Finite Group Databasebeta

← #10594141

Subgroups — #10594141

Order 576.

503 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64090C211yesmaximal, characteristic
2C3×#878C211yesmaximal, characteristic, complemented
3C3×#878C211yesmaximal, characteristic, complemented
4#63982C211yesmaximal, characteristic, complemented
5C2×Dic3⋊Dic3C211yesmaximal, characteristic, complemented
6#63952C211yesmaximal, characteristic
7#63952C211yesmaximal, characteristic
8#63895C211yesmaximal, characteristic, complemented
9#63888C211yesmaximal, characteristic, complemented
10#63883C211yesmaximal, characteristic, complemented
11#63871C211yesmaximal, characteristic, complemented
12#63870C211yesmaximal, characteristic, complemented
13#63847C211yesmaximal, characteristic, complemented
14#63843C211yesmaximal, characteristic, complemented
15#63839C211yesmaximal, characteristic, complemented
16#5636—31nomaximal
17#5635—31nomaximal
18C2×#638C2211yescharacteristic
19C22×C3⋊Dic3C2211yescharacteristic
20C2×C4×C3⋊D3C2211yescharacteristic
21C22×C3×Dic3C2211yescharacteristic
22C22×C3×Dic3C2211yescharacteristic
23C2×C3×Dic6C2211yescharacteristic
24C2×C3×Dic6C2211yescharacteristic
25C2×#627C2211yescharacteristic
26C2×#624C2211yescharacteristic
27C2×#624C2211yescharacteristic
28C32×C22⋊C4C2211yescharacteristic
29#3752C2211yescharacteristic
30#3750C2211yescharacteristic
31C4×C3⋊Dic3C2211yescharacteristic
32C3×#217C2211yescharacteristic
33C3×#217C2211yescharacteristic
34C3×C4⋊Dic3C2211yescharacteristic
35C3×C4⋊Dic3C2211yescharacteristic
36C3×Dic3⋊C4C2211yescharacteristic
37C3×Dic3⋊C4C2211yescharacteristic
38C3×Dic3⋊C4C2211yescharacteristic
39C3×Dic3⋊C4C2211yescharacteristic
40#3724C2211yescharacteristic
41#3724C2211yescharacteristic
42#3724C2211yescharacteristic
43Dic3⋊Dic3C2211yescharacteristic
44Dic3⋊Dic3C2211yescharacteristic
45Dic3⋊Dic3C2211yescharacteristic
46Dic3⋊Dic3C2211yescharacteristic
47Dic3⋊Dic3C2211yescharacteristic
48Dic3⋊Dic3C2211yescharacteristic
49#3722C2211yescharacteristic
50#3722C2211yescharacteristic
51Dic32C2211yescharacteristic
52Dic32C2211yescharacteristic
53C3×C22⋊Q8—31no—
54C3×C22⋊Q8—31no—
55#946—31no—
56#946—31no—
57Q8×Dic3—31no—
58#944—31no—
59C4×#67—31no—
60C4×#67—31no—
61C2×C4⋊Dic3—31no—
62C2×Dic3⋊C4—31no—
63#897—31no—
64#897—31no—
65#896—31no—
66#896—31no—
67D3×C4⋊C4—31no—
68D3×C4⋊C4—31no—
69#889—31no—
70#889—31no—
71#889—31no—
72#889—31no—
73#888—31no—
74#887—31no—
75#67⋊C4—31no—
76#67⋊C4—31no—
77#67⋊C4—31no—
78#67⋊C4—31no—
79#67⋊C4—61no—
80#878—31no—
81#878—31no—
82#878D311yescharacteristic, complemented
83#878D311yescharacteristic, complemented
