Finite Group Databasebeta

← #10691513

Subgroups — #10691513

Order 768.

478 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C2×#5003C211yesmaximal, characteristic
2#82210C211yesmaximal, characteristic, complemented
3C3×#1826C211yesmaximal, characteristic, complemented
4#69970C211yesmaximal, characteristic
5#69444C211yesmaximal, characteristic
6#67687C211yesmaximal, characteristic, complemented
7#67617C211yesmaximal, characteristic, complemented
8#9242—31nomaximal, Sylow
9C2×C3×#117C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C3×C4×C22⋊C4C2211yescharacteristic
13#5207C2211yescharacteristic
14C2×#830C2211yescharacteristic
15C2×#830C2211yescharacteristic
16#5003—21no—
17#5003—21no—
18#5003—21no—
19#5003—21no—
20#5000—22no—
21#4999—21no—
22#4997—21no—
23C2×#809C411yescharacteristic, complemented
24C2×#809C411yescharacteristic, complemented
25C2×#808C411yescharacteristic, complemented
26C2×#807C411yescharacteristic, complemented
27#4926—21no—
28#4925—21no—
29C3×#339—21no—
30C3×#339—21no—
31#4468—21no—
32#4468—21no—
33#4451—22no—
34C2×#482—31no—
35#2924—31no—
36#1826D311yescharacteristic, complemented
37#1507—31no—
38#1471—31no—
39#1293—31no—
40#1282—31no—
41C23×C3×C4C2311yescharacteristic
42C22×#67—21no—
43C22×D12C2×C411yescharacteristic
44C2×C3×M16—21no—
45C3×#117—21no—
46C3×#117—21no—
47C3×#117D411yescharacteristic
48C3×#117D411yescharacteristic
49C3×#117D411yescharacteristic
50C3×#117D411yescharacteristic
51C2×C3×C4⋊C4C2×C411yescharacteristic
52C2×C3×C4⋊C4C2×C411yescharacteristic
53C2×C3×C22⋊C4—21no—
54C2×C3×C22⋊C4—21no—
55C2×C3×C42—21no—
56C2×C3×C42—21no—
57#930—21no—
58#930—21no—
59#930—21no—
60#930—21no—
61C2×(C2×D3)⋊C4—21no—
62C2×C4⋊Dic3C2×C411yescharacteristic
63C2×#208—21no—
64C22×C3⋊C8—21no—
65C22×C3⋊C8C2×C411yescharacteristic
66C22×C3⋊C8C2×C411yescharacteristic
67C3×C22⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×#95—21no—
70#830—21no—
71#830—21no—
72#830—21no—
73#830—21no—
74#830D411yescharacteristic
75#830D411yescharacteristic
76#830D412yes—
77#809—21no—
78#809—21no—
79#809—21no—
80#809—21no—
81#808—21no—
82#808—21no—
83#807—21no—
84#807—21no—
85C4⋊(C3⋊C8)—22no—
86C4⋊(C3⋊C8)—22no—
87#803—22no—
88C4×C3⋊C8—22no—
89C2×#121—31no—
90C2×#117C2×D311yescharacteristic
91#482—61no—
92#482—61no—
93#482—61no—
94#482—61no—
95#433—62no—
96#432—61no—
97#431—61no—
98C2×#107—31no—
99C2×#106—31no—
100#424—61no—
101C2×#102—31no—
102C2×#102—31no—
103#407—31no—
104#407C2×D311yescharacteristic
105C2×C22⋊C8—31no—
106C2×C22⋊C8—31no—
107#405—61no—
108C4×C22⋊C4C2×D311yescharacteristic
109#339—21no—
110#339—21no—
111#327—62no—
112#323—61no—
113#323—61no—
114C24×C3#4111yescharacteristic
115C23×D3—21no—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4—41no—
121C22×C3×C4C22×C411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4#4111yescharacteristic
127C2×#67—41no—
128C2×#67—41no—
129C2×#67—82no—
130C22×Dic3—21no—
131C2×D12—21no—
132C2×D12—21no—
133C2×D12—41no—
134C2×D12—41no—
135C3×M16—41no—
136C3×M16—41no—
137C2×C3×C8—41no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4C22⋊C411yescharacteristic
141C3×C4⋊C4C22⋊C411yescharacteristic
142C3×C4⋊C4C22⋊C411yescharacteristic
143C3×C4⋊C4C22⋊C411yescharacteristic
144C3×C22⋊C4—21no—
145C3×C22⋊C4—21no—
146C3×C22⋊C4—21no—
147C3×C22⋊C4—21no—
148C3×C22⋊C4—22no—
149C3×C22⋊C4—41no—
150C3×C42—21no—
151C3×C42—21no—
152C3×C42—21no—
153C3×C42—21no—
154C3×C42—41no—
155C3×C42—41no—
156(C2×D3)⋊C4—41no—
157(C2×D3)⋊C4—41no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160#208—41no—
161#208—41no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8—21no—
164C2×C3⋊C8—21no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—21no—
167C2×C3⋊C8—41no—
168C2×C3⋊C8—41no—
169C2×C3⋊C8—41no—
170C2×C3⋊C8—41no—
171C22×D4—31no—
172C22×D4—61no—
173C23×C4C22×D311yescharacteristic
174C2×M16—21no—
175C2×M16—61no—
176C22×C8—31no—
177C22×C8—31no—
178C22×C8—61no—
179#121—61no—
180#121—61no—
181#121—61no—
182#121—61no—
183#117—21no—
184#117—21no—
185#117D1211yescharacteristic
186#117D1211yescharacteristic
187#117#6711yescharacteristic
188#117#6711yescharacteristic
189C2×C4⋊C4—31no—
190C2×C4⋊C4C4×D311yescharacteristic
191C2×C4⋊C4C4×D311yescharacteristic
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—61no—
195C2×C42—21no—
196C2×C42—21no—
197#107—61no—
198#107—61no—
199#106—61no—
200#106—61no—
201C4⋊C8—62no—
202C4⋊C8—62no—
203#102—61no—
