Finite Group Databasebeta

← #10594071

Subgroups — #10594071

Order 576.

597 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#627C211yesmaximal, characteristic, complemented
2C2×D3×Dic6C211yesmaximal, characteristic, complemented
3#64087C211yesmaximal, characteristic, complemented
4C3×#67⋊C4C211yesmaximal, characteristic
5C3×#878C211yesmaximal, characteristic, complemented
6#67⋊Dic3C211yesmaximal, characteristic
7Dic3×#67C211yesmaximal, characteristic
8#63964C211yesmaximal, characteristic, complemented
9#63918C211yesmaximal, characteristic, complemented
10#63899C211yesmaximal, characteristic, complemented
11#63893C211yesmaximal, characteristic, complemented
12#63851C211yesmaximal, characteristic, complemented
13#63840C211yesmaximal, characteristic, complemented
14#63839C211yesmaximal, characteristic, complemented
15#63837C211yesmaximal, characteristic, complemented
16#5632—31nomaximal
17D4×Dic6—31nomaximal
18C22×C3⋊Dic3C2211yescharacteristic
19C2×#634C2211yescharacteristic
20C2×C3×#67C2211yescharacteristic
21C22×C3×Dic3C2211yescharacteristic
22C22×C3×Dic3C2211yescharacteristic
23C2×C3×C4×D3C2211yescharacteristic
24C2×C3×Dic6C2211yescharacteristic
25C2×#627—24no—
26C2×#627C2211yescharacteristic
27C2×#627C2211yescharacteristic
28C2×#627C2211yescharacteristic
29C2×#627C2211yescharacteristic
30C2×#627C2211yescharacteristic
31C2×D3×Dic3C2211yescharacteristic
32C2×D3×Dic3C2211yescharacteristic
33D3×Dic6—24no—
34C32×C22⋊C4C2211yescharacteristic
35#3757C2211yescharacteristic
36C4⋊(C3⋊Dic3)C2211yescharacteristic
37#3750C2211yescharacteristic
38#3750C2211yescharacteristic
39C3×#217C2211yescharacteristic
40C3×(C2×D3)⋊C4C2211yescharacteristic
41C3×C4⋊Dic3C2211yescharacteristic
42C3×Dic3⋊C4C2211yescharacteristic
43C3×Dic3⋊C4C2211yescharacteristic
44C3×Dic3⋊C4C2211yescharacteristic
45C3×C4×Dic3C2211yescharacteristic
46#3724C2211yescharacteristic
47Dic3⋊Dic3C2211yescharacteristic
48Dic3⋊Dic3C2211yescharacteristic
49Dic3⋊Dic3C2211yescharacteristic
50Dic3⋊Dic3C2211yescharacteristic
51(C2×D3)⋊Dic3C2211yescharacteristic
52(C2×D3)⋊Dic3C2211yescharacteristic
53Dic32C2211yescharacteristic
54Dic32C2211yescharacteristic
55C2×D3×Q8—31no—
56C22×Dic6—31no—
57C22×Dic6—31no—
58C22×Dic6—31no—
59C3×C22⋊Q8—31no—
60C3×C4×D4—31no—
61#944—31no—
62D4×Dic3—31no—
63D4×Dic3—31no—
64C4×#67—31no—
65#924—31no—
66#924—31no—
67#897—31no—
68#896—31no—
69#896—31no—
70#888—31no—
71#888—31no—
72#888—31no—
73#888—31no—
74#887—31no—
75#67⋊C4—31no—
76#67⋊C4D311yescharacteristic
77#878—31no—
78#878—31no—
79#878—31no—
80#878—31no—
81#878—31no—
82#878—61no—
83#878D311yescharacteristic, complemented
84#869—31no—
85C4×Dic6—31no—
86C23×C32C2311yescharacteristic
87C22×C3×D3C2311yescharacteristic
88C2×C32×C4C2311yescharacteristic
89C2×C32×C4C2311yescharacteristic
