Finite Group Databasebeta

← #10854148

Subgroups — #10854148

Order 768.

750 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82337C211yesmaximal, characteristic
2#73716C211yesmaximal, characteristic, complemented
3C3×C4×#102C211yesmaximal, characteristic
4#71604C211yesmaximal, characteristic
5#70183C212yesmaximal
6#70180C212yesmaximal
7#69786C211yesmaximal, characteristic, complemented
8#69765C211yesmaximal, characteristic, complemented
9#69466C211yesmaximal, characteristic, complemented
10#68814C212yesmaximal, complemented
11#68795C212yesmaximal, complemented
12#26086—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C2×#926—21no—
15C2×#870—21no—
16C2×C3×#102C2211yescharacteristic
17C2×C3×#102C2211yescharacteristic
18C2×C3×C4×C8C2211yescharacteristic
19C3×C4×C4⋊C4C2211yescharacteristic
20#5221C2212yes—
21C2×#832C2212yes—
22#5212—21no—
23C2×#818C2212yes—
24C2×#818C2212yes—
25C2×Dic3⋊C8C2212yes—
26#4975C2212yes—
27Dic3×C4⋊C4C2212yes—
28#4972C2212yes—
29C2×#807C2212yes—
30#4951—22no—
31#4935C2211yescharacteristic
32#4935C2211yescharacteristic
33C4×C4⋊Dic3C2211yescharacteristic
34#4928—21no—
35C2×C4⋊(C3⋊C8)C2211yescharacteristic
36#4657—22no—
37C3×#340C2211yescharacteristic
38C3×#340C2211yescharacteristic
39#4550C2212yes—
40#4529C2212yes—
41#4529C2212yes—
42#2552—31no—
43#2354—31no—
44#2093—32no—
45#2067—32no—
46#1947—31no—
47#1947—32no—
48#1939—31no—
49#1918—32no—
50#1845—31no—
51#1802—31no—
52C4×#102D311yescharacteristic
53C22×C3×D4C2311yescharacteristic
54C23×C3×C4—21no—
55C22×C3×C8C2311yescharacteristic
56C22×C3×C8C2311yescharacteristic
57C3×C4×D4—22no—
58C3×C4×D4Q814yes—
59C2×C3×C4⋊C4C2311yescharacteristic
60C2×C3×C4⋊C4C2311yescharacteristic
61C2×C3×C4⋊C4C2311yescharacteristic
62C2×C3×C22⋊C4—21no—
63C2×C3×C42—21no—
64C2×C3×C42C2311yescharacteristic
65C2×#217—22no—
66C2×#217—22no—
67#926—44no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3—21no—
70C2×C4⋊Dic3—21no—
71C2×C4⋊Dic3—21no—
72C2×C4⋊Dic3D411yescharacteristic
73C2×C4⋊Dic3D411yescharacteristic
74C2×C4⋊Dic3C2312yes—
75C2×C4⋊Dic3C2312yes—
76C2×Dic3⋊C4—22no—
77C2×Dic3⋊C4—22no—
78C2×Dic3⋊C4—22no—
79C2×C4×Dic3—22no—
80C2×C4×Dic3C2312yes—
81C22×C3⋊C8C2312yes—
82#870—44no—
83C3×#102—22no—
84C3×#102—22no—
85C3×C4×C8—22no—
86C3×#95—21no—
87#832—42no—
88C3⋊#95—22no—
89C3⋊#95—22no—
90C3⋊#95D412yes—
91#818—24no—
92#818—42no—
93Dic3⋊C8—24no—
94#807—42no—
95C4⋊(C3⋊C8)—22no—
96C2×#125—61no—
97C2×C4×D4C2×D311yescharacteristic
98C2×#117—61no—
99C2×#107—32no—
100C2×#107—32no—
101C2×#107—32no—
102C2×C4⋊C8—32no—
103C2×C4⋊C8—31no—
104C2×#102—32no—
105C2×#102C2×D311yescharacteristic
106C2×#102C2×D311yescharacteristic
107C2×C4×C8C2×D311yescharacteristic
108#398—62no—
109#390—32no—
110#389—32no—
111#389—61no—
112#388—62no—
113#384—32no—
114#384—31no—
115#384—31no—
116#380—61no—
117C4×C4⋊C4—31no—
118C4×C4⋊C4—32no—
119C4×C4⋊C4C2×D311yescharacteristic
120#340—32no—
121#340—32no—
122#340C2×D311yescharacteristic
123#340C2×D311yescharacteristic
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—21no—
135C22×C3×C4—21no—
136C22×C3×C4—42no—
137C22×C3×C4—42no—
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2411yescharacteristic
141C22×Dic3—22no—