204#102—61no—
205#102—61no—
206#102—61no—
207C22⋊C8—21no—
208C22⋊C8—21no—
209C22⋊C8—31no—
210C22⋊C8—31no—
211C22⋊C8—32no—
212C22⋊C8—61no—
213C22⋊C8—61no—
214C22⋊C8—61no—
215C22⋊C8—61no—
216#97—62no—
217C4×C8—62no—
218#95—21no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3C4×D411yescharacteristic
224C23×C3#12111yescharacteristic
225C23×C3#12311yescharacteristic
226C22×D3—41no—
227C22×D3—41no—
228C22×D3—41no—
229C22×D3—81no—
230C22×D3—81no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—42no—
247C2×C3×C4—81no—
248C2×C3×C4—81no—
249C2×C3×C4—81no—
250C2×C3×C4C2×C22⋊C411yescharacteristic
251C2×C3×C4C4×D411yescharacteristic
252C2×C3×C4#12011yescharacteristic
253C2×C3×C4#12111yescharacteristic
254#67—82no—
255#67—82no—
256C2×Dic3—41no—
257C2×Dic3—81no—
258D12—41no—
259D12—41no—
260D12—81no—
261C3×C8—81no—
262C3⋊C8—41no—
263C3⋊C8—41no—
264C3⋊C8—42no—
265C3⋊C8—41no—
266C3⋊C8—41no—
267C24—61no—
268C24#23711yescharacteristic
269C2×D4—61no—
270C2×D4—61no—
271C2×D4—121no—
272C2×D4—121no—
273C2×D4—121no—
274C2×D4—121no—
275C2×D4—242no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—61no—
282C22×C4C2×C4×D311yescharacteristic
283C22×C4C2×D1211yescharacteristic
284C22×C4D3×D411yescharacteristic
285C22×C4D3×D411yescharacteristic
286C22×C4#23711yescharacteristic
287C22×C4C2×#6711yescharacteristic
288M16—41no—
289M16—41no—
290M16—121no—
291M16—121no—
292C2×C8—41no—
293C2×C8—61no—
294C2×C8—62no—
295C2×C8—61no—
296C2×C8—61no—
297C2×C8—61no—
298C2×C8—121no—
299C2×C8—121no—
300C2×C8—121no—
301C2×C8—121no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—61no—
305C4⋊C4—61no—
306C4⋊C4(C2×D3)⋊C411yescharacteristic
307C4⋊C4(C2×D3)⋊C411yescharacteristic
308C4⋊C4(C2×D3)⋊C411yescharacteristic
309C4⋊C4(C2×D3)⋊C411yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—22no—
315C22⋊C4—41no—
316C22⋊C4—121no—
317C22⋊C4—121no—
318C42—21no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—41no—
323C42—41no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—81no—
333C22×C3—81no—
334C22×C3#38611yescharacteristic
335C22×C3#44311yescharacteristic
336C22×C3#44411yescharacteristic
337C2×D3—81no—
338C2×D3—81no—
339C2×D3—81no—
340C2×D3—161no—
341C2×D3—161no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—41no—
346C3×C4—41no—
347C3×C4—41no—
348C3×C4—41no—
349C3×C4—82no—
350C3×C4#41011yescharacteristic
351C3×C4#41011yescharacteristic
352Dic3—81no—
353C23—21no—
354C23—21no—
355C23—41no—
356C23—41no—
357C23—121no—
358C23—121no—
359C23—121no—
360C23—241no—
361C23—241no—
362C23#67⋊C411yescharacteristic
363C23#88611yescharacteristic
364C23#93911yescharacteristic
365D4—121no—
366D4—121no—
367D4—241no—
368D4—242no—
369D4—242no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—121no—
390C2×C4—241no—
391C2×C4#67⋊C411yescharacteristic
392C2×C4#88211yescharacteristic
393C2×C4C2×(C2×D3)⋊C411yescharacteristic
394C2×C4#93911yescharacteristic
395C8—81no—
396C8—121no—
397C8—121no—
398C8—122no—
399C8—121no—
400C8—121no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—41no—
404C2×C3—41no—
405C2×C3#187611yescharacteristic
406C2×C3#189911yescharacteristic
407C2×C3#190311yescharacteristic
408D3—161no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—81no—
418C22—81no—
419C22—241no—
420C22—241no—
421C22—241no—
422C22—481no—
423C22—481no—
424C22#482311yescharacteristic
425C22#482411yescharacteristic
426C22#495811yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—41no—
434C4—82no—
435C4—241no—
436C4#500611yescharacteristic
437C4#500611yescharacteristic
438C3#924211yescharacteristic, complemented, Sylow
439C2—21no—
440C2—21no—
441C2—41no—
442C2—41no—
443C2—481no—
444C2#6953511yescharacteristic
445C2#7037711yescharacteristic
446C2#7040011yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1826C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1826D3
hypocenterC3#9242
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1826

Complements

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

NH ≅ G/Ntype
2. #82210C2semidirect
3. C3×#1826C2semidirect
6. #67687C2semidirect
7. #67617C2semidirect
23. C2×#809C4semidirect
24. C2×#809C4semidirect
25. C2×#808C4semidirect
26. C2×#807C4semidirect
36. #1826D3semidirect
438. C3#9242semidirect

Maximal subgroups

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

subgroupindex
C2×#50032
#822102
C3×#18262
#699702
#694442
#676872
#676172
#92423