Finite Group Databasebeta

← #86679

Subgroups — #86679

Order 384.

1476 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1010C212yesmaximal, complemented
2C3×#547C211yesmaximal, characteristic, complemented
3#5824C212yesmaximal, complemented
4#5727C211yesmaximal, characteristic, complemented
5D3×C4⋊Q8C211yesmaximal, characteristic, complemented
6#5655C214yesmaximal, complemented
7#5648C214yesmaximal, complemented
8#5632C212yesmaximal, complemented
9#5628C212yesmaximal, complemented
10#5611C212yesmaximal, complemented
11#5596C212yesmaximal, complemented
12#5555C211yesmaximal, characteristic, complemented
13#5548C211yesmaximal, characteristic, complemented
14D4×Dic6C211yesmaximal, characteristic, complemented
15C4×#237C211yesmaximal, characteristic, complemented
16#5507C212yesmaximal, complemented
17C2×#888C212yesmaximal, complemented
18#3499—31nomaximal, Sylow
19C2×C3×#41C2212yes—
20#1010—216no—
21C2×D3×Q8C2212yes—
22C2×D3×Q8C2212yes—
23C2×#237C2214yes—
24C2×#237C2214yes—
25C2×#237C2212yes—
26C2×#237C2211yescharacteristic
27C2×#235C2214yes—
28C2×#235C2212yes—
29C22×Dic6C2212yes—
30C22×Dic6C2214yes—
31C3×C4⋊Q8C2211yescharacteristic
32C3×#123C2214yes—
33C3×C22⋊Q8C2212yes—
34C3×#121C2212yes—
35C3×C4×D4C2211yescharacteristic
36C3×C4×D4C2211yescharacteristic
37C2×C3×C4⋊C4C2212yes—
38#946C2212yes—
39#944C2212yes—
40#940C2211yescharacteristic
41#939C2214yes—
42#938C2211yescharacteristic
43#935C2214yes—
44D4×Dic3C2211yescharacteristic
45D4×Dic3C2212yes—
46#930C2212yes—
47#929C2214yes—
48C4×#67C2212yes—
49C4×#67C2212yes—
50C2×C4⋊Dic3C2212yes—
51#924C2212yes—
52C2×Dic3⋊C4C2214yes—
53C2×C4×Dic3C2212yes—
54#897C2212yes—
55#896C2214yes—
56#896C2214yes—
57#895C2211yescharacteristic
58#894C2214yes—
59#893C2212yes—
60#892C2211yescharacteristic
61D3×C4⋊C4C2212yes—
62D3×C4⋊C4C2212yes—
63#888C2214yes—
64#888C2212yes—
65#888C2214yes—
66#888C2212yes—
67#888C2212yes—
68#888C2211yescharacteristic
69#888C2212yes—
70#887C2211yescharacteristic
71#886C2214yes—
72#885C2212yes—
73#885C2214yes—
74#884C2212yes—
75#883C2214yes—
76#67⋊C4C2214yes—
77#67⋊C4C2212yes—
78#878C2214yes—
79#878C2214yes—
80#877C2212yes—
81C4×D12C2211yescharacteristic
82C42×D3C2211yescharacteristic
83#869C2211yescharacteristic
84C4×Dic6C2211yescharacteristic
85C2×2−1+4—32no—
86D4×Q8—31no—
87D4×Q8—32no—
88D4×Q8—31no—
89#547—32no—
90#547—34no—
91#547—32no—
92#547—31no—
93#547—32no—
94#547D311yescharacteristic, complemented
95#536—34no—
96#536—32no—
97#532—32no—
98#532—31no—
99C2×C4⋊Q8—31no—
100C2×C4⋊Q8—32no—
101C4×#41—31no—
102C3×#41—28no—
103C2×C3×Q8C2312yes—
104C2×C3×D4C2311yescharacteristic
