×D4—34no—
226C4×D4—31no—
227C4×D4—32no—
228C4×D4C2×D311yescharacteristic
229#117—32no—
230#117—32no—
231#117—32no—
232#117C2×D312yes—
233C2×C42—31no—
234C23×C3C2412yes—
235C22×D3C2412yes—
236C22×D3C2411yescharacteristic
237C3×Q8—22no—
238C3×Q8—24no—
239C3×D4—22no—
240C2×C3×C4—24no—
241C2×C3×C4C2414yes—
242C2×C3×C4C2411yescharacteristic
243C2×C3×C4C2412yes—
244C2×C3×C4C2412yes—
245C2×C3×C4C2412yes—
246C2×C3×C4C2411yescharacteristic
247C2×C3×C4C2411yescharacteristic
248#67—24no—
249#67—42no—
250#67—42no—
251C2×Dic3—44no—
252C2×Dic3C2412yes—
253C2×Dic3C2411yescharacteristic
254C2×Dic3C2411yescharacteristic
255C2×Dic3C2412yes—
256C2×Dic3C2412yes—
257C2×Dic3C2414yes—
258C2×Dic3C2411yescharacteristic
259D12—22no—
260D12—42no—
261C4×D3—22no—
262C4×D3—24no—
263C4×D3—22no—
264C4×D3—22no—
265C4×D3—44no—
266Dic6—22no—
267Dic6—24no—
268Dic6—24no—
269Dic6—44no—
270Dic6—42no—
271Dic6—44no—
272#41—124no—
273C2×Q8—32no—
274C2×Q8—32no—
275C2×Q8—31no—
276C2×Q8—34no—
277C2×Q8—34no—
278C2×Q8—32no—
279C2×Q8—124no—
280C2×Q8—124no—
281C2×Q8C22×D312yes—
282C2×Q8C22×D311yescharacteristic
283C2×D4—32no—
284C2×D4—32no—
285C2×D4—32no—
286C2×D4—31no—
287C2×D4—32no—
288C2×D4C22×D311yescharacteristic
289C22×C4—34no—
290C22×C4—31no—
291C22×C4—31no—
292C22×C4—31no—
293C22×C4—32no—
294C22×C4—34no—
295C22×C4C22×D312yes—
296C4⋊C4—31no—
297C4⋊C4—32no—
298C4⋊C4—32no—
299C4⋊C4—32no—
300C4⋊C4—32no—
301C4⋊C4—32no—
302C4⋊C4—32no—
303C4⋊C4—31no—
304C4⋊C4—32no—
305C4⋊C4—34no—
306C4⋊C4—34no—
307C4⋊C4—34no—
308C4⋊C4—34no—
309C4⋊C4—32no—
310C4⋊C4—34no—
311C4⋊C4—32no—
312C4⋊C4C22×D312yes—
313C4⋊C4C22×D312yes—
314C4⋊C4C22×D311yescharacteristic
315C4⋊C4C22×D311yescharacteristic
316C4⋊C4C22×D312yes—
317C4⋊C4C22×D314yes—
318C22⋊C4—32no—
319C22⋊C4—34no—
320C22⋊C4—32no—
321C22⋊C4—34no—
322C22⋊C4—32no—
323C22⋊C4—34no—
324C22⋊C4—32no—
325C22⋊C4—32no—
326C22⋊C4—34no—
327C22⋊C4—34no—
328C22⋊C4C22×D314yes—
329C22⋊C4C22×D314yes—
330C22⋊C4C22×D312yes—
331C42—31no—
332C42—31no—
333C42—31no—
334C42—34no—
335C42—32no—
336C42—34no—
337C42C22×D311yescharacteristic
338C42C22×D312yes—
339C42C22×D314yes—
340C22×C3—22no—
341C22×C3—22no—
342C22×C3—22no—
343C22×C3C2511yescharacteristic
344C2×D3—22no—
345C2×D3—21no—
346C2×D3—21no—
347C2×D3—21no—
348C2×D3—22no—
349C2×D3—22no—
350C3×C4—22no—
351C3×C4—22no—
352C3×C4—22no—
353C3×C4—21no—
354C3×C4—24no—
355C3×C4H(5,𝔽2)12yes—
356C3×C42−1+412yes—
357Dic3—24no—
358Dic3—22no—
359Dic3—22no—
360Dic3—21no—
361Dic3—22no—
362Dic3—21no—
363Dic32−1+412yes—
364C23—32no—
365C23—31no—
366C23C23×D312yes—
367Q8—22no—
368Q8—24no—
369Q8—64no—
370Q8—62no—
371Q8—64no—
372Q8—124no—
373Q8—124no—
374Q8—122no—
375D4—22no—
376D4—64no—
377D4—62no—
378D4—122no—
379D4—122no—
380D4—122no—
381C2×C4—24no—
382C2×C4—31no—
383C2×C4—32no—
384C2×C4—31no—
385C2×C4—32no—
386C2×C4—31no—
387C2×C4—34no—
388C2×C4—32no—
389C2×C4—62no—
390C2×C4—64no—
391C2×C4—62no—
392C2×C4—62no—
393C2×C4—124no—
394C2×C4—124no—
395C2×C4C23×D314yes—
396C2×C4C23×D311yescharacteristic
397C2×C4C23×D312yes—
398C2×C4C23×D312yes—
399C2×C4C23×D312yes—
400C2×C4C23×D311yescharacteristic
401C2×C4C23×D311yescharacteristic
402C2×C3—42no—
403C2×C3C2×H(5,𝔽2)11yescharacteristic
404C2×C3C2×2−1+411yescharacteristic
405C2×C3C2×2−1+411yescharacteristic
406D3—42no—
407D3—41no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—62no—
412C22—61no—
413C22—61no—
414C22—62no—
415C22—61no—
416C22—62no—
417C22C24×D311yescharacteristic
418C4—22no—
419C4—22no—
420C4—24no—
421C4—21no—
422C4—22no—
423C4—32no—
424C4—61no—
425C4—62no—
426C4—64no—
427C4—62no—
428C4—61no—
429C4—62no—
430C4#100912yes—
431C4#101012yes—
432C3#355011yescharacteristic, complemented, Sylow
433C2—42no—
434C2—122no—
435C2—121no—
436C2C2×#100911yescharacteristic
437C2C2×#101011yescharacteristic
438C2D3×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#556C2
prime radical#5548C2
hypercenter#556D3
hypocenterC3#3550
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #556

Complements

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

NH ≅ G/Ntype
1. C3×#556C2semidirect
2. #5732C2semidirect
3. #5728C2semidirect
4. #5712C2semidirect
5. #5710C2semidirect
6. #5682C2semidirect
7. #5681C2semidirect
8. #5632C2semidirect
9. #5628C2semidirect
10. #5582C2semidirect
11. #5578C2semidirect
12. Q8×D12C2semidirect
13. Q8×Dic6C2semidirect
15. D4×Dic6C2semidirect
16. #5533C2semidirect
17. #5526C2semidirect
89. #556D3semidirect
432. C3#3550semidirect

Maximal subgroups

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

subgroupindex
C3×#5562
#57322
#57282
#57122
#57102
#56822
#56812
#56322
#56282
#55822
#55782
Q8×D122
Q8×Dic62
#55482
D4×Dic62
#55332
#55262
#35503