84C23×C32C2311yescharacteristic
85C22×C3⋊D3C2311yescharacteristic
86C2×C32×C4C2311yescharacteristic
87C2×C32×C4C2311yescharacteristic
88#638Q814yescomplemented
89C2×C3⋊Dic3—41no—
90C2×C3⋊Dic3C2311yescharacteristic
91C2×C3⋊Dic3C2311yescharacteristic
92C2×C3⋊Dic3C2311yescharacteristic
93C4×C3⋊D3—41no—
94C2×C3×Dic3—22no—
95C2×C3×Dic3—41no—
96C2×C3×Dic3C2311yescharacteristic
97C2×C3×Dic3C2311yescharacteristic
98C2×C3×Dic3C2311yescharacteristic
99C2×C3×Dic3C2311yescharacteristic
100C2×C3×Dic3C2311yescharacteristic
101C2×C3×Dic3C2311yescharacteristic
102C2×C3×Dic3C2311yescharacteristic
103C2×C3×Dic3C2311yescharacteristic
104C3×Dic6—22no—
105C3×Dic6—41no—
106#627—41no—
107#624—22no—
108#624—41no—
109#554—91noSylow
110C2×C3×Q8—31no—
111C2×C3×Q8—31no—
112C22×C3×C4—31no—
113C22×C3×C4—31no—
114C2×#67—31no—
115C2×#67—31no—
116C2×#67—61no—
117C22×Dic3—31no—
118C22×Dic3—31no—
119C22×Dic3—61no—
120C22×Dic3C2×D311yescharacteristic
121C22×Dic3C2×D311yescharacteristic
122C2×C4×D3—31no—
123C2×C4×D3—31no—
124C2×C4×D3—31no—
125C2×C4×D3—31no—
126C2×C4×D3—31no—
127C2×C4×D3—31no—
128C2×C4×D3—61no—
129C2×Dic6—31no—
130C2×Dic6—31no—
131C2×Dic6C2×D311yescharacteristic
132C2×Dic6C2×D311yescharacteristic
133C3×C4⋊C4—31no—
134C3×C4⋊C4—31no—
135C3×C4⋊C4—31no—
136C3×C4⋊C4—31no—
137C3×C4⋊C4—31no—
138C3×C4⋊C4—31no—
139C3×C22⋊C4—21no—
140C3×C22⋊C4—31no—
141C3×C22⋊C4—31no—
142C3×C22⋊C4C2×D311yescharacteristic
143C3×C22⋊C4C2×D311yescharacteristic
144#217C2×D311yescharacteristic
145#217C2×D311yescharacteristic
146(C2×D3)⋊C4—31no—
147(C2×D3)⋊C4—31no—
148(C2×D3)⋊C4—31no—
149(C2×D3)⋊C4—31no—
150(C2×D3)⋊C4—31no—
151(C2×D3)⋊C4—31no—
152(C2×D3)⋊C4—61no—
153C4⋊Dic3—31no—
154C4⋊Dic3—31no—
155C4⋊Dic3—31no—
156C4⋊Dic3—31no—
157C4⋊Dic3—31no—
158C4⋊Dic3—31no—
159C4⋊Dic3C2×D311yescharacteristic
160C4⋊Dic3C2×D311yescharacteristic
161Dic3⋊C4—31no—
162Dic3⋊C4—31no—
163Dic3⋊C4—31no—
164Dic3⋊C4—31no—
165Dic3⋊C4—31no—
166Dic3⋊C4—31no—
167Dic3⋊C4—31no—
168Dic3⋊C4—31no—
169Dic3⋊C4—31no—
170Dic3⋊C4—31no—
171Dic3⋊C4—31no—
172Dic3⋊C4—31no—
173Dic3⋊C4—31no—
174Dic3⋊C4—31no—
175Dic3⋊C4—61no—
176Dic3⋊C4C2×D311yescharacteristic
177Dic3⋊C4C2×D311yescharacteristic
178Dic3⋊C4C2×D311yescharacteristic
179Dic3⋊C4C2×D311yescharacteristic
180C4×Dic3—31no—
181C4×Dic3—31no—
182C4×Dic3—31no—
183C4×Dic3—31no—
184C4×Dic3—31no—
185C4×Dic3—31no—
186C4×Dic3—61no—
187C22×C32—21no—
188C22×C32—21no—
189C22×C32C2×Q812yes—
190C22×C32C2411yescharacteristic
191C2×C3⋊D3—21no—
192C2×C3⋊D3—21no—
193C2×C3⋊D3C2×Q812yes—
194C32×C4—21no—
195C32×C4—21no—
196C3⋊Dic3—21no—
197C3⋊Dic3—21no—
198C3⋊Dic3C2×Q812yes—
199C3×Dic3—21no—
200C3×Dic3—21no—
