Finite Group Databasebeta

← #80186

Subgroups — #80186

Order 384.

712 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1007C211yesmaximal, characteristic, complemented
2C2×#919C211yesmaximal, characteristic, complemented
3C2×#918C211yesmaximal, characteristic, complemented
4#5629C211yesmaximal, characteristic, complemented
5C3×#451C211yesmaximal, characteristic, complemented
6#5227C211yesmaximal, characteristic, complemented
7#5187C211yesmaximal, characteristic, complemented
8#5186C211yesmaximal, characteristic, complemented
9#5047C211yesmaximal, characteristic, complemented
10#4810C211yesmaximal, characteristic, complemented
11#4806C211yesmaximal, characteristic, complemented
12#4803C211yesmaximal, characteristic, complemented
13D3×#103C211yesmaximal, characteristic, complemented
14#4734C211yesmaximal, characteristic, complemented
15#4727C211yesmaximal, characteristic, complemented
16#3013—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18#1007—24no—
19#1007—24no—
20C2×#239—21no—
21C2×#239C2211yescharacteristic
22C2×#239C2211yescharacteristic
23C2×#239C2211yescharacteristic
24C2×D3×Q8—21no—
25C2×D3×Q8C2211yescharacteristic
26C2×D3×Q8C2211yescharacteristic
27C2×#235—21no—
28C2×#235—21no—
29C2×#235C2211yescharacteristic
30C2×#235C2211yescharacteristic
31C2×C3×Q16C2211yescharacteristic
32C2×C3×Q16C2211yescharacteristic
33C3×C22⋊Q8C2211yescharacteristic
34#946C2211yescharacteristic
35C2×#216C2211yescharacteristic
36C2×#215C2211yescharacteristic
37C2×#215C2211yescharacteristic
38C2×#215C2211yescharacteristic
39#930C2211yescharacteristic
40#919—24no—
41#918—24no—
42C2×D24C2211yescharacteristic
43C2×#204C2211yescharacteristic
44C2×ΓL(1,𝔽25)C2211yescharacteristic
45C2×D3×C8C2211yescharacteristic
46#896—21no—
47#895—21no—
48#895C2211yescharacteristic
49D3×C4⋊C4C2211yescharacteristic
50#884—21no—
51#884—21no—
52C3×#103C2211yescharacteristic
53C3×#103C2211yescharacteristic
54C3×C22⋊C8C2211yescharacteristic
55#835C2211yescharacteristic
56#835C2211yescharacteristic
57#830C2211yescharacteristic
58#821C2211yescharacteristic
59(C2×D3)⋊C8C2211yescharacteristic
60(C2×D3)⋊C8C2211yescharacteristic
61#816C2211yescharacteristic
62#810C2211yescharacteristic
63#809C2211yescharacteristic
64C2×2−1+4—31no—
65C2×#137—31no—
66C2×#135—31no—
67#537—31no—
68#452—31no—
69#452—31no—
70#451—31no—
71#451D311yescharacteristic, complemented
72#449—31no—
73#449—31no—
74#448—31no—
75#448—31no—
76#417—31no—
77C2×#103—31no—
78#408—31no—
79C2×C3×Q8—22no—
80C2×C3×Q8—22no—
81C2×C3×Q8—21no—
82C2×C3×Q8C2311yescharacteristic
83C2×C3×Q8C2311yescharacteristic
84C2×C3×Q8C2311yescharacteristic
85C22×C3×C4—21no—
86C22×C3×C4C2311yescharacteristic
87C2×#67—21no—
88C2×#67—21no—
89C2×#67D411yescharacteristic
90C2×#67D411yescharacteristic
91#239—28no—
92#239—22no—
93#239—22no—
94#239—22no—
95#239—22no—
96#239—22no—
97#239D414yescomplemented
98D3×Q8—28no—
99D3×Q8—22no—
100D3×Q8—22no—
101D3×Q8—22no—
102D3×Q8D414yescomplemented
103#235—28no—
104#235—28no—
105#235—22no—
106#235—22no—
107#235—22no—
108#235—22no—
109C2×D12—21no—
110C2×D12—21no—
111C2×D12—21no—
112C2×D12—21no—
113C2×D12D411yescharacteristic
114C2×D12C2311yescharacteristic
115C2×D12C2311yescharacteristic
116C2×C4×D3—21no—
117C2×C4×D3—21no—
118C2×C4×D3—21no—
119C2×C4×D3—21no—
120C2×C4×D3—21no—
