Finite Group Databasebeta

← #10853271

Subgroups — #10853271

Order 768.

786 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82337C211yesmaximal, characteristic
2#74335C211yesmaximal, characteristic, complemented
3C3×#1802C211yesmaximal, characteristic
4C4×#832C211yesmaximal, characteristic
5#70468C211yesmaximal, characteristic, complemented
6#70453C211yesmaximal, characteristic, complemented
7#70180C212yesmaximal
8#70163C212yesmaximal
9#69574C211yesmaximal, characteristic, complemented
10#68814C212yesmaximal, complemented
11#68800C212yesmaximal, complemented
12#26020—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C2×#926—21no—
15C2×#888—21no—
16C2×#888C2212yes—
17C2×C3×C4⋊C8C2211yescharacteristic
18C2×C3×#102C2212yes—
19C3×#384C2211yescharacteristic
20#5221C2212yes—
21C2×#832C2211yescharacteristic
22C2×#832C2211yescharacteristic
23#5212—21no—
24C2×#818C2212yes—
25C2×Dic3⋊C8C2212yes—
26#4979—22no—
27Dic3×C4⋊C4C2212yes—
28#4972—21no—
29#4972C2212yes—
30#4951—22no—
31#4935C2211yescharacteristic
32C4×C4⋊Dic3C2211yescharacteristic
33C4×C4⋊Dic3C2211yescharacteristic
34C2×C4×C3⋊C8C2211yescharacteristic
35C3×#340C2212yes—
36#4551C2212yes—
37#4550C2212yes—
38#4529C2212yes—
39#4529C2212yes—
40#4529C2211yescharacteristic
41#4529C2211yescharacteristic
42#2551—31no—
43#2354—31no—
44#2067—32no—
45#2023—32no—
46#1988—32no—
47#1947—31no—
48#1939—31no—
49#1918—32no—
50#1834—31no—
51#1802D311yescharacteristic
52C4×#102—31no—
53C22×C3×D4C2311yescharacteristic
54C23×C3×C4—21no—
55C22×Dic6—22no—
56C22×C3×C8C2312yes—
57C3×C4×D4—22no—
58C3×C4×D4Q814yescomplemented
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4C2312yes—
62C2×C3×C4⋊C4C2311yescharacteristic
63C2×C3×C22⋊C4—21no—
64C2×C3×C42C2311yescharacteristic
65C2×#217—22no—
66C2×#217—22no—
67#926—44no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3—21no—
70C2×C4⋊Dic3D411yescharacteristic
71C2×C4⋊Dic3D411yescharacteristic
72C2×C4⋊Dic3C2312yes—
73C2×C4⋊Dic3C2311yescharacteristic
74C2×C4⋊Dic3C2311yescharacteristic
75C2×Dic3⋊C4—22no—
76C2×Dic3⋊C4—22no—
77C2×C4×Dic3—21no—
78C2×C4×Dic3—22no—
79C2×C4×Dic3C2312yes—
80C22×C3⋊C8C2311yescharacteristic
81C22×C3⋊C8C2311yescharacteristic
82#888—24no—
83#888—42no—
84#888—44no—
85C3×C4⋊C8—22no—
86C3×#102—42no—
87#832—22no—
88#832—22no—
89C3⋊#95—21no—
90C3⋊#95—22no—
91C3⋊#95—22no—
92C3⋊#95D412yes—
93#818—24no—
94Dic3⋊C8—42no—
95C4×C3⋊C8—22no—
96C2×C4⋊Q8—32no—
97C2×C4⋊Q8—61no—
98C2×C4×D4C2×D311yescharacteristic
99C2×#117—61no—
100C2×#107—32no—
101C2×C4⋊C8—32no—
102C2×C4⋊C8C2×D311yescharacteristic
103C2×#102—31no—
104C2×#102—31no—
105C2×#102C2×D312yes—
106C2×C4×C8—31no—
107#395—62no—
108#390—32no—
109#389—32no—
110#389—61no—
111#388—62no—
112#384—31no—
113#384C2×D311yescharacteristic
114#380—61no—
115C4×C4⋊C4—31no—
116C4×C4⋊C4—32no—
117C4×C4⋊C4—31no—
118#340—32no—
119#340—32no—
120#340—32no—
121#340—31no—
122#340—31no—
123#340C2×D312yes—
124#336—32no—
125C24×C3—21no—
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4—21no—
129C2×C3×D4—21no—
130C2×C3×D4—21no—
131C2×C3×D4C2×Q812yes—
132C22×C3×C4—21no—
133C22×C3×C4—24no—
134C22×C3×C4—22no—
135C22×C3×C4—42no—
136C22×C3×C4—42no—
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2411yescharacteristic
140C22×Dic3—21no—
141C22×Dic3—21no—
142C22×Dic3—22no—
143C22×Dic3C2×D412yes—