201C3×Dic3—21no—
202C3×Dic3—21no—
203C3×Dic3—21no—
204C3×Dic3—21no—
205C3×Dic3—21no—
206C3×Dic3#4112yes—
207C4⋊Q8—91no—
208#125—91no—
209#125—91no—
210C22⋊Q8—31no—
211C22⋊Q8—31no—
212C22⋊Q8—91no—
213C22⋊Q8—91no—
214C22⋊Q8—91no—
215C22⋊Q8—91no—
216C4×Q8—91no—
217C4×D4—91no—
218C4×D4—91no—
219C4×D4—91no—
220C2×C4⋊C4—91no—
221C2×C4⋊C4—91no—
222C23×C3—21no—
223C23×C3C22×D311yescharacteristic
224C23×C3C22×D311yescharacteristic
225C22×D3—31no—
226C22×D3—31no—
227C22×D3—61no—
228C3×Q8—62no—
229C3×Q8—121no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—31no—
233C2×C3×C4—31no—
234C2×C3×C4—31no—
235C2×C3×C4—31no—
236C2×C3×C4—31no—
237C2×C3×C4—31no—
238C2×C3×C4—31no—
239C2×C3×C4—31no—
240C2×C3×C4—62no—
241C2×C3×C4—121no—
242C2×C3×C4C22×D311yescharacteristic
243C2×C3×C4C22×D311yescharacteristic
244C2×C3×C4C22×D311yescharacteristic
245C2×C3×C4C22×D311yescharacteristic
246#67—34no—
247#67—34no—
248#67—64no—
249C2×Dic3—22no—
250C2×Dic3—31no—
251C2×Dic3—31no—
252C2×Dic3—31no—
253C2×Dic3—31no—
254C2×Dic3—31no—
255C2×Dic3—31no—
256C2×Dic3—41no—
257C2×Dic3—61no—
258C2×Dic3—61no—
259C2×Dic3—61no—
260C2×Dic3—121no—
261C2×Dic3—121no—
262C2×Dic3—122no—
263C2×Dic3C22×D311yescharacteristic
264C2×Dic3C22×D311yescharacteristic
265C2×Dic3C22×D311yescharacteristic
266C2×Dic3C22×D311yescharacteristic
267C2×Dic3C22×D311yescharacteristic
268C2×Dic3C22×D311yescharacteristic
269C2×Dic3C22×D311yescharacteristic
270C2×Dic3C22×D311yescharacteristic
271C4×D3—62no—
272C4×D3—62no—
273C4×D3—121no—
274C4×D3—121no—
275C4×D3—121no—
276C4×D3—121no—
277C4×D3—122no—
278Dic6—22no—
279Dic6—41no—
280Dic6—121no—
281Dic6—121no—
282C2×C32—22no—
283C2×C32C22×Q811yescharacteristic
284C2×C32C2×#4111yescharacteristic
285C2×C32H(5,𝔽2)11yescharacteristic
286C3⋊D3—22no—
287C2×Q8—31no—
288C2×Q8—31no—
289C2×Q8—91no—
290C2×D4—91no—
291C22×C4—31no—
292C22×C4—31no—
293C22×C4—91no—
294C22×C4—91no—
295C22×C4—91no—
296C22×C4—91no—
297C4⋊C4—31no—
298C4⋊C4—31no—
299C4⋊C4—31no—
300C4⋊C4—31no—
301C4⋊C4—31no—
302C4⋊C4—31no—
303C4⋊C4—91no—
304C4⋊C4—91no—
305C4⋊C4—91no—
306C4⋊C4—91no—
307C4⋊C4—91no—
308C4⋊C4—91no—
309C4⋊C4—91no—
310C4⋊C4—91no—
311C4⋊C4—91no—
312C4⋊C4—91no—
313C22⋊C4—31no—
314C22⋊C4—31no—
315C22⋊C4—91no—
316C22⋊C4—91no—
317C22⋊C4—91no—
318C22⋊C4D3211yescharacteristic
319C42—91no—
320C42—91no—
321C42—91no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—22no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3D3×Q812yes—
331C22×C3D3×Q812yes—
332C22×C3C23×D311yescharacteristic
333C22×C3C23×D311yescharacteristic
334C2×D3—32no—
335C2×D3—32no—
336C2×D3—62no—
337C2×D3—61no—
338C2×D3—61no—
