Finite Group Databasebeta

← #10690168

Subgroups — #10690168

Order 768.

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

normal subgroups only

This group has 558 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
1#82274C211yesmaximal, characteristic
2#78634C211yesmaximal, characteristic
3C3×#1826C211yesmaximal, characteristic, complemented
4#71590C211yesmaximal, characteristic, complemented
5C4×#806C211yesmaximal, characteristic
6#67683C211yesmaximal, characteristic, complemented
7#67618C211yesmaximal, characteristic, complemented
8#9012—31nomaximal, Sylow
9C2×C3×#117C2211yescharacteristic
10C2×C4×#67C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C3×C4×C22⋊C4C2211yescharacteristic
13C4×#217C2211yescharacteristic
14C4×#217C2211yescharacteristic
15#5207C2211yescharacteristic
16#5128—22no—
17#4993C411yescharacteristic, complemented
18#4992C411yescharacteristic, complemented
19#4985C411yescharacteristic, complemented
20#4983C411yescharacteristic, complemented
21#4959—21no—
22C2×#806—21no—
23#4952—21no—
24C4×(C2×D3)⋊C4—21no—
25C4×Dic3⋊C4—21no—
26#4729—22no—
27#4726—21no—
28#4718—21no—
29C3×#339—22no—
30#4532—22no—
31#4527—21no—
32#2975—31no—
33#2395—31no—
34#1826—31no—
35#1826D311yescharacteristic, complemented
36C4×#99—31no—
37#1297—31no—
38#1297—31no—
39C23×C3×C4C2311yescharacteristic
40C22×#67D411yescharacteristic
41C22×C4×D3D411yescharacteristic
42C2×C3×M16D412yes—
43C3×#117—21no—
44C3×#117—21no—
45C3×#117—21no—
46C3×#117—21no—
47C2×C3×C4⋊C4C2×C411yescharacteristic
48C2×C3×C4⋊C4C2×C411yescharacteristic
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4—21no—
51C2×C3×C42—21no—
52C2×C3×C42—21no—
53C2×#217—21no—
54C2×#217—21no—
55C2×#217D411yescharacteristic
56C4×#67—21no—
57C4×#67—21no—
58C4×#67—21no—
59C4×#67—21no—
60C2×(C2×D3)⋊C4—21no—
61C2×(C2×D3)⋊C4—21no—
62C2×(C2×D3)⋊C4C2×C411yescharacteristic
63C2×Dic3⋊C4—21no—
64C2×Dic3⋊C4—21no—
65C2×Dic3⋊C4C2×C411yescharacteristic
66C2×C4×Dic3—21no—
67C2×C4×Dic3—21no—
68C2×C4×Dic3D411yescharacteristic
69C2×#208D412yes—
70C2×ΓL(1,𝔽25)—22no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95C2×C411yescharacteristic
80C3⋊#95C2×C411yescharacteristic
81#830—21no—
82#830—21no—
83(C2×D3)⋊C8—22no—
84Dic3⋊C8—22no—
85C8×Dic3—22no—
86#806—24no—
87C2×C4×D4—31no—
88C2×#117C2×D311yescharacteristic
89#436—62no—
90#436—62no—
91#431—61no—
92C2×#99—61no—
93#407—31no—
94#407—61no—
95#407C2×D311yescharacteristic
96#400—31no—
97#399—31no—
98#398—31no—
99#397—31no—
100#397—61no—
101#394—61no—
102C4×C4⋊C4—61no—
103C4×C22⋊C4—31no—
104C4×C22⋊C4—31no—
105C4×C22⋊C4—61no—
106C4×C22⋊C4C2×D311yescharacteristic
107#342—61no—
108#339—22no—
109#339—62no—
110C24×C3C2×D411yescharacteristic
111C23×D3C22⋊C411yescharacteristic
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—42no—
115C22×C3×C4—41no—
116C22×C3×C4C22×C411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C22×C3×C4#4111yescharacteristic
122C2×#67—21no—
123C2×#67—21no—
124C2×#67—22no—
125C2×#67—41no—
126C22×Dic3—21no—
127C22×Dic3—21no—
128C22×Dic3—41no—
129C22×Dic3C22⋊C411yescharacteristic