142C22×Dic3—22no—
143C22×Dic3C2×D412yes—
144C22×Dic3C2×D412yes—
145C2×C3×C8—21no—
146C2×C3×C8—21no—
147C2×C3×C8—21no—
148C2×C3×C8—21no—
149C2×C3×C8#4112yes—
150C2×C3×C8#4112yes—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4C2×Q812yes—
157C3×C22⋊C4—24no—
158C3×C42—42no—
159C3×C42C2×Q812yes—
160C3×C42#4112yes—
161#217—44no—
162#217—82no—
163C4⋊Dic3—22no—
164C4⋊Dic3—22no—
165C4⋊Dic3—22no—
166C4⋊Dic3—22no—
167C4⋊Dic3—22no—
168C4⋊Dic3—42no—
169C4⋊Dic3—42no—
170C4⋊Dic3—81no—
171C4⋊Dic3—81no—
172C4⋊Dic3D814yes—
173Dic3⋊C4—44no—
174Dic3⋊C4—44no—
175Dic3⋊C4—82no—
176C4×Dic3—22no—
177C4×Dic3—22no—
178C4×Dic3—44no—
179C2×C3⋊C8—22no—
180C2×C3⋊C8—22no—
181C2×C3⋊C8—22no—
182C22×D4C22×D311yescharacteristic
183C23×C4—21no—
184C22×C8—32no—
185C22×C8C22×D311yescharacteristic
186C22×C8C22×D311yescharacteristic
187#125—124no—
188C4×D4—22no—
189C4×D4Dic614yes—
190#117—124no—
191C2×C4⋊C4—31no—
192C2×C4⋊C4—31no—
193C2×C4⋊C4—32no—
194C2×C4⋊C4—32no—
195C2×C4⋊C4—62no—
196C2×C4⋊C4—62no—
197C2×C4⋊C4—61no—
198C2×C4⋊C4—61no—
199C2×C4⋊C4—62no—
200C2×C4⋊C4—61no—
201C2×C4⋊C4—61no—
202C2×C4⋊C4C22×D311yescharacteristic
203C2×C4⋊C4C22×D311yescharacteristic
204C2×C4⋊C4C22×D311yescharacteristic
205C2×C22⋊C4—21no—
206C2×C22⋊C4—62no—
207C2×C22⋊C4—62no—
208C2×C42—21no—
209C2×C42—32no—
210C2×C42—62no—
211C2×C42C22×D311yescharacteristic
212#107—64no—
213#107—122no—
214#107—122no—
215C4⋊C8—64no—
216C4⋊C8—62no—
217#102—22no—
218#102—22no—
219#102—122no—
220C4×C8—22no—
221#95—21no—
222#95—32no—
223#95—62no—
224#95—62no—
225C23×C3—22no—
226C23×C3—41no—
227C23×C3—41no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3C22×D411yescharacteristic
233C3×D4—22no—
234C3×D4—22no—
235C3×D4—41no—
236C3×D4—41no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—22no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—48no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4C22⋊Q812yes—
250C2×C3×C4C22⋊Q812yes—
251C2×C3×C4C2×D812yes—
252C2×C3×C4#13712yes—
253C2×C3×C4C22×Q811yescharacteristic
254C2×C3×C4C2×#4111yescharacteristic
255C2×C3×C4C2×#4111yescharacteristic
256C2×C3×C4C2×#4111yescharacteristic
257C2×C3×C4H(5,𝔽2)11yescharacteristic
258C2×C3×C42−1+411yescharacteristic
259C2×Dic3—22no—
260C2×Dic3—22no—
261C2×Dic3—22no—
262C2×Dic3—22no—
263C2×Dic3—22no—
264C2×Dic3—42no—
265C2×Dic3—42no—
266C2×Dic3—42no—
267C2×Dic3—42no—
268C2×Dic3—42no—
269C2×Dic3—42no—
270C2×Dic3C2×D814yes—
271C3×C8—22no—
272C3×C8—22no—
273C3⋊C8—42no—
274C24—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4C2×Dic612yes—
281C22×C4—21no—
282C22×C4—24no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—32no—
286C22×C4—32no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—62no—
290C22×C4—62no—
291C22×C4D3×D411yescharacteristic
292C22×C4D3×D411yescharacteristic
293C22×C4C23×D311yescharacteristic
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—62no—
299C2×C8—62no—
300C2×C8—62no—
301C2×C8#23512yes—
302C2×C8#23512yes—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—34no—
309C4⋊C4—62no—
310C4⋊C4—62no—
311C4⋊C4—62no—
312C4⋊C4—62no—
313C4⋊C4—62no—
314C4⋊C4—124no—
315C4⋊C4—124no—
316C4⋊C4—122no—
317C4⋊C4—122no—
318C4⋊C4—241no—
319C4⋊C4—241no—
320C4⋊C4—242no—
321C4⋊C4C2×Dic612yes—
