Finite Group Databasebeta

← #10705643

Subgroups — #10705643

Order 768.

778 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 778 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2404C211yesmaximal, characteristic
2C2×#4526C211yesmaximal, characteristic, complemented
3#71539C211yesmaximal, characteristic, complemented
4#67646C212yesmaximal, complemented
5#67645C212yesmaximal, complemented
6#11753—31nomaximal, Sylow
7C22×C3×#41C411yescharacteristic, complemented
8C2×C3×#120—21no—
9C22×C3×C22⋊C4C2211yescharacteristic
10C22×#217—21no—
11C3×#393C411yescharacteristic, complemented
12C3×#393C2211yescharacteristic
13C3×#392C411yescharacteristic, complemented
14C3×#392C2211yescharacteristic
15C3×#386—21no—
16C3×#386—21no—
17C3×#386—21no—
18C3×#381C411yescharacteristic, complemented
19#5207—21no—
20#4526—21no—
21#4526C2212yes—
22#4525—21no—
23#4525C2212yes—
24#2404D311yescharacteristic
25C2×#342—31no—
26#1462—31no—
27#1351—32no—
28#1343—32no—
29C25×C3D411yescharacteristic
30C2×C3×#41—41no—
31C2×C3×#41—41no—
32C2×C3×#41—44no—
33C22×C3×Q8C2×C411yescharacteristic, complemented
34C22×C3×D4—21no—
35C22×C3×D4—21no—
36C22×C3×D4—21no—
37C22×C3×D4C2×C411yescharacteristic, complemented
38C23×C3×C4—21no—
39C23×C3×C4C2×C411yescharacteristic
40C23×C3×C4D411yescharacteristic
41C23×Dic3—41no—
42C3×#120—44no—
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—41no—
49C2×C3×C22⋊C4—41no—
50C2×C3×C22⋊C4C2×C411yescharacteristic
51C2×C3×C22⋊C4D411yescharacteristic
52C2×C3×C22⋊C4D411yescharacteristic
53C2×C3×C22⋊C4C2311yescharacteristic
54C2×#217—24no—
55C2×#217—22no—
56C2×#217—41no—
57C2×#217—42no—
58C2×#208—41no—
59C3×#95—21no—
60C3×#95—21no—
61C3×#95—21no—
62C3×#95C2×C411yescharacteristic, complemented
63C3×#95C2×C411yescharacteristic, complemented
64#830—22no—
65#830—41no—
66C22×#41Dic311yescharacteristic, complemented
67C2×#120—21no—
68C22×C22⋊C4—61no—
69C22×C22⋊C4C2×D311yescharacteristic
70#407—61no—
71#393Dic311yescharacteristic, complemented
72#393C2×D311yescharacteristic
73#392Dic311yescharacteristic, complemented
74#392C2×D311yescharacteristic
75#386—21no—
76#386—21no—
77#386—21no—
78#381Dic311yescharacteristic, complemented
79#342—32no—
80#342—61no—
81#323—32no—
82#323—61no—
83C24×C3—21no—
84C24×C3—21no—
85C24×C3—22no—
86C24×C3—21no—
87C24×C3—21no—
88C24×C3—21no—
89C24×C3—42no—
90C24×C3—81no—
91C24×C3C22⋊C411yescharacteristic
92C24×C3C22⋊C411yescharacteristic
93C24×C3C2×D411yescharacteristic
94C3×#41—44no—
95C3×#41—44no—
96C3×#41—44no—
97C3×#41—44no—
98C2×C3×Q8—42no—
99C2×C3×Q8—41no—
100C2×C3×D4—24no—
101C2×C3×D4—41no—
102C2×C3×D4—42no—
103C2×C3×D4—41no—
104C2×C3×D4—41no—
105C2×C3×D4—42no—
106C2×C3×D4—42no—
107C2×C3×D4—42no—
108C2×C3×D4—42no—
109C2×C3×D4—41no—
110C2×C3×D4—41no—
111C2×C3×D4—81no—
112C2×C3×D4—81no—
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4—21no—
120C22×C3×C4—42no—
121C22×C3×C4—42no—
122C22×C3×C4—42no—
123C22×C3×C4—42no—
124C22×C3×C4—81no—
125C22×C3×C4C22⋊C411yescharacteristic
126C22×C3×C4C22⋊C411yescharacteristic
127C22×C3×C4C22×C411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×Dic3—44no—
130C22×Dic3—42no—
131C22×Dic3—82no—
132C22×Dic3—82no—
133C3×C22⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C22⋊C4—42no—