130C22×Dic3C22⋊C411yescharacteristic
131C22×Dic3C22⋊C411yescharacteristic
132C2×C4×D3—21no—
133C2×C4×D3—21no—
134C2×C4×D3—42no—
135C3×M16—42no—
136C2×C3×C8—22no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C22⋊C4—22no—
142C3×C22⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C42—42no—
146C3×C42—41no—
147C3×C42—41no—
148#217—21no—
149#217—21no—
150#217—22no—
151#217—42no—
152#217—41no—
153(C2×D3)⋊C4—21no—
154(C2×D3)⋊C4—21no—
155(C2×D3)⋊C4—42no—
156(C2×D3)⋊C4—42no—
157Dic3⋊C4—21no—
158Dic3⋊C4—21no—
159Dic3⋊C4—42no—
160Dic3⋊C4—42no—
161C4×Dic3—21no—
162C4×Dic3—21no—
163C4×Dic3—42no—
164C4×Dic3—42no—
165#208—42no—
166C2×C3⋊C8—22no—
167ΓL(1,𝔽25)—82no—
168C22×D4—31no—
169C23×C4—31no—
170C23×C4C22×D311yescharacteristic
171C2×M16—32no—
172C2×M16—62no—
173C2×M16#6712yes—
174C4×D4—61no—
175C4×D4—61no—
176C4×D4—61no—
177C4×D4—61no—
178#117—21no—
179#117—21no—
180#117—21no—
181#117—21no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4C4×D311yescharacteristic
186C2×C4⋊C4C4×D311yescharacteristic
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—31no—
190C2×C22⋊C4—31no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4—61no—
193C2×C22⋊C4—61no—
194C2×C22⋊C4—61no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—31no—
198C2×C42—61no—
199C2×C42—61no—
200C4⋊C8—62no—
201#99—64no—
202C22⋊C8—21no—
203C22⋊C8—21no—
204C22⋊C8—62no—
205C22⋊C8—61no—
206C22⋊C8—61no—
207C4×C8—62no—
208#95—21no—
209#95—31no—
210#95—31no—
211#95—61no—
212#95—61no—
213#95—61no—
214#95—61no—
215#95—61no—
216C23×C3—21no—
217C23×C3—21no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3C2×C22⋊C411yescharacteristic
221C23×C3#12111yescharacteristic
222C23×C3#12111yescharacteristic
223C22×D3—21no—
224C22×D3—21no—
225C22×D3—21no—
226C22×D3—41no—
227C22×D3—41no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—42no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—81no—
242C2×C3×C4—81no—
243C2×C3×C4C4×D411yescharacteristic
244C2×C3×C4C4×D411yescharacteristic
245C2×C3×C4#12011yescharacteristic
246C2×C3×C4#12311yescharacteristic
247#67—42no—
248#67—42no—
249C2×Dic3—21no—
250C2×Dic3—21no—
251C2×Dic3—22no—
252C2×Dic3—21no—
253C2×Dic3—21no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258C2×Dic3—41no—
259C2×Dic3—42no—
260C2×Dic3—41no—
261C2×Dic3—41no—
262C2×Dic3—81no—
263C2×Dic3—81no—
264C2×Dic3#10412yes—
265C4×D3—42no—
266C4×D3—81no—
267C3×C8—42no—
268C3⋊C8—42no—
269C24—31no—
270C24D3×D411yescharacteristic
271C2×D4—62no—
272C2×D4—61no—
273C2×D4—61no—
274C2×D4—121no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—31no—
278C22×C4—31no—
279C22×C4—31no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—61no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4—122no—
287C22×C4—121no—
288C22×C4C2×C4×D311yescharacteristic
289C22×C4#23511yescharacteristic
290C22×C4D3×D411yescharacteristic
291C22×C4D3×D411yescharacteristic
292C22×C4#23711yescharacteristic
293C22×C4C2×#6711yescharacteristic
294M16—42no—
295M16—122no—
296M16—242no—
297C2×C8—22no—
298C2×C8—62no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—61no—
