Finite Group Databasebeta

← #10594516

Subgroups — #10594516

Order 576.

623 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3799C211yesmaximal, characteristic, complemented
2C2×#3798C211yesmaximal, characteristic, complemented
3#64103C211yesmaximal, characteristic, complemented
4C3×#896C211yesmaximal, characteristic, complemented
5C3×D3×C4⋊C4C211yesmaximal, characteristic, complemented
6#63910C211yesmaximal, characteristic, complemented
7#63906C211yesmaximal, characteristic, complemented
8#63905C211yesmaximal, characteristic, complemented
9#63904C211yesmaximal, characteristic, complemented
10#63904C211yesmaximal, characteristic, complemented
11#63895C211yesmaximal, characteristic, complemented
12#63883C211yesmaximal, characteristic, complemented
13#63854C211yesmaximal, characteristic, complemented
14#63850C211yesmaximal, characteristic, complemented
15#63839C211yesmaximal, characteristic, complemented
16#5631—31nomaximal
17#5507—31nomaximal
18C2×C4⋊(C3⋊D3)C2211yescharacteristic
19C2×C4×C3⋊D3C2211yescharacteristic
20C2×C3×C4×D3C2211yescharacteristic
21C2×C3×C4×D3C2211yescharacteristic
22C2×C3×C4×D3C2211yescharacteristic
23C2×C3×C4×D3C2211yescharacteristic
24C2×C3×Dic6C2211yescharacteristic
25C2×#627C2211yescharacteristic
26C2×#626C2211yescharacteristic
27C2×#626C2211yescharacteristic
28C2×#626C2211yescharacteristic
29C2×#626C2211yescharacteristic
30C2×#625C2211yescharacteristic
31C2×#624C2211yescharacteristic
32C2×#624C2211yescharacteristic
33C2×#624C2211yescharacteristic
34#3799—24no—
35#3798—24no—
36C32×C4⋊C4C2211yescharacteristic
37#3752C2211yescharacteristic
38#3752C2211yescharacteristic
39#3752C2211yescharacteristic
40#3750C2211yescharacteristic
41C3×(C2×D3)⋊C4C2211yescharacteristic
42C3×(C2×D3)⋊C4C2211yescharacteristic
43C3×C4⋊Dic3C2211yescharacteristic
44C3×Dic3⋊C4C2211yescharacteristic
45C3×Dic3⋊C4C2211yescharacteristic
46C3×Dic3⋊C4C2211yescharacteristic
47C3×Dic3⋊C4C2211yescharacteristic
48#3724C2211yescharacteristic
49Dic3⋊Dic3C2211yescharacteristic
50Dic3⋊Dic3C2211yescharacteristic
51#3722C2211yescharacteristic
52(C2×D3)⋊Dic3C2211yescharacteristic
53(C2×D3)⋊Dic3C2211yescharacteristic
54Dic32C2211yescharacteristic
55C2×#239—31no—
56C2×#235—31no—
57C2×#235—31no—
58C2×#235—31no—
59C3×C22⋊Q8—31no—
60C2×C3×C4⋊C4—31no—
61#944—31no—
62#930—31no—
63#929—31no—
64#929—31no—
65C4×#67—31no—
66C4×#67—31no—
67#897—31no—
68#896—31no—
69#896D311yescharacteristic, complemented
70#895—31no—
71#894—31no—
72#894—31no—
73#894—31no—
74#894—31no—
75#894—61no—
76#893—31no—
77#893—31no—
78D3×C4⋊C4—31no—
79D3×C4⋊C4—31no—
80D3×C4⋊C4D311yescharacteristic, complemented
81#888—31no—
82#884—31no—
83#884—31no—
84#883—31no—
85#883—31no—
86C22×C3⋊D3C2311yescharacteristic
87C22×C3⋊D3C2311yescharacteristic
88C22×C3×D3C2311yescharacteristic
89C22×C3×D3C2311yescharacteristic
90C2×C32×C4C2311yescharacteristic