138C3×C22⋊C4—42no—
139C3×C22⋊C4—42no—
140C3×C22⋊C4—81no—
141C3×C22⋊C4—81no—
142#217—24no—
143#217—44no—
144#217—42no—
145#208—82no—
146C2×C3⋊C8—42no—
147C25#6711yescharacteristic
148C2×#41—41no—
149C2×#41—41no—
150C2×#41—44no—
151C22×Q8C2×Dic311yescharacteristic, complemented
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4C2×Dic311yescharacteristic, complemented
156C23×C4—21no—
157C23×C4—121no—
158C23×C4C2×Dic311yescharacteristic
159C23×C4#6711yescharacteristic
160C2×M16—121no—
161#120—44no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—64no—
170C2×C22⋊C4—62no—
171C2×C22⋊C4—121no—
172C2×C22⋊C4—122no—
173C2×C22⋊C4C2×Dic311yescharacteristic
174C2×C22⋊C4#6711yescharacteristic
175C2×C22⋊C4#6711yescharacteristic
176C2×C22⋊C4C22×D311yescharacteristic
177C22⋊C8—62no—
178C22⋊C8—121no—
179#95—21no—
180#95—21no—
181#95—21no—
182#95C2×Dic311yescharacteristic, complemented
183#95C2×Dic311yescharacteristic, complemented
184C23×C3—22no—
185C23×C3—22no—
186C23×C3—22no—
187C23×C3—22no—
188C23×C3—22no—
189C23×C3—21no—
190C23×C3—21no—
191C23×C3—21no—
192C23×C3—21no—
193C23×C3—21no—
194C23×C3—21no—
195C23×C3—22no—
196C23×C3—41no—
197C23×C3—42no—
198C23×C3—41no—
199C23×C3—41no—
200C23×C3—41no—
201C23×C3—41no—
202C23×C3—41no—
203C23×C3—41no—
204C23×C3—41no—
205C23×C3—42no—
206C23×C3—41no—
207C23×C3—41no—
208C23×C3—41no—
209C23×C3—81no—
210C23×C3—81no—
211C23×C3—84no—
212C23×C3—82no—
213C23×C3—82no—
214C23×C3#9911yescharacteristic
215C23×C3#9911yescharacteristic
216C23×C3#10412yes—
217C23×C3C2×C22⋊C411yescharacteristic
218C23×C3AGL(1,ℤ/8)11yescharacteristic
219C23×C3#13711yescharacteristic
220C3×Q8—42no—
221C3×Q8—42no—
222C3×D4—44no—
223C3×D4—42no—
224C3×D4—42no—
225C3×D4—44no—
226C3×D4—84no—
227C3×D4—82no—
228C3×D4—82no—
229C2×C3×C4—22no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—42no—
235C2×C3×C4—41no—
236C2×C3×C4—42no—
237C2×C3×C4—48no—
238C2×C3×C4—42no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
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×C4—41no—
250C2×C3×C4—81no—
251C2×Dic3—44no—
252C2×Dic3—88no—
253C2×Dic3—82no—
254C2×Dic3—82no—
255C3⋊C8—82no—
256C24—21no—
257C24—21no—
258C24—22no—
259C24—21no—
260C24—21no—
261C24—21no—
262C24—42no—
263C24—81no—
264C24#21711yescharacteristic
265C24#21711yescharacteristic
266C24C2×#6711yescharacteristic
267#41—44no—
268#41—44no—
269#41—44no—
270#41—44no—
271C2×Q8—42no—
272C2×Q8—41no—
273C2×D4—24no—
274C2×D4—41no—
275C2×D4—42no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—81no—
285C2×D4—81no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—81no—
298C22×C4—124no—
299C22×C4—122no—
300C22×C4—242no—
301C22×C4—242no—
302C22×C4#21711yescharacteristic
303C22×C4#21711yescharacteristic
304C22×C4C22×Dic311yescharacteristic
305C22×C4C2×#6711yescharacteristic
306M16—242no—
307C2×C8—122no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C22⋊C4—64no—
316C22⋊C4—81no—
317C22⋊C4—81no—
318C22⋊C4—124no—
319C22⋊C4—122no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—22no—
329C22×C3—21no—
330C22×C3—42no—
331C22×C3—42no—
332C22×C3—42no—
333C22×C3—41no—
334C22×C3—41no—
335C22×C3—44no—
336C22×C3—44no—
337C22×C3—42no—
338C22×C3—42no—
339C22×C3—41no—
340C22×C3—41no—
341C22×C3—41no—
342C22×C3—41no—