105C2×C3×D4C2314yes—
106C2×C3×D4C2311yescharacteristic
107C22×C3×C4C2312yes—
108C22×C3×C4C2314yes—
109C22×C3×C4C2312yes—
110C2×#67C2312yes—
111C2×#67C2312yes—
112C2×#67C2312yes—
113C2×#67C2314yes—
114C2×#67C2312yes—
115C22×Dic3C2314yes—
116C22×Dic3C2312yes—
117C22×Dic3C2314yes—
118C22×Dic3C2312yes—
119D3×Q8—28no—
120D3×Q8—28no—
121#237—28no—
122#237—216no—
123#237—216no—
124#237D418yescomplemented
125#235—216no—
126#235—28no—
127C2×D12C2311yescharacteristic
128C2×D12C2312yes—
129C2×C4×D3C2311yescharacteristic
130C2×C4×D3C2311yescharacteristic
131C2×C4×D3C2312yes—
132C2×C4×D3C2314yes—
133C2×C4×D3C2314yes—
134C2×C4×D3C2311yescharacteristic
135C2×Dic6—216no—
136C2×Dic6—28no—
137C2×Dic6C2312yes—
138C2×Dic6C2314yes—
139C2×Dic6C2314yes—
140C2×Dic6C2312yes—
141C2×Dic6C2314yes—
142C2×Dic6C2311yescharacteristic
143C3×C4⋊C4C2311yescharacteristic
144C3×C4⋊C4C2311yescharacteristic
145C3×C4⋊C4C2312yes—
146C3×C4⋊C4C2314yes—
147C3×C4⋊C4C2312yes—
148C3×C22⋊C4C2312yes—
149C3×C22⋊C4C2312yes—
150C3×C22⋊C4C2314yes—
151C3×C42C2311yescharacteristic
152#217C2312yes—
153#217C2312yes—
154#217C2314yes—
155(C2×D3)⋊C4C2312yes—
156(C2×D3)⋊C4C2312yes—
157(C2×D3)⋊C4C2314yes—
158(C2×D3)⋊C4C2314yes—
159(C2×D3)⋊C4C2312yes—
160C4⋊Dic3C2311yescharacteristic
161C4⋊Dic3C2312yes—
162C4⋊Dic3C2312yes—
163C4⋊Dic3C2311yescharacteristic
164C4⋊Dic3C2314yes—
165Dic3⋊C4C2312yes—
166Dic3⋊C4C2312yes—
167Dic3⋊C4C2314yes—
168Dic3⋊C4C2314yes—
169Dic3⋊C4C2314yes—
170Dic3⋊C4C2314yes—
171Dic3⋊C4C2312yes—
172C4×Dic3C2312yes—
173C4×Dic3C2311yescharacteristic
174C4×Dic3C2311yescharacteristic
175C4×Dic3C2311yescharacteristic
176C4×Dic3C2312yes—
177C4×Dic3C2312yes—
1782−1+4—616no—
179C2×#41—31no—
180C2×#41—34no—
181C2×#41—32no—
182C2×#41—34no—
183C2×#41—32no—
184C2×#41—34no—
185C2×#41C2×D312yes—
186C22×Q8—32no—
187C22×Q8—32no—
188C22×Q8—34no—
189C22×Q8—32no—
190C4⋊Q8—32no—
191C4⋊Q8—32no—
192C4⋊Q8—31no—
193C4⋊Q8—32no—
194C4⋊Q8—34no—
195C4⋊Q8—34no—
196C4⋊Q8—32no—
197C4⋊Q8—32no—
198C4⋊Q8—31no—
199C4⋊Q8C2×D311yescharacteristic
200#127—31no—
201#124—34no—
202#124—32no—
203#123—34no—
204#123—34no—
205#123—34no—
206#123—34no—
207#123—34no—
208#123C2×D314yes—
209C22⋊Q8—32no—
210C22⋊Q8—32no—
211C22⋊Q8—34no—
212C22⋊Q8—34no—
213C22⋊Q8—32no—
214C22⋊Q8—34no—
215C22⋊Q8—34no—
216C22⋊Q8C2×D312yes—
217#121—31no—
218#121—34no—
219#121—32no—
220#121—31no—
221#121—32no—
222#121C2×D312yes—
223C4×Q8—31no—
224C4×Q8—31no—
225C4×D4—31no—
226C4×D4—32no—
227C4×D4—32no—
228C4×D4—32no—
229C4×D4—32no—
230C4×D4—32no—
231C4×D4—34no—
232C4×D4—31no—
233C4×D4C2×D311yescharacteristic
234C4×D4C2×D311yescharacteristic
