Finite Group Databasebeta

← #10594917

Subgroups — #10594917

Order 576.

666 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3828C211yesmaximal, characteristic, complemented
2C2×#3804C211yesmaximal, characteristic, complemented
3C3×#929C211yesmaximal, characteristic, complemented
4C2×C3×Dic3⋊C4C211yesmaximal, characteristic, complemented
5#63978C211yesmaximal, characteristic, complemented
6#63966C211yesmaximal, characteristic, complemented
7#63952C211yesmaximal, characteristic, complemented
8#63910C211yesmaximal, characteristic, complemented
9#63897C211yesmaximal, characteristic, complemented
10#63896C211yesmaximal, characteristic, complemented
11#63883C211yesmaximal, characteristic, complemented
12#63881C211yesmaximal, characteristic, complemented
13#63865C211yesmaximal, characteristic, complemented
14#63853C211yesmaximal, characteristic, complemented
15#63840C211yesmaximal, characteristic, complemented
16#5655—31nomaximal
17#5507—31nomaximal
18C22×C32×C4C2211yescharacteristic
19C2×#638C2211yescharacteristic
20C2×#638C2211yescharacteristic
21#3828—24no—
22C2×C4⋊(C3⋊D3)C2211yescharacteristic
23C2×C4×C3⋊D3C2211yescharacteristic
24C2×C4×C3⋊D3C2211yescharacteristic
25C2×#634C2211yescharacteristic
26C2×C3×#67C2211yescharacteristic
27C22×C3×Dic3C2211yescharacteristic
28C22×C3×Dic3C2211yescharacteristic
29C2×#627C2211yescharacteristic
30C2×#626C2211yescharacteristic
31C2×#626C2211yescharacteristic
32C2×#624C2211yescharacteristic
33C2×#624C2211yescharacteristic
34#3804—24no—
35C2×D3×Dic3C2211yescharacteristic
36C3×#217C2211yescharacteristic
37C3×(C2×D3)⋊C4C2211yescharacteristic
38C3×(C2×D3)⋊C4C2211yescharacteristic
39C3×Dic3⋊C4C2211yescharacteristic
40C3×Dic3⋊C4C2211yescharacteristic
41C3×Dic3⋊C4C2211yescharacteristic
42C3×Dic3⋊C4C2211yescharacteristic
43C3×Dic3⋊C4C2211yescharacteristic
44C3×Dic3⋊C4C2211yescharacteristic
45#3724C2211yescharacteristic
46Dic3⋊Dic3C2211yescharacteristic
47Dic3⋊Dic3C2211yescharacteristic
48Dic3⋊Dic3C2211yescharacteristic
49#3722C2211yescharacteristic
50#3722C2211yescharacteristic
51#3722C2211yescharacteristic
52#3722C2211yescharacteristic
53(C2×D3)⋊Dic3C2211yescharacteristic
54Dic32C2211yescharacteristic
55C2×#237—31no—
56C2×#235—31no—
57C2×#235—31no—
58C2×#235—31no—
59C2×#235—61no—
60C3×#123—31no—
61C2×C3×C4⋊C4—31no—
62#939—31no—
63#930—31no—
64#929—31no—
65#929D311yescharacteristic, complemented
66C4×#67—31no—
67C2×Dic3⋊C4D311yescharacteristic, complemented
68#896—31no—
69#896—31no—
70#896—31no—
71#895—31no—
72#894—31no—
73#894—31no—
74#894—31no—
75#893—31no—
76D3×C4⋊C4—31no—
77D3×C4⋊C4—31no—
78#888—31no—
79#888—31no—
80#886—31no—
81#884—31no—
82#883—31no—
83#67⋊C4—31no—
84#67⋊C4—31no—
85#878—31no—
86C23×C32C2311yescharacteristic
87C22×C3⋊D3C2311yescharacteristic
88C22×C3⋊D3C2311yescharacteristic
89C22×C3×D3C2311yescharacteristic
90C2×C32×C4—22no—
91C2×C32×C4C2311yescharacteristic