121C2×C4×D3—21no—
122C2×C4×D3—21no—
123C2×C4×D3C2311yescharacteristic
124C2×C4×D3C2311yescharacteristic
125C2×Dic6—21no—
126C2×Dic6—21no—
127C2×Dic6D411yescharacteristic
128C2×Dic6C2311yescharacteristic
129C3×Q16—22no—
130C3×Q16—22no—
131C2×C3×C8C2311yescharacteristic
132C2×C3×C8C2311yescharacteristic
133C3×C4⋊C4—21no—
134C3×C4⋊C4C2311yescharacteristic
135C3×C22⋊C4—21no—
136#216—22no—
137#215—22no—
138#215—22no—
139#215—22no—
140(C2×D3)⋊C4—21no—
141(C2×D3)⋊C4—21no—
142(C2×D3)⋊C4—21no—
143C4⋊Dic3C2311yescharacteristic
144Dic3⋊C4—21no—
145Dic3⋊C4—21no—
146C2×C3⋊C8C2311yescharacteristic
147C2×C3⋊C8C2311yescharacteristic
148D24—22no—
149#204—22no—
150ΓL(1,𝔽25)—22no—
151D3×C8—22no—
1522−1+4—64no—
1532−1+4—64no—
154C2×#41—31no—
155C2×#41—31no—
156C2×#41—31no—
157C2×#41—31no—
158C2×#41—31no—
159C2×#41—61no—
160C2×#41—61no—
161C2×#41—61no—
162C22×Q8—31no—
163C22×Q8—31no—
164C22×Q8—61no—
165C22×Q8C2×D311yescharacteristic
166#137—64no—
167#135—64no—
168C2×Q16—31no—
169C2×Q16C2×D311yescharacteristic
170C2×Q16C2×D311yescharacteristic
171C2×SD16—31no—
172C2×SD16—31no—
173C2×SD16—31no—
174C2×SD16—31no—
175C2×D8—31no—
176C2×M16—31no—
177C22×C8—31no—
178C22⋊Q8—31no—
179C22⋊Q8—61no—
180C22⋊Q8C2×D311yescharacteristic
181#121—31no—
182#121—31no—
183#121—61no—
184#121—61no—
185#121—61no—
186C2×C4⋊C4—31no—
187#103—31no—
188#103—31no—
189#103—31no—
190#103—31no—
191#103C2×D311yescharacteristic
192#103C2×D311yescharacteristic
193#102—31no—
194#102—31no—
195C22⋊C8—31no—
196C22⋊C8—31no—
197C22⋊C8—31no—
198C22⋊C8C2×D311yescharacteristic
199C23×C3C2×D411yescharacteristic
200C22×D3—21no—
201C22×D3—21no—
202C22×D3C2×D411yescharacteristic
203C22×D3C2×D411yescharacteristic
204C3×Q8—24no—
205C3×Q8—21no—
206C3×Q8—21no—
207C3×Q8—21no—
208C3×Q8—21no—
209C3×Q8C2×D412yes—
210C3×Q8C2×D412yes—
211C2×C3×C4—24no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4C2×D411yescharacteristic
219C2×C3×C4C2411yescharacteristic
220#67—24no—
221#67—22no—
222#67—22no—
223#67—81no—
224C2×Dic3—21no—
225C2×Dic3—21no—
226C2×Dic3C2×D411yescharacteristic
227C2×Dic3C2×D411yescharacteristic
228D12—24no—
229D12—24no—
230D12—22no—
231D12—21no—
232D12—21no—
233D12—21no—
234D12—42no—
235D12—81no—
236D12C2×D412yes—
237C4×D3—24no—
238C4×D3—24no—
239C4×D3—24no—
240C4×D3—24no—
241C4×D3—21no—
242C4×D3—21no—
243C4×D3—24no—
244C4×D3—42no—
245C4×D3—81no—
246C4×D3C2×D412yes—
247C4×D3C2×D412yes—
248Dic6—24no—
249Dic6—21no—
250Dic6—24no—
251Dic6—22no—
252Dic6C2×D412yes—
253C3×C8—21no—
254C3×C8—21no—
255C3⋊C8—21no—
256C3⋊C8—21no—
257#41—34no—
258#41—62no—
259#41—62no—
260#41—62no—
261#41—62no—
262#41—68no—
263#41—68no—
264#41—62no—
265#41—62no—
266#41—62no—
267#41—62no—
268#41—68no—
269#41—62no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—21no—
273C2×Q8—31no—
274C2×Q8—34no—
275C2×Q8—31no—
276C2×Q8—61no—
277C2×Q8—61no—
278C2×Q8—62no—
279C2×Q8—62no—
280C2×Q8—68no—
281C2×Q8—62no—
282C2×Q8C22×D311yescharacteristic
283C2×Q8C22×D311yescharacteristic
284C2×Q8C22×D311yescharacteristic
285C2×D4—31no—
286C2×D4—31no—
287C2×D4—31no—
288C2×D4—31no—
289C2×D4—31no—
290C2×D4—61no—
291C2×D4—61no—
292C2×D4—61no—
293C2×D4—61no—
294C2×D4—61no—
295C2×D4—61no—
296C22×C4—21no—
297C22×C4—31no—
298C22×C4—31no—
299C22×C4—61no—
300C22×C4—61no—
301C22×C4—61no—
302C22×C4—61no—
303C22×C4—61no—
304C22×C4—61no—