339C2×D3—61no—
340C2×D3—61no—
341C2×D3—121no—
342C2×D3—121no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—21no—
347C3×C4—32no—
348C3×C4—41no—
349C3×C4—41no—
350C3×C4—61no—
351C3×C4—61no—
352C3×C4—61no—
353C3×C4—61no—
354C3×C4—61no—
355C3×C4—61no—
356C3×C4—61no—
357Dic3—21no—
358Dic3—21no—
359Dic3—21no—
360Dic3—21no—
361Dic3—21no—
362Dic3—21no—
363Dic3—21no—
364Dic3—32no—
365Dic3—32no—
366Dic3—62no—
367Dic3—62no—
368Dic3—61no—
369Dic3—61no—
370Dic3—61no—
371Dic3—61no—
372Dic3—121no—
373Dic3#23712yes—
374C32#55411yescharacteristic, complemented, Sylow
375C23—91no—
376C23C2×D3211yescharacteristic
377Q8—62no—
378Q8—121no—
379Q8—361no—
380D4—94no—
381C2×C4—31no—
382C2×C4—31no—
383C2×C4—31no—
384C2×C4—31no—
385C2×C4—31no—
386C2×C4—31no—
387C2×C4—31no—
388C2×C4—31no—
389C2×C4—62no—
390C2×C4—91no—
391C2×C4—91no—
392C2×C4—91no—
393C2×C4—121no—
394C2×C4—182no—
395C2×C4—361no—
396C2×C4—361no—
397C2×C4—361no—
398C2×C4C2×D3211yescharacteristic
399C2×C4C2×D3211yescharacteristic
400C2×C3—22no—
401C2×C3—22no—
402C2×C3—21no—
403C2×C3—21no—
404C2×C3—21no—
405C2×C3—42no—
406C2×C3C2×#23711yescharacteristic
407C2×C3#100411yescharacteristic
408C2×C3C2×D3×Q811yescharacteristic
409C2×C3C2×D3×Q811yescharacteristic
410C2×C3D3×#4111yescharacteristic
411C2×C3#100911yescharacteristic
412D3—62no—
413D3—62no—
414D3—122no—
415C22—21no—
416C22—21no—
417C22—92no—
418C22—181no—
419C22—181no—
420C22#379712yes—
421C22C22×D3211yescharacteristic
422C4—21no—
423C4—21no—
424C4—32no—
425C4—61no—
426C4—61no—
427C4—61no—
428C4—61no—
429C4—61no—
430C4—61no—
431C4—61no—
432C4—92no—
433C4—181no—
434C4—181no—
435C3—21no—
436C3#563511yescharacteristic, complemented
437C3#563611yescharacteristic, complemented
438C2—22no—
439C2—182no—
440C2C2×#379711yescharacteristic
441C2D3×#23711yescharacteristic
442C2#6431411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊C4C22
prime radicalC2×#638C22
hypercenterC22⋊C4D32
hypocenterC32#554
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#878C2semidirect
3. C3×#878C2semidirect
4. #63982C2semidirect
5. C2×Dic3⋊Dic3C2semidirect
8. #63895C2semidirect
9. #63888C2semidirect
10. #63883C2semidirect
11. #63871C2semidirect
12. #63870C2semidirect
13. #63847C2semidirect
14. #63843C2semidirect
15. #63839C2semidirect
82. #878D3semidirect
83. #878D3semidirect
88. #638Q8semidirect
374. C32#554semidirect
436. C3#5635semidirect
437. C3#5636semidirect

Maximal subgroups

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

subgroupindex
#640902
C3×#8782
#639822
C2×Dic3⋊Dic32
#639522
#638952
#638882
#638832
#638712
#638702
#638472
#638432
#638392
#56363
#56353