90C2×C3⋊Dic3—22no—
91C2×C3⋊Dic3C2311yescharacteristic
92C2×C3⋊Dic3C2311yescharacteristic
93C2×C3⋊Dic3C2311yescharacteristic
94C2×C3⋊Dic3C2311yescharacteristic
95#634—22no—
96C3×#67Q814yescomplemented
97C2×C3×Dic3—22no—
98C2×C3×Dic3—22no—
99C2×C3×Dic3C2311yescharacteristic
100C2×C3×Dic3C2311yescharacteristic
101C2×C3×Dic3C2311yescharacteristic
102C2×C3×Dic3C2311yescharacteristic
103C2×C3×Dic3C2311yescharacteristic
104C2×C3×Dic3C2311yescharacteristic
105C2×C3×Dic3C2311yescharacteristic
106C3×C4×D3—22no—
107C3×Dic6—22no—
108#627—22no—
109#627—22no—
110#627—22no—
111#627—22no—
112#627D414yescomplemented
113D3×Dic3—22no—
114D3×Dic3—22no—
115D4×Q8—91noSylow
116C2×C3×Q8—31no—
117C2×C3×D4—31no—
118C22×C3×C4—31no—
119C22×C3×C4—31no—
120C22×C3×C4—31no—
121C2×#67C2×D311yescharacteristic
122C22×Dic3—31no—
123C22×Dic3—31no—
124C22×Dic3—31no—
125C22×Dic3—31no—
126C22×Dic3—61no—
127C22×Dic3C2×D311yescharacteristic
128C22×Dic3C2×D311yescharacteristic
129D3×Q8—64no—
130C2×C4×D3—31no—
131C2×C4×D3—31no—
132C2×C4×D3C2×D311yescharacteristic
133C2×Dic6—31no—
134C2×Dic6—31no—
135C2×Dic6—31no—
136C2×Dic6—31no—
137C2×Dic6—31no—
138C2×Dic6—31no—
139C2×Dic6—31no—
140C2×Dic6—31no—
141C2×Dic6—31no—
142C2×Dic6—31no—
143C2×Dic6—31no—
144C2×Dic6—31no—
145C2×Dic6—61no—
146C2×Dic6—64no—
147C2×Dic6—64no—
148C2×Dic6—64no—
149C2×Dic6C2×D311yescharacteristic
150C3×C4⋊C4—31no—
151C3×C4⋊C4—31no—
152C3×C4⋊C4—31no—
153C3×C4⋊C4—31no—
154C3×C22⋊C4—21no—
155C3×C22⋊C4—31no—
156C3×C22⋊C4—31no—
157C3×C22⋊C4C2×D311yescharacteristic
158C3×C22⋊C4C2×D311yescharacteristic
159C3×C42—31no—
160#217—31no—
161#217—31no—
162#217—31no—
163#217—31no—
164#217—61no—
165#217C2×D311yescharacteristic
166(C2×D3)⋊C4—31no—
167(C2×D3)⋊C4—31no—
168(C2×D3)⋊C4C2×D311yescharacteristic
169C4⋊Dic3—31no—
170C4⋊Dic3—31no—
171C4⋊Dic3—31no—
172C4⋊Dic3—31no—
173C4⋊Dic3—31no—
174C4⋊Dic3—31no—
175C4⋊Dic3—61no—
176C4⋊Dic3C2×D311yescharacteristic
177Dic3⋊C4—31no—
178Dic3⋊C4—31no—
179Dic3⋊C4—31no—
180Dic3⋊C4—31no—
181Dic3⋊C4—31no—
182Dic3⋊C4—31no—
183Dic3⋊C4—31no—
184Dic3⋊C4—31no—
185Dic3⋊C4—31no—
186Dic3⋊C4—31no—
187Dic3⋊C4—61no—
188Dic3⋊C4—61no—
189Dic3⋊C4C2×D311yescharacteristic
190Dic3⋊C4C2×D311yescharacteristic
191Dic3⋊C4C2×D311yescharacteristic
192C4×Dic3—31no—
193C4×Dic3—31no—
194C4×Dic3—31no—
195C4×Dic3—31no—
196C4×Dic3C2×D311yescharacteristic
197C22×C32—21no—
198C22×C32—21no—
199C22×C32C2×Q812yescomplemented
200C22×C32C2411yescharacteristic
201C2×C3×D3—21no—
202C2×C3×D3—21no—
203C2×C3×D3C2×Q812yescomplemented
204C32×C4—21no—
205C32×C4—21no—
206C3⋊Dic3—21no—
207C3⋊Dic3—21no—
208C3⋊Dic3—21no—
209C3⋊Dic3C2×D412yes—
210C3×Dic3—21no—
211C3×Dic3—21no—
212C3×Dic3—21no—
213C3×Dic3—21no—
214C3×Dic3C2×D412yes—
215C3×Dic3C2×D412yes—
216C3×Dic3C2×Q812yescomplemented
217C22×Q8—91no—
218C22×Q8—91no—