304C4⋊C4—61no—
305C4⋊C4—122no—
306C4⋊C4—122no—
307C22⋊C4—22no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—61no—
312C22⋊C4—61no—
313C22⋊C4—61no—
314C22⋊C4—61no—
315C22⋊C4—62no—
316C22⋊C4—122no—
317C22⋊C4—122no—
318C22⋊C4—122no—
319C22⋊C4—121no—
320C42—42no—
321C42—41no—
322C42—41no—
323C42—61no—
324C42—61no—
325C42—122no—
326C42—122no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—81no—
336C22×C3—81no—
337C22×C3#38611yescharacteristic
338C22×C3C2×#10411yescharacteristic
339C22×C3#42111yescharacteristic
340C2×D3—42no—
341C2×D3—41no—
342C2×D3—41no—
343C2×D3—41no—
344C2×D3—81no—
345C3×C4—21no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—82no—
349C3×C4—81no—
350C3×C4—81no—
351C3×C4#45812yes—
352Dic3—22no—
353Dic3—42no—
354Dic3—41no—
355Dic3—41no—
356Dic3—81no—
357Dic3—82no—
358C23—21no—
359C23—21no—
360C23—41no—
361C23—41no—
362C23—61no—
363C23—61no—
364C23—61no—
365C23—121no—
366C23—121no—
367C23D3×C22⋊C411yescharacteristic
368C23#88411yescharacteristic
369C23#93811yescharacteristic
370D4—122no—
371D4—122no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—32no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—61no—
387C2×C4—61no—
388C2×C4—62no—
389C2×C4—61no—
390C2×C4—61no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—122no—
396C2×C4—121no—
397C2×C4—121no—
398C2×C4—121no—
399C2×C4—122no—
400C2×C4—121no—
401C2×C4—121no—
402C2×C4—241no—
403C2×C4—241no—
404C2×C4—241no—
405C2×C4#67⋊C411yescharacteristic
406C2×C4#88311yescharacteristic
407C2×C4C4×#6711yescharacteristic
408C2×C4#93711yescharacteristic
409C8—42no—
410C8—122no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3—41no—
414C2×C3—41no—
415C2×C3#187111yescharacteristic
416C2×C3#188411yescharacteristic
417C2×C3#189011yescharacteristic
418D3—81no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—81no—
428C22—81no—
429C22—122no—
430C22—121no—
431C22—121no—
432C22—121no—
433C22—241no—
434C22D3×#10411yescharacteristic
435C22#482111yescharacteristic
436C22#495611yescharacteristic
437C4—21no—
438C4—21no—
439C4—21no—
440C4—62no—
441C4—82no—
442C4—81no—
443C4—81no—
444C4—122no—
445C4—121no—
446C4—121no—
447C4—241no—
448C4—242no—
449C4#505312yes—
450C3#901211yescharacteristic, complemented, Sylow
451C2—21no—
452C2—21no—
453C2—41no—
454C2—41no—
455C2—241no—
456C2#6953811yescharacteristic
457C2#7038111yescharacteristic
458C2#7041711yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1826C2
socleC22×C3#386
prime radicalC22×#67D4
cosocleC23×C3×C4C23
hypercenter#1826D3
hypocenterC3#9012
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1826

Complements

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

NH ≅ G/Ntype
3. C3×#1826C2semidirect
4. #71590C2semidirect
6. #67683C2semidirect
7. #67618C2semidirect
17. #4993C4semidirect
18. #4992C4semidirect
19. #4985C4semidirect
20. #4983C4semidirect
35. #1826D3semidirect
450. C3#9012semidirect

Maximal subgroups

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

subgroupindex
#822742
#786342
C3×#18262
#715902
C4×#8062
#676832
#676182
#90123