91C2×C32×C4C2311yescharacteristic
92C2×C32×C4C2311yescharacteristic
93C2×C3⋊Dic3C2311yescharacteristic
94C2×C3⋊Dic3C2311yescharacteristic
95C4⋊(C3⋊D3)—22no—
96C4×C3⋊D3—22no—
97C2×C3×Dic3C2311yescharacteristic
98C2×C3×Dic3C2311yescharacteristic
99C2×C3×Dic3C2311yescharacteristic
100C2×C3×Dic3C2311yescharacteristic
101C2×C3×Dic3C2311yescharacteristic
102C2×C3×Dic3C2311yescharacteristic
103C3×C4×D3—22no—
104C3×C4×D3—22no—
105C3×C4×D3—22no—
106C3×C4×D3—22no—
107C3×Dic6—22no—
108#627—22no—
109#626—22no—
110#626—22no—
111#626—22no—
112#626D414yescomplemented
113#625—22no—
114#624—22no—
115#624—22no—
116#624—22no—
117#547—91noSylow
118C2×C3×Q8—31no—
119C22×C3×C4—31no—
120C22×C3×C4—31no—
121C22×C3×C4—31no—
122C22×C3×C4—31no—
123C2×#67—31no—
124C2×#67—31no—
125C2×#67—31no—
126C2×#67—31no—
127C2×#67—31no—
128C2×#67—31no—
129#239—64no—
130#235—64no—
131#235—64no—
132#235—64no—
133C2×D12—31no—
134C2×D12—31no—
135C2×D12—31no—
136C2×D12—31no—
137C2×D12—31no—
138C2×D12—31no—
139C2×D12—61no—
140C2×C4×D3—31no—
141C2×C4×D3—31no—
142C2×C4×D3—31no—
143C2×C4×D3—31no—
144C2×C4×D3—31no—
145C2×C4×D3—31no—
146C2×C4×D3—31no—
147C2×C4×D3—31no—
148C2×C4×D3—61no—
149C2×C4×D3C2×D311yescharacteristic
150C2×C4×D3C2×D311yescharacteristic
151C2×C4×D3C2×D311yescharacteristic
152C2×C4×D3C2×D311yescharacteristic
153C2×Dic6—31no—
154C2×Dic6—31no—
155C2×Dic6C2×D311yescharacteristic
156C3×C4⋊C4—21no—
157C3×C4⋊C4—31no—
158C3×C4⋊C4—31no—
159C3×C4⋊C4—31no—
160C3×C4⋊C4—31no—
161C3×C4⋊C4—31no—
162C3×C4⋊C4C2×D311yescharacteristic
163C3×C4⋊C4C2×D311yescharacteristic
164C3×C22⋊C4—31no—
165C3×C22⋊C4—31no—
166#217—31no—
167#217—31no—
168(C2×D3)⋊C4—31no—
169(C2×D3)⋊C4—31no—
170(C2×D3)⋊C4—31no—
171(C2×D3)⋊C4—31no—
172(C2×D3)⋊C4—31no—
173(C2×D3)⋊C4—31no—
174(C2×D3)⋊C4—31no—
175(C2×D3)⋊C4—31no—
176(C2×D3)⋊C4—31no—
177(C2×D3)⋊C4—31no—
178(C2×D3)⋊C4—61no—
179(C2×D3)⋊C4—61no—
180(C2×D3)⋊C4—61no—
181(C2×D3)⋊C4C2×D311yescharacteristic
182(C2×D3)⋊C4C2×D311yescharacteristic
183C4⋊Dic3—31no—
184C4⋊Dic3—31no—
185C4⋊Dic3C2×D311yescharacteristic
186Dic3⋊C4—31no—
187Dic3⋊C4—31no—
188Dic3⋊C4—31no—
189Dic3⋊C4—31no—
190Dic3⋊C4—31no—
191Dic3⋊C4—31no—
192Dic3⋊C4—61no—
193Dic3⋊C4C2×D311yescharacteristic
194Dic3⋊C4C2×D311yescharacteristic
195Dic3⋊C4C2×D311yescharacteristic
196Dic3⋊C4C2×D311yescharacteristic
197C4×Dic3—31no—
198C4×Dic3—31no—
199C22×C32C2411yescharacteristic
200C2×C3⋊D3—21no—
201C2×C3⋊D3—21no—
202C2×C3⋊D3—21no—
203C2×C3⋊D3—21no—
204C2×C3⋊D3—21no—
205C2×C3⋊D3C2×D412yes—
206C2×C3×D3—21no—
207C2×C3×D3—21no—
208C2×C3×D3—21no—
209C2×C3×D3—21no—
210C2×C3×D3—21no—
211C2×C3×D3C2×D412yes—
212C32×C4—21no—
213C32×C4—21no—
214C32×C4—21no—
215C3⋊Dic3—21no—
216C3⋊Dic3—21no—
217C3×Dic3—21no—
218C3×Dic3—21no—
219C3×Dic3—21no—
220C3×Dic3—21no—
221C3×Dic3C2×D412yes—
222C3×Dic3#4112yescomplemented
223C2×#41—91no—
224C2×#41—91no—
225C4⋊Q8—91no—
