Finite Group Databasebeta

← #10691660

Subgroups — #10691660

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
1#82934C211yesmaximal, characteristic
2#82920C211yesmaximal, characteristic
3C3×#1504C211yesmaximal, characteristic
4#69972C211yesmaximal, characteristic, complemented
5C2×#4452C211yesmaximal, characteristic, complemented
6#67688C211yesmaximal, characteristic, complemented
7#67613C211yesmaximal, characteristic, complemented
8#9263—31nomaximal, Sylow
9C3×#525C2211yescharacteristic
10C2×#924C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C2×C3×#95C2211yescharacteristic
13#5239—21no—
14#5234—21no—
15C2×#830C2211yescharacteristic
16C2×#830C2211yescharacteristic
17C2×#830C2211yescharacteristic
18#5051—21no—
19#5050C411yescharacteristic, complemented
20#5043C411yescharacteristic, complemented
21#5041—21no—
22#5040C411yescharacteristic, complemented
23#5033C411yescharacteristic, complemented
24#5011—21no—
25#5011—21no—
26D4×C3⋊C8—21no—
27D4×C3⋊C8—21no—
28#5000—21no—
29#4996—21no—
30C3×#328—21no—
31C3×#327—21no—
32#4469—21no—
33#4469—21no—
34#4452—21no—
35#4452—21no—
36#3190—31no—
37#2977—31no—
38#1504D311yescharacteristic
39C2×#328—31no—
40#1472—31no—
41#1298—31no—
42#1280—31no—
43C2×C3×#41D411yescharacteristic
44C23×C3×C4—21no—
45C23×C3×C4C2311yescharacteristic
46C22×Dic6C2×C411yescharacteristic
47C2×C3×M16—21no—
48C3×#123—21no—
49C3×C22⋊Q8C2×C411yescharacteristic
50C3×C22⋊Q8D411yescharacteristic
51C3×#121C2×C411yescharacteristic
52C3×#121D411yescharacteristic
53C3×#120—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C3×#117D411yescharacteristic
57C2×#217—21no—
58C2×#216—21no—
59C2×#214—21no—
60C2×C4⋊Dic3C2×C411yescharacteristic
61#924—21no—
62#924—21no—
63#924—21no—
64#924—21no—
65C2×Dic3⋊C4—21no—
66C2×#208—21no—
67C22×C3⋊C8—21no—
68C22×C3⋊C8—21no—
69C22×C3⋊C8—21no—
70C22×C3⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×#95—21no—
74C3×#95—21no—
75#835—21no—
76#832—21no—
77#830—21no—
78#830—21no—
79#830—21no—
80#830—21no—
81#830—21no—
82#830C2×C411yescharacteristic
83#830C2×C411yescharacteristic
84#830D411yescharacteristic
85#830D411yescharacteristic
86#830D411yescharacteristic
87#830D411yescharacteristic
88#810—21no—
89#810—21no—
90#808—21no—
91#807—21no—
92C4⋊(C3⋊C8)—21no—
93C4⋊(C3⋊C8)—21no—
94#803—21no—
95C4×C3⋊C8—21no—
96#525C2×D311yescharacteristic
97C2×C22⋊Q8—31no—
98#485—61no—
99#483—31no—
100#480—31no—
101#467—61no—
102#465—61no—
103#452—61no—
104#451—31no—
105#448—31no—
106#435—61no—
107#435—61no—
108C8×D4—61no—
109C8×D4—61no—
110#433—61no—
111#408—61no—
112#407C2×D311yescharacteristic
113C2×C22⋊C8—31no—
114C2×C22⋊C8—31no—
115C2×C22⋊C8—31no—
116C2×#95C2×D311yescharacteristic
117#328—21no—
118#328—61no—
119#328—61no—
120#327—21no—
121#324—61no—
122#324—61no—
123C24×C3#4111yescharacteristic
124C3×#41—81no—
125C2×C3×Q8C22⋊C411yescharacteristic