144C22×Dic3C2×D412yes—
145C2×Dic6—44no—
146C2×Dic6—44no—
147C2×Dic6—82no—
148C2×C3×C8—22no—
149C2×C3×C8—22no—
150C2×C3×C8—22no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—42no—
155C3×C4⋊C4—81no—
156C3×C4⋊C4C2×Q812yes—
157C3×C22⋊C4—24no—
158C3×C42C2×Q812yes—
159C3×C42#4112yes—
160#217—44no—
161#217—82no—
162C4⋊Dic3—22no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165C4⋊Dic3—22no—
166C4⋊Dic3—22no—
167C4⋊Dic3—42no—
168C4⋊Dic3—81no—
169C4⋊Dic3D814yes—
170C4⋊Dic3#4112yes—
171C4⋊Dic3#4112yes—
172Dic3⋊C4—44no—
173Dic3⋊C4—44no—
174C4×Dic3—22no—
175C4×Dic3—22no—
176C4×Dic3—42no—
177C4×Dic3—44no—
178C2×C3⋊C8—21no—
179C2×C3⋊C8—21no—
180C2×C3⋊C8—21no—
181C2×C3⋊C8—21no—
182C2×C3⋊C8—21no—
183C2×C3⋊C8—21no—
184C22×Q8—62no—
185C22×D4C22×D311yescharacteristic
186C23×C4—21no—
187C22×C8—31no—
188C22×C8—31no—
189C22×C8C22×D312yes—
190C4⋊Q8—64no—
191C4⋊Q8—122no—
192C4⋊Q8—124no—
193C4×D4—22no—
194C4×D4Dic614yescomplemented
195#117—124no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—31no—
199C2×C4⋊C4—31no—
200C2×C4⋊C4—31no—
201C2×C4⋊C4—31no—
202C2×C4⋊C4—32no—
203C2×C4⋊C4—62no—
204C2×C4⋊C4—62no—
205C2×C4⋊C4—61no—
206C2×C4⋊C4—61no—
207C2×C4⋊C4C22×D312yes—
208C2×C4⋊C4C22×D311yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—62no—
211C2×C22⋊C4—62no—
212C2×C42—32no—
213C2×C42—62no—
214C2×C42—61no—
215C2×C42C22×D311yescharacteristic
216#107—64no—
217C4⋊C8—22no—
218C4⋊C8—122no—
219#102—42no—
220#102—62no—
221#102—62no—
222C4×C8—62no—
223#95—32no—
224#95—62no—
225#95—62no—
226#95—61no—
227C23×C3—22no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3—41no—
233C23×C3—41no—
234C23×C3C22×D411yescharacteristic
235C3×D4—22no—
236C3×D4—22no—
237C3×D4—41no—
238C3×D4—41no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—22no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—48no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4C22⋊Q812yes—
249C2×C3×C4C22⋊Q812yes—
250C2×C3×C4C2×D812yes—
251C2×C3×C4#13712yes—
252C2×C3×C4C22×Q811yescharacteristic
253C2×C3×C4C2×#4111yescharacteristic
254C2×C3×C4C2×#4111yescharacteristic
255C2×C3×C4C2×#4111yescharacteristic
256C2×C3×C4H(5,𝔽2)11yescharacteristic
257C2×C3×C42−1+411yescharacteristic
258C2×Dic3—22no—
259C2×Dic3—22no—
260C2×Dic3—22no—
261C2×Dic3—22no—
262C2×Dic3—22no—
263C2×Dic3—22no—
264C2×Dic3—22no—
265C2×Dic3—41no—
266C2×Dic3—41no—
267C2×Dic3—41no—
268C2×Dic3—41no—
269C2×Dic3—42no—
270C2×Dic3—42no—
271C2×Dic3—42no—
272C2×Dic3C2×D814yes—
273Dic6—88no—
274C3×C8—42no—
275C3⋊C8—41no—
276C3⋊C8—41no—
277C24—21no—
278C2×Q8—124no—
279C2×Q8—124no—
280C2×Q8—242no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4C2×Dic612yes—
287C22×C4—21no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—32no—
291C22×C4—32no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—61no—
295C22×C4—61no—
296C22×C4—62no—
297C22×C4D3×D411yescharacteristic
298C22×C4D3×D411yescharacteristic
299C22×C4C23×D311yescharacteristic
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—61no—
304C2×C8—61no—
305C2×C8—61no—
306C2×C8—61no—
307C2×C8—61no—
308C2×C8—61no—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—34no—
313C4⋊C4—32no—
314C4⋊C4—32no—
315C4⋊C4—42no—
316C4⋊C4—62no—
317C4⋊C4—61no—
318C4⋊C4—61no—
319C4⋊C4—62no—
320C4⋊C4—62no—
321C4⋊C4—81no—
322C4⋊C4—124no—
323C4⋊C4—124no—
324C4⋊C4—122no—
325C4⋊C4—241no—
