Finite Group Databasebeta

← #10694897

Subgroups — #10694897

Order 768.

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

normal subgroups only

This group has 548 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×#5038C211yesmaximal, characteristic
2C3×#1507C211yesmaximal, characteristic, complemented
3#71676C211yesmaximal, characteristic, complemented
4#69982C211yesmaximal, characteristic, complemented
5#69970C211yesmaximal, characteristic, complemented
6C22⋊C4×C3⋊C8C211yesmaximal, characteristic, complemented
7#69444C211yesmaximal, characteristic, complemented
8#9815—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C22×#213C411yescharacteristic, complemented
11C2×#930C2211yescharacteristic
12C23×C3⋊C8C2211yescharacteristic
13C3×#407C2211yescharacteristic
14C3×C4×C22⋊C4C2211yescharacteristic
15C2×#832C411yescharacteristic, complemented
16#5207C2211yescharacteristic
17C2×#830C2211yescharacteristic
18C2×#822—21no—
19#5038—21no—
20#5038—21no—
21#5038—21no—
22#5038—21no—
23C2×#809C411yescharacteristic, complemented
24C2×#807C411yescharacteristic, complemented
25C2×C4×C3⋊C8—21no—
26C3×#327—22no—
27#4558—21no—
28#4525—22no—
29#4524—21no—
30#4468—22no—
31#4451—22no—
32C2×#466—31no—
33#1810—31no—
34C8×C22⋊C4—31no—
35#1507—31no—
36#1507D311yescharacteristic, complemented
37#1471—31no—
38#1471—31no—
39C22×C3×D4—21no—
40C22×C3×D4C2×C411yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C22×#67—21no—
43C22×D12C2×C411yescharacteristic
44C2×C3×M16—21no—
45C3×#121—21no—
46C3×#121—21no—
47C3×#121—21no—
48C3×#121—21no—
49C2×C3×C4⋊C4C2×C411yescharacteristic
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C42—21no—
53C2×#213—21no—
54C2×#213—21no—
55C2×#213—42no—
56#930—21no—
57#930—21no—
58#930—21no—
59#930—21no—
60C2×(C2×D3)⋊C4—21no—
61C2×C4⋊Dic3C2×C411yescharacteristic
62C2×#208—21no—
63C22×C3⋊C8—21no—
64C22×C3⋊C8—42no—
65C22×C3⋊C8C2×C411yescharacteristic
66C22×C3⋊C8C2×C411yescharacteristic
67C22×C3⋊C8D411yescharacteristic
68C22×C3⋊C8D411yescharacteristic
69C22×C3⋊C8D411yescharacteristic
70C22×C3⋊C8D411yescharacteristic
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×#95D412yes—
74#832—21no—
75#832—21no—
76#830—21no—
77#830—21no—
78#830—21no—
79#830D412yes—
80#822—42no—
81#809—21no—
82#809—21no—
83#807—21no—
84#807—21no—
85C4×C3⋊C8—42no—
86C22×D8—31no—
87C23×C8—31no—
88C2×#121—31no—
89C2×#121C2×D311yescharacteristic
90#466—61no—
91#466—61no—
92#466—61no—
93#466—61no—
94C2×#108—61no—
95C2×#107—31no—
96C2×#102—31no—
97C2×#102—31no—
98#407—31no—
99#407C2×D311yescharacteristic
100C2×C22⋊C8—31no—
101C2×C4×C8—61no—
102C4×C22⋊C4C2×D311yescharacteristic
103#341—61no—
104#336—61no—
105#327—22no—
106#327—62no—
107#323—62no—
108#323—62no—
109C24×C3—21no—
110C24×C3#4111yescharacteristic
111C23×D3—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—82no—
119C22×C3×C4—22no—
120C22×C3×C4—21no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4C22×C411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4#4111yescharacteristic
129C2×#67—41no—
130C2×#67—41no—