322C22⋊C4—24no—
323C22⋊C4—124no—
324C22⋊C4—242no—
325C42—42no—
326C42—62no—
327C42—62no—
328C42—124no—
329C42C2×Dic612yes—
330C42#23712yes—
331C22×C3—42no—
332C22×C3—42no—
333C22×C3—42no—
334C22×C3—81no—
335C22×C3—81no—
336C22×C3—81no—
337C22×C3—81no—
338C22×C3C2×C22⋊Q811yescharacteristic
339C22×C3#54711yescharacteristic
340C22×C3#55011yescharacteristic
341C22×C3C22×D811yescharacteristic
342C22×C3C2×#13711yescharacteristic
343C22×C3#57611yescharacteristic
344C22×C3#57811yescharacteristic
345C3×C4—22no—
346C3×C4—42no—
347C3×C4—81no—
348C3×C4—81no—
349C3×C4#47414yes—
350C3×C4#53311yescharacteristic
351C3×C4#54411yescharacteristic
352C3×C4#55211yescharacteristic
353C3×C4#55311yescharacteristic
354Dic3—24no—
355Dic3—42no—
356Dic3—82no—
357Dic3—82no—
358C23—22no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23C2×D3×D411yescharacteristic
366D4—22no—
367D4—22no—
368D4—41no—
369D4—41no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—34no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—48no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—62no—
384C2×C4—62no—
385C2×C4—62no—
386C2×C4—62no—
387C2×C4—62no—
388C2×C4—122no—
389C2×C4—122no—
390C2×C4—122no—
391C2×C4—122no—
392C2×C4—122no—
393C2×C4—122no—
394C2×C4#87812yes—
395C2×C4#87812yes—
396C2×C4D3×D812yes—
397C2×C4#91512yes—
398C2×C4C22×Dic611yescharacteristic
399C2×C4C2×#23511yescharacteristic
400C2×C4C2×#23511yescharacteristic
401C2×C4C2×#23711yescharacteristic
402C2×C4#100911yescharacteristic
403C2×C4#101011yescharacteristic
404C8—22no—
405C8—22no—
406C8—122no—
407C2×C3—82no—
408C2×C3#244111yescharacteristic
409C2×C3C2×#47411yescharacteristic
410C2×C3#308311yescharacteristic
411C2×C3#333411yescharacteristic
412C2×C3#335711yescharacteristic
413C2×C3#339111yescharacteristic
414C2×C3#339811yescharacteristic
415C22—42no—
416C22—42no—
417C22—42no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22C2×#87811yescharacteristic
423C22#555111yescharacteristic
424C22#555311yescharacteristic
425C22C2×D3×D811yescharacteristic
426C22C2×#91511yescharacteristic
427C22#576911yescharacteristic
428C22#577111yescharacteristic
429C4—22no—
430C4—42no—
431C4—64no—
432C4—81no—
433C4—81no—
434C4—122no—
435C4—242no—
436C4—242no—
437C4#475914yes—
438C4#547011yescharacteristic
439C4#551911yescharacteristic
440C4#554111yescharacteristic
441C4#556511yescharacteristic
442C3#2608611yescharacteristic, complemented, Sylow
443C2—82no—
444C2#7349411yescharacteristic
445C2C2×#475911yescharacteristic
446C2#7882311yescharacteristic
447C2#7957811yescharacteristic
448C2#7961411yescharacteristic
449C2#7964011yescharacteristic
450C2#7969311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#102C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC4×#102D3
hypocenterC3#26086
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
2. #73716C2semidirect
7. #69786C2semidirect
8. #69765C2semidirect
9. #69466C2semidirect
10. #68814C2semidirect
11. #68795C2semidirect
442. C3#26086semidirect

Maximal subgroups

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

subgroupindex
#823372
#737162
C3×C4×#1022
#716042
#701832
#701802
#697862
#697652
#694662
#688142
#687952
#260863