226#123—91no—
227#123—91no—
228#123—91no—
229#123—91no—
230C22⋊Q8—31no—
231C22⋊Q8—91no—
232#121—91no—
233#121—91no—
234C4×D4—91no—
235C4×D4—91no—
236C2×C4⋊C4—31no—
237C2×C4⋊C4—91no—
238C23×C3—31no—
239C23×C3—31no—
240C22×D3—31no—
241C22×D3—31no—
242C22×D3—31no—
243C22×D3—31no—
244C22×D3—61no—
245C22×D3—61no—
246C22×D3C22×D311yescharacteristic
247C22×D3C22×D311yescharacteristic
248C3×Q8—62no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—31no—
253C2×C3×C4—31no—
254C2×C3×C4—31no—
255C2×C3×C4—31no—
256C2×C3×C4—31no—
257C2×C3×C4—31no—
258C2×C3×C4—62no—
259C2×C3×C4—62no—
260C2×C3×C4—62no—
261C2×C3×C4—62no—
262C2×C3×C4C22×D311yescharacteristic
263C2×C3×C4C22×D311yescharacteristic
264C2×C3×C4C22×D311yescharacteristic
265C2×C3×C4C22×D311yescharacteristic
266C2×C3×C4C22×D311yescharacteristic
267C2×C3×C4C22×D311yescharacteristic
268#67—34no—
269#67—62no—
270#67—62no—
271#67—62no—
272#67—62no—
273#67—62no—
274C2×Dic3—31no—
275C2×Dic3—31no—
276C2×Dic3—31no—
277C2×Dic3—31no—
278C2×Dic3—61no—
279C2×Dic3—61no—
280C2×Dic3C22×D311yescharacteristic
281C2×Dic3C22×D311yescharacteristic
282C2×Dic3C22×D311yescharacteristic
283C2×Dic3C22×D311yescharacteristic
284C2×Dic3C22×D311yescharacteristic
285C2×Dic3C22×D311yescharacteristic
286D12—34no—
287D12—62no—
288D12—62no—
289D12—62no—
290D12—62no—
291D12—62no—
292D12—122no—
293C4×D3—22no—
294C4×D3—22no—
295C4×D3—22no—
296C4×D3—22no—
297C4×D3—62no—
298C4×D3—62no—
299C4×D3—62no—
300C4×D3—62no—
301C4×D3—62no—
302C4×D3—62no—
303C4×D3—62no—
304C4×D3—62no—
305C4×D3—122no—
306Dic6—22no—
307Dic6—62no—
308Dic6—62no—
309C2×C32C22×D411yescharacteristic
310C2×C32C2×#4111yescharacteristic
311C2×C322−1+411yescharacteristic
312C3⋊D3—22no—
313C3⋊D3—41no—
314C3×D3—22no—
315C3×D3—41no—
316#41—184no—
317#41—184no—
318C2×Q8—31no—
319C2×Q8—91no—
320C2×D4—91no—
321C2×D4—91no—
322C2×D4—91no—
323C2×D4—91no—
324C2×D4—91no—
325C2×D4—91no—
326C22×C4—31no—
327C22×C4—31no—
328C22×C4—31no—
329C22×C4—31no—
330C22×C4—91no—
331C22×C4—91no—
332C22×C4—91no—
333C22×C4—91no—
334C4⋊C4—31no—
335C4⋊C4—31no—
336C4⋊C4—31no—
337C4⋊C4—31no—
338C4⋊C4—31no—
339C4⋊C4—91no—
340C4⋊C4—91no—
341C4⋊C4—91no—
342C4⋊C4—91no—
343C4⋊C4D3211yescharacteristic
344C22⋊C4—31no—
345C22⋊C4—31no—
346C22⋊C4—91no—
347C22⋊C4—91no—
348C22⋊C4—91no—
349C22⋊C4—91no—
350C22⋊C4—91no—
351C22⋊C4—91no—
352C42—91no—
353C22×C3—21no—
354C22×C3—32no—
355C22×C3—61no—
356C22×C3—61no—
357C22×C3—61no—
358C22×C3—61no—
359C22×C3—61no—
360C22×C3C23×D311yescharacteristic
361C22×C3C23×D311yescharacteristic
362C2×D3—21no—
363C2×D3—21no—
364C2×D3—21no—
365C2×D3—21no—
366C2×D3—21no—
367C2×D3—32no—
368C2×D3—32no—
369C2×D3—62no—
370C2×D3—61no—
371C2×D3—61no—
372C2×D3—61no—
373C2×D3—61no—
374C2×D3—61no—
375C2×D3—61no—
376C2×D3—61no—
377C2×D3—61no—
378C2×D3—61no—
379C2×D3—61no—
380C2×D3—121no—
381C2×D3—121no—
382C2×D3—121no—
383C2×D3—121no—