343C22×C3—42no—
344C22×C3—81no—
345C22×C3—81no—
346C22×C3—81no—
347C22×C3—81no—
348C22×C3—82no—
349C22×C3—82no—
350C22×C3—81no—
351C22×C3—81no—
352C22×C3AΣL(1,𝔽16)12yes—
353C22×C3C2×#9911yescharacteristic
354C22×C3#41711yescharacteristic
355C22×C3C2×#10411yescharacteristic
356C3×C4—41no—
357C3×C4—42no—
358C3×C4—41no—
359C3×C4—41no—
360C3×C4—41no—
361C3×C4—81no—
362C3×C4—81no—
363Dic3—84no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—22no—
373C23—21no—
374C23—21no—
375C23—22no—
376C23—41no—
377C23—42no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—42no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—81no—
390C23—81no—
391C23—84no—
392C23—82no—
393C23—82no—
394C23#83411yescharacteristic
395C23#83411yescharacteristic
396C23#83712yes—
397C23#94911yescharacteristic
398C23#95111yescharacteristic
399C23C2×#21711yescharacteristic
400Q8—42no—
401Q8—42no—
402D4—44no—
403D4—42no—
404D4—42no—
405D4—44no—
406D4—84no—
407D4—82no—
408D4—82no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—48no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—81no—
431C2×C4—124no—
432C2×C4—248no—
433C2×C4—242no—
434C2×C4—242no—
435C8—242no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—22no—
439C2×C3—41no—
440C2×C3—41no—
441C2×C3—42no—
442C2×C3—41no—
443C2×C3—82no—
444C2×C3#150411yescharacteristic
445C2×C3C2×AΣL(1,𝔽16)11yescharacteristic
446C2×C3#212711yescharacteristic
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—22no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—41no—
461C22—41no—
462C22—44no—
463C22—44no—
464C22—42no—
465C22—42no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—42no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22—81no—
475C22—82no—
476C22—82no—
477C22—81no—
478C22—81no—
479C22#453612yes—
480C22C2×#83411yescharacteristic
481C22#523211yescharacteristic
482C22C2×#83711yescharacteristic
483C4—41no—
484C4—41no—
485C4—41no—
486C4—42no—
487C4—41no—
488C4—81no—
489C4—81no—
490C4—244no—
491C3#1175311yescharacteristic, complemented, Sylow
492C2—21no—
493C2—21no—
494C2—22no—
495C2—41no—
496C2—41no—
497C2—42no—
498C2—41no—
499C2—82no—
500C2C2×#453611yescharacteristic
501C2#7170811yescharacteristic
502C2#7186011yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC22×C3×C4C22×C4
FrattiniC2×C22⋊C4C22×D3
Fitting, prime radicalC3×#2404C2
socleC22×C3C2×#99
cosocleC2×C3×C22⋊C4C23
hypercenter#2404D3
hypocenterC3#11753
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2404

Complements

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

NH ≅ G/Ntype
2. C2×#4526C2semidirect
3. #71539C2semidirect
4. #67646C2semidirect
5. #67645C2semidirect
7. C22×C3×#41C4semidirect
11. C3×#393C4semidirect
13. C3×#392C4semidirect
18. C3×#381C4semidirect
33. C22×C3×Q8C2×C4semidirect
37. C22×C3×D4C2×C4semidirect
62. C3×#95C2×C4semidirect
63. C3×#95C2×C4semidirect
66. C22×#41Dic3semidirect
71. #393Dic3semidirect
73. #392Dic3semidirect
78. #381Dic3semidirect
151. C22×Q8C2×Dic3semidirect
155. C22×D4C2×Dic3semidirect
182. #95C2×Dic3semidirect
183. #95C2×Dic3semidirect
491. C3#11753semidirect

Maximal subgroups

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

subgroupindex
C3×#24042
C2×#45262
#715392
#676462
#676452
#117533