131C2×#67—82no—
132C22×Dic3—21no—
133C2×D12—21no—
134C2×D12—21no—
135C2×D12—41no—
136C2×D12—41no—
137C3×M16—41no—
138C3×M16—41no—
139C2×C3×C8—41no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C22⋊C4—24no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C42—41no—
146C3×C42—41no—
147#213—42no—
148#213—42no—
149(C2×D3)⋊C4—41no—
150(C2×D3)⋊C4—41no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153#208—41no—
154#208—41no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—41no—
160C2×C3⋊C8—41no—
161C2×C3⋊C8—41no—
162C2×C3⋊C8—41no—
163C2×C3⋊C8—41no—
164C2×C3⋊C8—41no—
165C2×C3⋊C8C22⋊C411yescharacteristic
166C2×C3⋊C8C22⋊C411yescharacteristic
167C2×C3⋊C8C22⋊C411yescharacteristic
168C2×C3⋊C8C22⋊C411yescharacteristic
169C22×D4—21no—
170C22×D4—31no—
171C22×D4—61no—
172C22×D4C4×D311yescharacteristic
173C23×C4C22×D311yescharacteristic
174C2×D8—61no—
175C2×D8—61no—
176C2×D8—122no—
177C2×M16—21no—
178C2×M16—61no—
179C22×C8—31no—
180C22×C8—31no—
181C22×C8—31no—
182C22×C8—31no—
183C22×C8—31no—
184C22×C8—31no—
185C22×C8—61no—
186C22×C8—122no—
187#121—21no—
188#121—21no—
189#121—21no—
190#121—21no—
191#121—61no—
192#121—61no—
193#121—61no—
194#121—61no—
195C2×C4⋊C4—31no—
196C2×C4⋊C4C4×D311yescharacteristic
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—61no—
200C2×C42—21no—
201#108—122no—
202#107—61no—
203#107—61no—
204#102—61no—
205#102—61no—
206#102—61no—
207#102—61no—
208C22⋊C8—21no—
209C22⋊C8—21no—
210C22⋊C8—32no—
211C22⋊C8—61no—
212C22⋊C8—61no—
213C22⋊C8—61no—
214C4×C8—122no—
215#95#6712yes—
216C23×C3—22no—
217C23×C3—22no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3—42no—
221C23×C3—81no—
222C23×C3—81no—
223C23×C3C4×D411yescharacteristic
224C23×C3#12111yescharacteristic
225C23×C3#12411yescharacteristic
226C22×D3—41no—
227C22×D3—42no—
228C22×D3—81no—
229C22×D3—81no—
230C3×D4—41no—
231C3×D4—41no—
232C3×D4—81no—
233C3×D4—82no—
234C3×D4—82no—
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—24no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—41no—
248C2×C3×C4—81no—
249C2×C3×C4C2×C22⋊C411yescharacteristic
250C2×C3×C4C4×D411yescharacteristic
251C2×C3×C4#12111yescharacteristic
252C2×C3×C4#12711yescharacteristic
253#67—82no—
254#67—82no—
255C2×Dic3—41no—
256C2×Dic3—81no—
257D12—41no—
258D12—41no—
259D12—81no—
260C3×C8—81no—
261C3⋊C8—21no—
262C3⋊C8—21no—
263C3⋊C8—21no—
264C3⋊C8—42no—
265C3⋊C8—81no—
266C3⋊C8#9912yes—
267C24—21no—
268C24—61no—
269C24#23711yescharacteristic
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—61no—
277C2×D4—61no—
278C2×D4—82no—
279C2×D4—121no—
280C2×D4—121no—
281C2×D4—121no—
282C2×D4—121no—
283C2×D4—242no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—61no—
289C22×C4C2×C4×D311yescharacteristic
290C22×C4#23511yescharacteristic
291C22×C4D3×D411yescharacteristic
292C22×C4D3×D411yescharacteristic
293C22×C4D3×D411yescharacteristic
294C22×C4C2×#6711yescharacteristic
295D8—122no—
296D8—122no—
297M16—41no—
298M16—41no—
299M16—121no—
300M16—121no—
301C2×C8—31no—
302C2×C8—31no—
303C2×C8—31no—
304C2×C8—31no—
305C2×C8—41no—