326C4⋊C4C2×Dic612yes—
327C22⋊C4—24no—
328C22⋊C4—124no—
329C22⋊C4—242no—
330C42—62no—
331C42—62no—
332C42—124no—
333C42—122no—
334C42C2×Dic612yes—
335C42#23712yes—
336C22×C3—42no—
337C22×C3—42no—
338C22×C3—42no—
339C22×C3—81no—
340C22×C3—81no—
341C22×C3—81no—
342C22×C3—81no—
343C22×C3C2×C22⋊Q811yescharacteristic
344C22×C3#54611yescharacteristic
345C22×C3#54811yescharacteristic
346C22×C3C22×D811yescharacteristic
347C22×C3C2×#13711yescharacteristic
348C22×C3#57711yescharacteristic
349C22×C3#57711yescharacteristic
350C3×C4—22no—
351C3×C4—42no—
352C3×C4—82no—
353C3×C4#47414yes—
354C3×C4#53311yescharacteristic
355C3×C4#54411yescharacteristic
356C3×C4#55511yescharacteristic
357C3×C4#55611yescharacteristic
358Dic3—24no—
359Dic3—42no—
360Dic3—42no—
361Dic3—42no—
362Dic3—82no—
363C23—22no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23C2×D3×D411yescharacteristic
371Q8—248no—
372D4—22no—
373D4—22no—
374D4—41no—
375D4—41no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—34no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—48no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—62no—
387C2×C4—62no—
388C2×C4—62no—
389C2×C4—62no—
390C2×C4—62no—
391C2×C4—62no—
392C2×C4—62no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4—121no—
397C2×C4—122no—
398C2×C4—122no—
399C2×C4—122no—
400C2×C4#87812yes—
401C2×C4#87812yes—
402C2×C4D3×D812yes—
403C2×C4#91512yes—
404C2×C4C22×Dic611yescharacteristic
405C2×C4C2×#23711yescharacteristic
406C2×C4#100411yescharacteristic
407C2×C4#100711yescharacteristic
408C2×C4D3×#4111yescharacteristic
409C2×C4D3×#4111yescharacteristic
410C8—42no—
411C8—121no—
412C8—121no—
413C2×C3—82no—
414C2×C3#243911yescharacteristic
415C2×C3C2×#47411yescharacteristic
416C2×C3#308411yescharacteristic
417C2×C3#329411yescharacteristic
418C2×C3#332911yescharacteristic
419C2×C3#335811yescharacteristic
420C2×C3#336911yescharacteristic
421C22—42no—
422C22—42no—
423C22—42no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22C2×#87811yescharacteristic
429C22#561211yescharacteristic
430C22#563311yescharacteristic
431C22C2×D3×D811yescharacteristic
432C22C2×#91511yescharacteristic
433C22#577511yescharacteristic
434C22#577811yescharacteristic
435C4—22no—
436C4—42no—
437C4—64no—
438C4—82no—
439C4—122no—
440C4—122no—
441C4—122no—
442C4—242no—
443C4#475914yes—
444C4#550011yescharacteristic
445C4#551711yescharacteristic
446C4#561611yescharacteristic
447C4#564111yescharacteristic
448C3#2602011yescharacteristic, complemented, Sylow
449C2—82no—
450C2#7389811yescharacteristic
451C2C2×#475911yescharacteristic
452C2#7887311yescharacteristic
453C2#8035211yescharacteristic
454C2#8045411yescharacteristic
455C2#8099211yescharacteristic
456C2#8109111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1802C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#1802D3
hypocenterC3#26020
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. #74335C2semidirect
5. #70468C2semidirect
6. #70453C2semidirect
9. #69574C2semidirect
10. #68814C2semidirect
11. #68800C2semidirect
58. C3×C4×D4Q8semidirect
194. C4×D4Dic6semidirect
448. C3#26020semidirect

Maximal subgroups

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

subgroupindex
#823372
#743352
C3×#18022
C4×#8322
#704682
#704532
#701802
#701632
#695742
#688142
#688002
#260203