235#117—31no—
236#117—32no—
237C2×C4⋊C4—34no—
238C2×C4⋊C4—32no—
239C2×C4⋊C4—32no—
240C2×C4⋊C4—32no—
241C2×C4⋊C4C2×D312yes—
242C2×C42—32no—
243C2×C42—31no—
244C23×C3C2412yes—
245C23×C3C2412yes—
246C22×D3C2412yes—
247C22×D3C2411yescharacteristic
248C3×Q8—24no—
249C3×D4—28no—
250C3×D4—22no—
251C3×D4C2×D414yes—
252C2×C3×C4—28no—
253C2×C3×C4—24no—
254C2×C3×C4—24no—
255C2×C3×C4C2412yes—
256C2×C3×C4C2411yescharacteristic
257C2×C3×C4C2412yes—
258C2×C3×C4C2414yes—
259C2×C3×C4C2411yescharacteristic
260C2×C3×C4C2411yescharacteristic
261#67—28no—
262#67—24no—
263#67—24no—
264#67—24no—
265#67C2×D418yes—
266C2×Dic3—28no—
267C2×Dic3—28no—
268C2×Dic3—24no—
269C2×Dic3C2×D418yes—
270C2×Dic3C2411yescharacteristic
271C2×Dic3C2412yes—
272C2×Dic3C2414yes—
273C2×Dic3C2412yes—
274C2×Dic3C2411yescharacteristic
275C2×Dic3C2412yes—
276C2×Dic3C2411yescharacteristic
277D12—24no—
278D12—22no—
279C4×D3—28no—
280C4×D3—24no—
281C4×D3—22no—
282C4×D3—28no—
283C4×D3—22no—
284C4×D3C2×D414yes—
285Dic6—24no—
286Dic6—24no—
287Dic6—28no—
288Dic6—28no—
289Dic6—28no—
290Dic6C2×D414yes—
291#41—28no—
292#41—38no—
293#41—68no—
294#41—616no—
295#41—616no—
296#41—616no—
297#41—68no—
298C2×Q8—31no—
299C2×Q8—34no—
300C2×Q8—34no—
301C2×Q8—32no—
302C2×Q8—34no—
303C2×Q8—32no—
304C2×Q8—616no—
305C2×Q8—68no—
306C2×Q8—68no—
307C2×Q8—68no—
308C2×Q8C22×D312yes—
309C2×D4—32no—
310C2×D4—32no—
311C2×D4—32no—
312C2×D4—34no—
313C2×D4—32no—
314C2×D4—31no—
315C2×D4—32no—
316C2×D4C22×D311yescharacteristic
317C2×D4C22×D314yes—
318C2×D4C22×D311yescharacteristic
319C22×C4—32no—
320C22×C4—31no—
321C22×C4—32no—
322C22×C4—34no—
323C22×C4—34no—
324C22×C4—31no—
325C22×C4—31no—
326C22×C4—34no—
327C22×C4—34no—
328C22×C4—32no—
329C22×C4C22×D312yes—
330C22×C4C22×D314yes—
331C22×C4C22×D312yes—
332C4⋊C4—34no—
333C4⋊C4—32no—
334C4⋊C4—31no—
335C4⋊C4—32no—
336C4⋊C4—34no—
337C4⋊C4—34no—
338C4⋊C4—34no—
339C4⋊C4—34no—
340C4⋊C4—32no—
341C4⋊C4—32no—
342C4⋊C4—32no—
343C4⋊C4—31no—
344C4⋊C4C22×D312yes—
345C4⋊C4C22×D312yes—
346C4⋊C4C22×D311yescharacteristic
347C4⋊C4C22×D314yes—
348C4⋊C4C22×D311yescharacteristic
349C22⋊C4—32no—
350C22⋊C4—34no—
351C22⋊C4—32no—
352C22⋊C4—32no—
353C22⋊C4—34no—
354C22⋊C4—32no—
355C22⋊C4—34no—
356C22⋊C4—32no—
357C22⋊C4C22×D312yes—
358C22⋊C4C22×D314yes—
359C22⋊C4C22×D312yes—
360C42—31no—
361C42—32no—
362C42—31no—
363C42—32no—
364C42—31no—
365C42—32no—
366C42C22×D311yescharacteristic
367C22×C3—22no—
368C22×C3—22no—
369C22×C3—22no—
370C22×C3—22no—
371C22×C3—22no—
372C22×C3C22×D414yes—
373C22×C3C2511yescharacteristic
374C2×D3—21no—
375C2×D3—21no—
376C2×D3—22no—
377C2×D3—22no—
378C2×D3—22no—
379C2×D3C22×D412yes—