92C2×C32×C4C2311yescharacteristic
93#638—22no—
94#638D414yescomplemented
95C2×C3⋊Dic3C2311yescharacteristic
96C2×C3⋊Dic3C2311yescharacteristic
97C4⋊(C3⋊D3)—22no—
98C4×C3⋊D3—22no—
99C4×C3⋊D3—22no—
100#634—22no—
101C3×#67—22no—
102C2×C3×Dic3—22no—
103C2×C3×Dic3—22no—
104C2×C3×Dic3C2311yescharacteristic
105C2×C3×Dic3C2311yescharacteristic
106C2×C3×Dic3C2311yescharacteristic
107C2×C3×Dic3C2311yescharacteristic
108C2×C3×Dic3C2311yescharacteristic
109C2×C3×Dic3C2311yescharacteristic
110C2×C3×Dic3C2311yescharacteristic
111#627—22no—
112#626—22no—
113#626—22no—
114#624—22no—
115#624—22no—
116D3×Dic3—22no—
117#547—91noSylow
118C2×C3×D4—31no—
119C22×C3×C4—21no—
120C22×C3×C4—31no—
121C22×C3×C4—31no—
122C22×C3×C4C2×D311yescharacteristic
123C22×C3×C4C2×D311yescharacteristic
124C2×#67—31no—
125C2×#67—31no—
126C2×#67—31no—
127C2×#67—31no—
128C2×#67—31no—
129C2×#67—31no—
130C2×#67—61no—
131C2×#67—61no—
132C2×#67C2×D311yescharacteristic
133C22×Dic3—31no—
134C22×Dic3C2×D311yescharacteristic
135C22×Dic3C2×D311yescharacteristic
136#237—64no—
137#235—64no—
138#235—64no—
139#235—64no—
140#235—68no—
141C2×D12—31no—
142C2×D12—31no—
143C2×D12—31no—
144C2×D12—31no—
145C2×D12—61no—
146C2×C4×D3—31no—
147C2×C4×D3—31no—
148C2×C4×D3—31no—
149C2×C4×D3—31no—
150C2×C4×D3—31no—
151C2×C4×D3—31no—
152C2×C4×D3—31no—
153C2×C4×D3—31no—
154C2×C4×D3—31no—
155C2×C4×D3—61no—
156C2×C4×D3—61no—
157C2×Dic6—31no—
158C2×Dic6—31no—
159C2×Dic6—31no—
160C2×Dic6—31no—
161C2×Dic6—61no—
162C3×C4⋊C4—31no—
163C3×C4⋊C4—31no—
164C3×C4⋊C4—31no—
165C3×C4⋊C4—31no—
166C3×C4⋊C4—31no—
167C3×C4⋊C4—31no—
168C3×C22⋊C4—31no—
169C3×C22⋊C4—31no—
170C3×C22⋊C4—31no—
171#217—31no—
172#217C2×D311yescharacteristic
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—31no—
179(C2×D3)⋊C4—31no—
180(C2×D3)⋊C4—31no—
181(C2×D3)⋊C4—31no—
182(C2×D3)⋊C4C2×D311yescharacteristic
183(C2×D3)⋊C4C2×D311yescharacteristic
184C4⋊Dic3—31no—
185C4⋊Dic3—31no—
186C4⋊Dic3—31no—
187Dic3⋊C4—31no—
188Dic3⋊C4—31no—
189Dic3⋊C4—31no—
190Dic3⋊C4—31no—
191Dic3⋊C4—31no—
192Dic3⋊C4C2×D311yescharacteristic
193Dic3⋊C4C2×D311yescharacteristic
194Dic3⋊C4C2×D311yescharacteristic
195Dic3⋊C4C2×D311yescharacteristic
196Dic3⋊C4C2×D311yescharacteristic
197Dic3⋊C4C2×D311yescharacteristic
198C4×Dic3—31no—
199C4×Dic3—31no—
200C22×C32—21no—
201C22×C32—21no—
202C22×C32C2×D412yes—
203C22×C32C2411yescharacteristic
204C2×C3⋊D3—21no—
205C2×C3⋊D3—21no—
206C2×C3⋊D3—21no—
207C2×C3⋊D3—21no—
208C2×C3⋊D3—21no—
209C2×C3⋊D3C2×D412yes—
210C2×C3×D3—21no—
211C2×C3×D3—21no—
212C2×C3×D3—21no—
213C32×C4—21no—
214C32×C4—21no—
215C3⋊Dic3—21no—
216C3⋊Dic3C2×D412yes—
217C3×Dic3—21no—
218C3×Dic3—21no—
219C3×Dic3—21no—
220C3×Dic3—21no—
221C3×Dic3—21no—
222C3×Dic3—21no—
223C3×Dic3#4112yescomplemented
224C2×#41—91no—
225C2×#41—91no—
226C4⋊Q8—91no—
227#123—31no—
228#123—91no—
229#123—91no—
230#123—91no—
231C22⋊Q8—91no—
232C22⋊Q8—91no—