305C22×C4—61no—
306C22×C4C22×D311yescharacteristic
307Q16—22no—
308Q16—22no—
309Q16—62no—
310SD16—62no—
311SD16—62no—
312SD16—62no—
313SD16—62no—
314D8—62no—
315M16—62no—
316C2×C8—31no—
317C2×C8—31no—
318C2×C8—62no—
319C2×C8C22×D311yescharacteristic
320C2×C8C22×D311yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—31no—
323C4⋊C4—61no—
324C4⋊C4—61no—
325C4⋊C4C22×D311yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—61no—
328C22⋊C4—61no—
329C22⋊C4—61no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3C22×D411yescharacteristic
334C2×D3—21no—
335C2×D3—21no—
336C2×D3—21no—
337C2×D3—22no—
338C2×D3—21no—
339C2×D3—21no—
340C2×D3—41no—
341C2×D3—41no—
342C2×D3—41no—
343C2×D3—41no—
344C2×D3—41no—
345C2×D3#12012yes—
346C3×C4—22no—
347C3×C4—22no—
348C3×C4—21no—
349C3×C4—41no—
350C3×C4—41no—
351C3×C4C22×D411yescharacteristic
352C3×C4C22×D411yescharacteristic
353Dic3—21no—
354Dic3—22no—
355Dic3—41no—
356Dic3#12012yes—
357C23—31no—
358C23—31no—
359C23—61no—
360C23—61no—
361C23D3×D411yescharacteristic
362Q8—24no—
363Q8—21no—
364Q8—21no—
365Q8—21no—
366Q8—21no—
367Q8—32no—
368Q8—64no—
369Q8—62no—
370Q8—64no—
371Q8—61no—
372Q8D3×D412yes—
373Q8D3×D412yes—
374D4—32no—
375D4—62no—
376D4—64no—
377D4—64no—
378D4—62no—
379D4—62no—
380D4—64no—
381D4—61no—
382D4—61no—
383D4—61no—
384D4—122no—
385D4—241no—
386D4—241no—
387C2×C4—24no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—31no—
395C2×C4—32no—
396C2×C4—32no—
397C2×C4—31no—
398C2×C4—64no—
399C2×C4—64no—
400C2×C4—64no—
401C2×C4—64no—
402C2×C4—64no—
403C2×C4—61no—
404C2×C4—61no—
405C2×C4—61no—
406C2×C4—61no—
407C2×C4—122no—
408C2×C4—241no—
409C2×C4D3×D411yescharacteristic
410C2×C4C23×D311yescharacteristic
411C8—21no—
412C8—21no—
413C8—61no—
414C8—61no—
415C2×C3—41no—
416C2×C3C2×#12011yescharacteristic
417C2×C3#57711yescharacteristic
418C2×C3#57811yescharacteristic
419D3—22no—
420D3—41no—
421D3—42no—
422D3—81no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—32no—
427C22—61no—
428C22—61no—
429C22—61no—
430C22—61no—
431C22—62no—
432C22—61no—
433C22—121no—
434C22—121no—
435C22—121no—
436C22—121no—
437C22—121no—
438C22C2×D3×D411yescharacteristic
439C4—22no—
440C4—22no—
441C4—21no—
442C4—32no—
443C4—41no—
444C4—41no—
445C4—61no—
446C4—62no—
447C4—121no—
448C4C2×D3×D411yescharacteristic
449C4C2×D3×D411yescharacteristic
450C3#301311yescharacteristic, complemented, Sylow
451C2—41no—
452C2—62no—
453C2—121no—
454C2—122no—
455C2—241no—
456C2D3×#12011yescharacteristic
457C2#576611yescharacteristic
458C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#451C2
socleC22×C3C22×D4
hypercenter#451D3
hypocenterC3#3013
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C2×#1007C2semidirect
2. C2×#919C2semidirect
3. C2×#918C2semidirect
4. #5629C2semidirect
5. C3×#451C2semidirect
6. #5227C2semidirect
7. #5187C2semidirect
8. #5186C2semidirect
9. #5047C2semidirect
10. #4810C2semidirect
11. #4806C2semidirect
12. #4803C2semidirect
13. D3×#103C2semidirect
14. #4734C2semidirect
15. #4727C2semidirect
71. #451D3semidirect
97. #239D4semidirect
102. D3×Q8D4semidirect
450. C3#3013semidirect

Maximal subgroups

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

subgroupindex
C2×#10072
C2×#9192
C2×#9182
#56292
C3×#4512
#52272
#51872
#51862
#50472
#48102
#48062
#48032
D3×#1032
#47342
#47272
#30133