219C4⋊Q8—91no—
220C4⋊Q8—91no—
221C4⋊Q8—91no—
222C22⋊Q8—31no—
223C22⋊Q8—91no—
224C22⋊Q8—91no—
225C22⋊Q8—91no—
226C22⋊Q8—91no—
227C22⋊Q8—91no—
228C4×Q8—91no—
229C4×D4—31no—
230C4×D4—91no—
231C4×D4—91no—
232C23×C3—21no—
233C23×C3—31no—
234C23×C3C22×D311yescharacteristic
235C23×C3C22×D311yescharacteristic
236C22×D3C22×D311yescharacteristic
237C3×Q8—62no—
238C3×D4—34no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—31no—
242C2×C3×C4—31no—
243C2×C3×C4—31no—
244C2×C3×C4—31no—
245C2×C3×C4—31no—
246C2×C3×C4—31no—
247C2×C3×C4—31no—
248C2×C3×C4—62no—
249C2×C3×C4—62no—
250C2×C3×C4—62no—
251C2×C3×C4C22×D311yescharacteristic
252C2×C3×C4C22×D311yescharacteristic
253C2×C3×C4C22×D311yescharacteristic
254C2×C3×C4C22×D311yescharacteristic
255#67Dic614yescomplemented
256C2×Dic3—22no—
257C2×Dic3—22no—
258C2×Dic3—31no—
259C2×Dic3—31no—
260C2×Dic3—31no—
261C2×Dic3—31no—
262C2×Dic3—31no—
263C2×Dic3—31no—
264C2×Dic3—31no—
265C2×Dic3—31no—
266C2×Dic3—61no—
267C2×Dic3—61no—
268C2×Dic3—61no—
269C2×Dic3—61no—
270C2×Dic3—62no—
271C2×Dic3—62no—
272C2×Dic3—62no—
273C2×Dic3—62no—
274C2×Dic3—122no—
275C2×Dic3C22×D311yescharacteristic
276C2×Dic3C22×D311yescharacteristic
277C2×Dic3C22×D311yescharacteristic
278C2×Dic3C22×D311yescharacteristic
279C2×Dic3C22×D311yescharacteristic
280C2×Dic3C22×D311yescharacteristic
281C2×Dic3C22×D311yescharacteristic
282C4×D3—22no—
283C4×D3—62no—
284C4×D3—62no—
285Dic6—22no—
286Dic6—34no—
287Dic6—34no—
288Dic6—62no—
289Dic6—62no—
290Dic6—62no—
291Dic6—62no—
292Dic6—62no—
293Dic6—62no—
294Dic6—62no—
295Dic6—62no—
296Dic6—62no—
297Dic6—62no—
298Dic6—122no—
299C2×C32—22no—
300C2×C32C22×D411yescharacteristic
301C2×C32C22×Q811yescharacteristic
302C2×C322−1+411yescharacteristic
303C3×D3—22no—
304C2×Q8—31no—
305C2×Q8—91no—
306C2×Q8—91no—
307C2×Q8—91no—
308C2×Q8—91no—
309C2×Q8—91no—
310C2×Q8—91no—
311C2×Q8—184no—
312C2×Q8—184no—
313C2×D4—31no—
314C22×C4—31no—
315C22×C4—31no—
316C22×C4—31no—
317C22×C4—91no—
318C22×C4—91no—
319C22×C4—91no—
320C4⋊C4—31no—
321C4⋊C4—31no—
322C4⋊C4—31no—
323C4⋊C4—31no—
324C4⋊C4—91no—
325C4⋊C4—91no—
326C4⋊C4—91no—
327C4⋊C4—91no—
328C4⋊C4—91no—
329C4⋊C4—91no—
330C4⋊C4—91no—
331C4⋊C4—91no—
332C22⋊C4—31no—
333C22⋊C4—31no—
334C22⋊C4—91no—
335C22⋊C4—91no—
336C22⋊C4—91no—
337C22⋊C4D3211yescharacteristic
338C42—31no—
339C42—91no—
340C42—91no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—22no—
347C22×C3—32no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—61no—
351C22×C3—61no—
352C22×C3C2×Dic612yescomplemented
353C22×C3D3×Q812yescomplemented
354C22×C3C23×D311yescharacteristic
355C22×C3C23×D311yescharacteristic
356C2×D3—21no—
357C2×D3—21no—
358C2×D3C2×Dic612yescomplemented
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—32no—
364C3×C4—32no—
365C3×C4—32no—
366C3×C4—41no—
367C3×C4—41no—