233#121—91no—
234#121—91no—
235C4×D4—91no—
236C4×D4—91no—
237C2×C4⋊C4—31no—
238C2×C4⋊C4—91no—
239C23×C3—21no—
240C23×C3—31no—
241C23×C3C22×D311yescharacteristic
242C23×C3C22×D311yescharacteristic
243C22×D3—31no—
244C22×D3—31no—
245C22×D3—31no—
246C22×D3—31no—
247C22×D3—61no—
248C22×D3—61no—
249C22×D3C22×D311yescharacteristic
250C3×D4—62no—
251C2×C3×C4—24no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—31no—
257C2×C3×C4—31no—
258C2×C3×C4—31no—
259C2×C3×C4—31no—
260C2×C3×C4—31no—
261C2×C3×C4—31no—
262C2×C3×C4—31no—
263C2×C3×C4—62no—
264C2×C3×C4—62no—
265C2×C3×C4C22×D311yescharacteristic
266C2×C3×C4C22×D311yescharacteristic
267C2×C3×C4C22×D311yescharacteristic
268C2×C3×C4C22×D311yescharacteristic
269#67—22no—
270#67—34no—
271#67—34no—
272#67—64no—
273#67—62no—
274#67—64no—
275#67—62no—
276#67—62no—
277#67—62no—
278C2×Dic3—22no—
279C2×Dic3—22no—
280C2×Dic3—31no—
281C2×Dic3—31no—
282C2×Dic3—31no—
283C2×Dic3—31no—
284C2×Dic3—61no—
285C2×Dic3—61no—
286C2×Dic3—62no—
287C2×Dic3C22×D311yescharacteristic
288C2×Dic3C22×D311yescharacteristic
289C2×Dic3C22×D311yescharacteristic
290C2×Dic3C22×D311yescharacteristic
291C2×Dic3C22×D311yescharacteristic
292C2×Dic3C22×D311yescharacteristic
293C2×Dic3C22×D311yescharacteristic
294D12—62no—
295D12—62no—
296D12—64no—
297D12—62no—
298D12—62no—
299C4×D3—62no—
300C4×D3—62no—
301C4×D3—62no—
302C4×D3—62no—
303C4×D3—62no—
304C4×D3—62no—
305C4×D3—62no—
306C4×D3—62no—
307C4×D3—64no—
308C4×D3—64no—
309C4×D3—62no—
310Dic6—62no—
311Dic6—62no—
312Dic6—62no—
313Dic6—62no—
314Dic6—64no—
315C2×C32—22no—
316C2×C32C22×D411yescharacteristic
317C2×C32C2×#4111yescharacteristic
318C2×C322−1+411yescharacteristic
319C3⋊D3—22no—
320C3⋊D3—41no—
321C3×D3—41no—
322#41—184no—
323#41—184no—
324C2×Q8—91no—
325C2×Q8—91no—
326C2×D4—31no—
327C2×D4—91no—
328C2×D4—91no—
329C2×D4—91no—
330C2×D4—91no—
331C2×D4—91no—
332C22×C4—31no—
333C22×C4—31no—
334C22×C4—91no—
335C22×C4—91no—
336C22×C4—91no—
337C22×C4—91no—
338C22×C4—91no—
339C22×C4D3211yescharacteristic
340C4⋊C4—31no—
341C4⋊C4—31no—
342C4⋊C4—31no—
343C4⋊C4—31no—
344C4⋊C4—31no—
345C4⋊C4—31no—
346C4⋊C4—91no—
347C4⋊C4—91no—
348C4⋊C4—91no—
349C4⋊C4—91no—
350C22⋊C4—31no—
351C22⋊C4—31no—
352C22⋊C4—31no—
353C22⋊C4—91no—
354C22⋊C4—91no—
355C22⋊C4—91no—
356C22⋊C4—91no—
357C22⋊C4—91no—
358C42—91no—
359C22×C3—21no—
360C22×C3—21no—
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—22no—
365C22×C3—41no—
366C22×C3—41no—
367C22×C3—61no—
368C22×C3—61no—
369C22×C3—61no—
370C22×C3D3×D412yes—
371C22×C3D3×D412yes—
372C22×C3C23×D311yescharacteristic
373C22×C3C23×D311yescharacteristic
374C2×D3—21no—
375C2×D3—21no—
376C2×D3—21no—
377C2×D3—32no—
378C2×D3—32no—
379C2×D3—62no—
380C2×D3—61no—
381C2×D3—61no—
382C2×D3—61no—
383C2×D3—61no—
384C2×D3—62no—
385C2×D3—61no—
386C2×D3—61no—
387C2×D3—61no—
388C2×D3—61no—
389C2×D3—61no—
390C2×D3—61no—
391C2×D3—121no—
392C2×D3—121no—