306C2×C8—62no—
307C2×C8—62no—
308C2×C8—62no—
309C2×C8—62no—
310C2×C8—121no—
311C2×C8—121no—
312C2×C8—121no—
313C2×C8—121no—
314C2×C8—121no—
315C2×C8—121no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—61no—
319C4⋊C4—61no—
320C22⋊C4—24no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—121no—
324C22⋊C4—121no—
325C42—41no—
326C42—41no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—44no—
335C22×C3—81no—
336C22×C3—81no—
337C22×C3—81no—
338C22×C3—162no—
339C22×C3#39011yescharacteristic
340C22×C3#44312yes—
341C22×C3#47212yes—
342C2×D3—81no—
343C2×D3—81no—
344C2×D3—81no—
345C2×D3—162no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—44no—
350C3×C4—81no—
351C3×C4C2×#9911yescharacteristic
352C3×C4#41011yescharacteristic
353Dic3—81no—
354C23—22no—
355C23—22no—
356C23—41no—
357C23—41no—
358C23—42no—
359C23—81no—
360C23—81no—
361C23—121no—
362C23—122no—
363C23—241no—
364C23—241no—
365C23#67⋊C411yescharacteristic
366C23#88511yescharacteristic
367C23#93911yescharacteristic
368D4—41no—
369D4—41no—
370D4—81no—
371D4—82no—
372D4—82no—
373D4—121no—
374D4—121no—
375D4—241no—
376D4—242no—
377D4—242no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—24no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—81no—
392C2×C4—121no—
393C2×C4—241no—
394C2×C4D3×C22⋊C411yescharacteristic
395C2×C4#88411yescharacteristic
396C2×C4C4×#6711yescharacteristic
397C2×C4#94011yescharacteristic
398C8—32no—
399C8—61no—
400C8—61no—
401C8—61no—
402C8—81no—
403C8—122no—
404C8—241no—
405C2×C3—22no—
406C2×C3—22no—
407C2×C3—41no—
408C2×C3—161no—
409C2×C3#196411yescharacteristic
410C2×C3#197411yescharacteristic
411C2×C3#197811yescharacteristic
412D3—161no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—44no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—162no—
425C22—241no—
426C22—241no—
427C22—241no—
428C22—482no—
429C22#482412yes—
430C22#495711yescharacteristic
431C22#519912yes—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—44no—
436C4—81no—
437C4—241no—
438C4#474011yescharacteristic
439C4D3×#9911yescharacteristic
440C3#981511yescharacteristic, complemented, Sylow
441C2—22no—
442C2—22no—
443C2—41no—
444C2—161no—
445C2—481no—
446C2#7003011yescharacteristic
447C2#7031611yescharacteristic
448C2#7037311yescharacteristic
subgroupG/H
centerC22#4957
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1507C2
socleC22×C3#390
prime radicalC2×#5038C2
cosocleC23×C3×C4C23
hypercenter#1507D3
hypocenterC3#9815
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1507

Complements

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

NH ≅ G/Ntype
2. C3×#1507C2semidirect
3. #71676C2semidirect
4. #69982C2semidirect
5. #69970C2semidirect
6. C22⋊C4×C3⋊C8C2semidirect
7. #69444C2semidirect
10. C22×#213C4semidirect
15. C2×#832C4semidirect
23. C2×#809C4semidirect
24. C2×#807C4semidirect
36. #1507D3semidirect
440. C3#9815semidirect

Maximal subgroups

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

subgroupindex
C2×#50382
C3×#15072
#716762
#699822
#699702
C22⋊C4×C3⋊C82
#694442
#98153