380C3×C4—22no—
381C3×C4—24no—
382C3×C4—21no—
383C3×C4—22no—
384C3×C4C22×D412yes—
385C3×C42−1+412yescomplemented
386Dic3—24no—
387Dic3—22no—
388Dic3—21no—
389Dic3—22no—
390Dic3C22×D412yes—
391Dic3C22×D414yes—
392Dic32−1+412yescomplemented
393C23—31no—
394C23—32no—
395C23C23×D312yes—
396C23C23×D312yes—
397Q8—24no—
398Q8—34no—
399Q8—64no—
400Q8—68no—
401Q8—68no—
402Q8—68no—
403Q8—64no—
404D4—28no—
405D4—22no—
406D4—38no—
407D4—64no—
408D4—64no—
409D4—64no—
410D4—64no—
411D4—68no—
412D4—62no—
413D4D3×D414yes—
414C2×C4—28no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—38no—
418C2×C4—31no—
419C2×C4—31no—
420C2×C4—32no—
421C2×C4—34no—
422C2×C4—31no—
423C2×C4—32no—
424C2×C4—32no—
425C2×C4—34no—
426C2×C4—68no—
427C2×C4—62no—
428C2×C4—68no—
429C2×C4—68no—
430C2×C4—64no—
431C2×C4—68no—
432C2×C4—64no—
433C2×C4—62no—
434C2×C4C23×D312yes—
435C2×C4C23×D311yescharacteristic
436C2×C4C23×D314yes—
437C2×C4C23×D312yes—
438C2×C4C23×D311yescharacteristic
439C2×C4C23×D311yescharacteristic
440C2×C3—24no—
441C2×C3—42no—
442C2×C3C23×D411yescharacteristic
443C2×C3C2×2−1+411yescharacteristic
444C2×C3C2×2−1+411yescharacteristic
445D3—22no—
446D3—42no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—32no—
453C22—61no—
454C22—61no—
455C22—62no—
456C22—62no—
457C22—62no—
458C22C2×D3×D414yes—
459C22C24×D311yescharacteristic
460C4—22no—
461C4—24no—
462C4—22no—
463C4—21no—
464C4—32no—
465C4—34no—
466C4—32no—
467C4—64no—
468C4—62no—
469C4—61no—
470C4—62no—
471C4C2×D3×D412yes—
472C4#101012yescomplemented
473C3#349911yescharacteristic, complemented, Sylow
474C2—24no—
475C2—42no—
476C2—62no—
477C2—122no—
478C2C22×D3×D411yescharacteristic
479C2C2×#101011yescharacteristic
480C2D3×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#547C2
hypercenter#547D3
hypocenterC3#3499
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#1010C2semidirect
2. C3×#547C2semidirect
3. #5824C2semidirect
4. #5727C2semidirect
5. D3×C4⋊Q8C2semidirect
6. #5655C2semidirect
7. #5648C2semidirect
8. #5632C2semidirect
9. #5628C2semidirect
10. #5611C2semidirect
11. #5596C2semidirect
12. #5555C2semidirect
13. #5548C2semidirect
14. D4×Dic6C2semidirect
15. C4×#237C2semidirect
16. #5507C2semidirect
17. C2×#888C2semidirect
94. #547D3semidirect
124. #237D4semidirect
385. C3×C42−1+4semidirect
392. Dic32−1+4semidirect
472. C4#1010semidirect
473. C3#3499semidirect

Maximal subgroups

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

subgroupindex
C2×#10102
C3×#5472
#58242
#57272
D3×C4⋊Q82
#56552
#56482
#56322
#56282
#56112
#55962
#55552
#55482
D4×Dic62
C4×#2372
#55072
C2×#8882
#34993