393C2×D3—121no—
394C2×D3—121no—
395C3×C4—22no—
396C3×C4—22no—
397C3×C4—21no—
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—21no—
401C3×C4—32no—
402C3×C4—61no—
403C3×C4—61no—
404C3×C4—61no—
405C3×C4—61no—
406C3×C4—61no—
407C3×C4—61no—
408Dic3—21no—
409Dic3—21no—
410Dic3—21no—
411Dic3—21no—
412Dic3—21no—
413Dic3—21no—
414Dic3—32no—
415Dic3—32no—
416Dic3—62no—
417Dic3—62no—
418Dic3—61no—
419Dic3—61no—
420Dic3#23512yescomplemented
421C32#54711yescharacteristic, complemented, Sylow
422C23—31no—
423C23—91no—
424C23—91no—
425C23C2×D3211yescharacteristic
426Q8—182no—
427Q8—182no—
428D4—62no—
429D4—94no—
430D4—182no—
431D4—182no—
432D4—182no—
433D4—182no—
434C2×C4—22no—
435C2×C4—31no—
436C2×C4—31no—
437C2×C4—31no—
438C2×C4—31no—
439C2×C4—31no—
440C2×C4—31no—
441C2×C4—31no—
442C2×C4—62no—
443C2×C4—62no—
444C2×C4—91no—
445C2×C4—91no—
446C2×C4—182no—
447C2×C4—182no—
448C2×C4—182no—
449C2×C4—182no—
450C2×C4—182no—
451C2×C4C2×D3211yescharacteristic
452C2×C4C2×D3211yescharacteristic
453C2×C3—22no—
454C2×C3—22no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—42no—
459C2×C3—121no—
460C2×C3C2×#23511yescharacteristic
461C2×C3C2×D3×D411yescharacteristic
462C2×C3C2×D3×D411yescharacteristic
463C2×C3#100711yescharacteristic
464C2×C3D3×#4111yescharacteristic
465C2×C3#101011yescharacteristic
466D3—41no—
467D3—62no—
468D3—62no—
469D3—122no—
470D3—122no—
471D3—121no—
472D3—121no—
473C22—21no—
474C22—21no—
475C22—61no—
476C22—61no—
477C22—61no—
478C22—92no—
479C22—181no—
480C22—181no—
481C22—181no—
482C22—181no—
483C22—181no—
484C22#381112yes—
485C22C22×D3211yescharacteristic
486C4—21no—
487C4—21no—
488C4—32no—
489C4—61no—
490C4—61no—
491C4—61no—
492C4—61no—
493C4—61no—
494C4—61no—
495C4—92no—
496C4—181no—
497C3—21no—
498C3#550711yescharacteristic, complemented
499C3#565511yescharacteristic, complemented
500C2—22no—
501C2—121no—
502C2—182no—
503C2—361no—
504C2#6429711yescharacteristic
505C2D3×#23511yescharacteristic
506C2C2×#381111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
hypercenterC22×C4D32
hypocenterC32#547
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#3828C2semidirect
2. C2×#3804C2semidirect
3. C3×#929C2semidirect
4. C2×C3×Dic3⋊C4C2semidirect
5. #63978C2semidirect
6. #63966C2semidirect
7. #63952C2semidirect
8. #63910C2semidirect
9. #63897C2semidirect
10. #63896C2semidirect
11. #63883C2semidirect
12. #63881C2semidirect
13. #63865C2semidirect
14. #63853C2semidirect
15. #63840C2semidirect
65. #929D3semidirect
67. C2×Dic3⋊C4D3semidirect
94. #638D4semidirect
223. C3×Dic3#41semidirect
420. Dic3#235semidirect
421. C32#547semidirect
498. C3#5507semidirect
499. C3#5655semidirect

Maximal subgroups

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

subgroupindex
C2×#38282
C2×#38042
C3×#9292
C2×C3×Dic3⋊C42
#639782
#639662
#639522
#639102
#638972
#638962
#638832
#638812
#638652
#638532
#638402
#56553
#55073