368C3×C4—61no—
369C3×C4—61no—
370C3×C4—61no—
371C3×C4—61no—
372Dic3—21no—
373Dic3—21no—
374Dic3—21no—
375Dic3—21no—
376Dic3—32no—
377Dic3—32no—
378Dic3—62no—
379Dic3—61no—
380Dic3—61no—
381Dic3—61no—
382Dic3—61no—
383Dic3—61no—
384Dic3—61no—
385Dic3—121no—
386Dic3—121no—
387Dic3—121no—
388Dic3C2×Dic612yescomplemented
389Dic3D3×D412yes—
390Dic3D3×D412yes—
391C32D4×Q811yescharacteristic, complemented, Sylow
392C23—31no—
393C23C2×D3211yescharacteristic
394Q8—62no—
395Q8—94no—
396Q8—182no—
397Q8—182no—
398Q8—182no—
399Q8—182no—
400Q8—182no—
401D4—34no—
402C2×C4—31no—
403C2×C4—31no—
404C2×C4—31no—
405C2×C4—31no—
406C2×C4—31no—
407C2×C4—31no—
408C2×C4—31no—
409C2×C4—62no—
410C2×C4—62no—
411C2×C4—62no—
412C2×C4—91no—
413C2×C4—91no—
414C2×C4—91no—
415C2×C4—91no—
416C2×C4—182no—
417C2×C4—182no—
418C2×C4—182no—
419C2×C4C2×D3211yescharacteristic
420C2×C4C2×D3211yescharacteristic
421C2×C3—22no—
422C2×C3—22no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—42no—
427C2×C3—62no—
428C2×C3C22×Dic611yescharacteristic
429C2×C3C2×D3×D411yescharacteristic
430C2×C3C2×D3×D411yescharacteristic
431C2×C3C2×D3×Q811yescharacteristic
432C2×C3#101011yescharacteristic
433C2×C3#101011yescharacteristic
434D3—22no—
435C22—21no—
436C22—21no—
437C22—32no—
438C22—61no—
439C22—61no—
440C22D3×Dic612yescomplemented
441C22C22×D3211yescharacteristic
442C4—21no—
443C4—21no—
444C4—32no—
445C4—32no—
446C4—32no—
447C4—61no—
448C4—61no—
449C4—61no—
450C4—61no—
451C4—92no—
452C4—181no—
453C4—181no—
454C4—181no—
455C3—21no—
456C3D4×Dic611yescharacteristic, complemented
457C3#563211yescharacteristic, complemented
458C2—22no—
459C2—62no—
460C2C2×D3×Dic611yescharacteristic
461C2#6430911yescharacteristic
462C2D32×D411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊C4C22
prime radicalC2×C3×#67C22
hypercenterC22⋊C4D32
hypocenterC32D4×Q8
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#627C2semidirect
2. C2×D3×Dic6C2semidirect
3. #64087C2semidirect
5. C3×#878C2semidirect
8. #63964C2semidirect
9. #63918C2semidirect
10. #63899C2semidirect
11. #63893C2semidirect
12. #63851C2semidirect
13. #63840C2semidirect
14. #63839C2semidirect
15. #63837C2semidirect
83. #878D3semidirect
96. C3×#67Q8semidirect
112. #627D4semidirect
199. C22×C32C2×Q8semidirect
203. C2×C3×D3C2×Q8semidirect
216. C3×Dic3C2×Q8semidirect
255. #67Dic6semidirect
352. C22×C3C2×Dic6semidirect
353. C22×C3D3×Q8semidirect
358. C2×D3C2×Dic6semidirect
388. Dic3C2×Dic6semidirect
391. C32D4×Q8semidirect
440. C22D3×Dic6semidirect
456. C3D4×Dic6semidirect
457. C3#5632semidirect

Maximal subgroups

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

subgroupindex
C22×#6272
C2×D3×Dic62
#640872
C3×#67⋊C42
C3×#8782
#67⋊Dic32
Dic3×#672
#639642
#639182
#638992
#638932
#638512
#638402
#638392
#638372
#56323
D4×Dic63