384C2×D3—121no—
385C2×D3D3×D412yes—
386C3×C4—21no—
387C3×C4—21no—
388C3×C4—21no—
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—32no—
393C3×C4—32no—
394C3×C4—41no—
395C3×C4—41no—
396C3×C4—41no—
397C3×C4—61no—
398C3×C4—61no—
399C3×C4—61no—
400C3×C4—61no—
401Dic3—21no—
402Dic3—21no—
403Dic3—21no—
404Dic3—21no—
405Dic3—61no—
406Dic3—61no—
407Dic3—61no—
408Dic3—61no—
409Dic3—121no—
410Dic3—121no—
411Dic3#23512yescomplemented
412Dic3D3×D412yes—
413C32#54711yescharacteristic, complemented, Sylow
414C23—31no—
415C23—31no—
416C23—91no—
417C23—91no—
418Q8—62no—
419Q8—182no—
420D4—94no—
421D4—182no—
422D4—182no—
423D4—182no—
424D4—182no—
425D4—182no—
426C2×C4—31no—
427C2×C4—31no—
428C2×C4—31no—
429C2×C4—31no—
430C2×C4—31no—
431C2×C4—31no—
432C2×C4—62no—
433C2×C4—62no—
434C2×C4—62no—
435C2×C4—62no—
436C2×C4—91no—
437C2×C4—91no—
438C2×C4—182no—
439C2×C4—182no—
440C2×C4—182no—
441C2×C4—182no—
442C2×C4C2×D3211yescharacteristic
443C2×C4C2×D3211yescharacteristic
444C2×C4C2×D3211yescharacteristic
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—62no—
449C2×C3—121no—
450C2×C3C2×#23511yescharacteristic
451C2×C3C2×D3×D411yescharacteristic
452C2×C3C2×D3×D411yescharacteristic
453C2×C3#100711yescharacteristic
454C2×C3#100711yescharacteristic
455C2×C3D3×#4111yescharacteristic
456D3—22no—
457D3—41no—
458D3—62no—
459D3—62no—
460D3—122no—
461D3—121no—
462D3—121no—
463D3—241no—
464C22—32no—
465C22—61no—
466C22—61no—
467C22—61no—
468C22—61no—
469C22—61no—
470C22—92no—
471C22—181no—
472C22—181no—
473C22—181no—
474C22—181no—
475C22—181no—
476C22C22×D3211yescharacteristic
477C4—21no—
478C4—21no—
479C4—21no—
480C4—32no—
481C4—32no—
482C4—61no—
483C4—61no—
484C4—61no—
485C4—61no—
486C4—181no—
487C4—181no—
488C3—21no—
489C3#550711yescharacteristic, complemented
490C3#563111yescharacteristic, complemented
491C2—62no—
492C2—121no—
493C2—182no—
494C2—361no—
495C2D3×#23511yescharacteristic
496C2D32×D411yescharacteristic
497C2#6431611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
hypercenterC4⋊C4D32
hypocenterC32#547
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#3799C2semidirect
2. C2×#3798C2semidirect
3. #64103C2semidirect
4. C3×#896C2semidirect
5. C3×D3×C4⋊C4C2semidirect
6. #63910C2semidirect
7. #63906C2semidirect
8. #63905C2semidirect
9. #63904C2semidirect
10. #63904C2semidirect
11. #63895C2semidirect
12. #63883C2semidirect
13. #63854C2semidirect
14. #63850C2semidirect
15. #63839C2semidirect
69. #896D3semidirect
80. D3×C4⋊C4D3semidirect
112. #626D4semidirect
222. C3×Dic3#41semidirect
411. Dic3#235semidirect
413. C32#547semidirect
489. C3#5507semidirect
490. C3#5631semidirect

Maximal subgroups

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

subgroupindex
C2×#37992
C2×#37982
#641032
C3×#8962
C3×D3×C4⋊C42
#639102
#639062
#639052
#639042
#638952
#638832
#638542
#638502
#638392
#56313
#55073