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4—21no—
129C2×C3×D4C22⋊C411yescharacteristic
130C22×C3×C4—21no—
131C22×C3×C4—22no—
132C22×C3×C4—41no—
133C22×C3×C4—41no—
134C22×C3×C4—81no—
135C22×C3×C4—81no—
136C22×C3×C4—81no—
137C22×C3×C4C22×C411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4#4111yescharacteristic
143C22×Dic3—21no—
144C22×Dic3—21no—
145C2×Dic6—21no—
146C2×Dic6—21no—
147C2×Dic6—81no—
148C3×M16—81no—
149C2×C3×C8—41no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4C22⋊C411yescharacteristic
153C3×C4⋊C4C22⋊C411yescharacteristic
154C3×C22⋊C4—21no—
155C3×C22⋊C4—21no—
156C3×C22⋊C4—21no—
157C3×C22⋊C4—21no—
158C3×C22⋊C4—21no—
159C3×C42—21no—
160#217—41no—
161#217—41no—
162#216—81no—
163#214—81no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166Dic3⋊C4—41no—
167Dic3⋊C4—41no—
168#208—81no—
169C2×C3⋊C8—21no—
170C2×C3⋊C8—21no—
171C2×C3⋊C8—21no—
172C2×C3⋊C8—21no—
173C2×C3⋊C8—21no—
174C2×C3⋊C8—21no—
175C2×C3⋊C8—81no—
176C2×C3⋊C8—81no—
177C2×C3⋊C8—81no—
178C2×C3⋊C8—81no—
179C2×#41D1211yescharacteristic
180C22×Q8—31no—
181C23×C4—21no—
182C23×C4C22×D311yescharacteristic
183C2×Q16—61no—
184C2×SD16—61no—
185C2×M16—21no—
186C2×M16—61no—
187C22×C8—61no—
188C22×C8—61no—
189C22×C8—61no—
190C22×C8—61no—
191#123—21no—
192C22⋊Q8—61no—
193C22⋊Q8—61no—
194C22⋊Q8—61no—
195C22⋊Q8—61no—
196C22⋊Q8C4×D311yescharacteristic
197C22⋊Q8#6711yescharacteristic
198#121C4×D311yescharacteristic
199#121#6711yescharacteristic
200#120—21no—
201C4×D4—21no—
202C4×D4—21no—
203#117D1211yescharacteristic
204C2×C4⋊C4—31no—
205C2×C4⋊C4—61no—
206C2×C22⋊C4—61no—
207#107—61no—
208#106—61no—
209C4⋊C8—61no—
210C4⋊C8—61no—
211#103—61no—
212#103—61no—
213#103—61no—
214#102—61no—
215C22⋊C8—21no—
216C22⋊C8—21no—
217C22⋊C8—31no—
218C22⋊C8—31no—
219C22⋊C8—31no—
220C22⋊C8—31no—
221C22⋊C8—31no—
222C22⋊C8—31no—
223C22⋊C8—61no—
224C22⋊C8—61no—
225C22⋊C8—61no—
226C22⋊C8—61no—
227C22⋊C8—61no—
228#97—61no—
229C4×C8—61no—
230#95—21no—
231#95—21no—
232C23×C3—21no—
233C23×C3—21no—
234C23×C3—21no—
235C23×C3—21no—
236C23×C3—21no—
237C23×C3—41no—
238C23×C3C4×D411yescharacteristic
239C23×C3#12111yescharacteristic
240C23×C3#12311yescharacteristic
241C3×Q8—41no—
242C3×D4—41no—
243C3×D4—41no—
244C3×D4—41no—
245C3×D4—41no—
246C3×D4—41no—
247C3×D4—81no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—42no—
260C2×C3×C4—81no—
261C2×C3×C4—82no—
262C2×C3×C4—81no—
263C2×C3×C4—82no—
264C2×C3×C4C2×C22⋊C411yescharacteristic
265C2×C3×C4C4×D411yescharacteristic
266C2×C3×C4#12011yescharacteristic
267C2×C3×C4#12111yescharacteristic
268C2×Dic3—41no—
269C2×Dic3—41no—
270C2×Dic3—41no—
271C2×Dic3—41no—
272C2×Dic3—81no—
273Dic6—41no—
274Dic6—41no—
275Dic6—81no—
276C3×C8—81no—
277C3⋊C8—41no—
278C3⋊C8—41no—
279C3⋊C8—41no—
280C3⋊C8—41no—
281C3⋊C8—41no—
282C3⋊C8—41no—
283C24#23711yescharacteristic
284#41—81no—
285C2×Q8—61no—
286C2×Q8—61no—
287C2×Q8—241no—
288C2×Q8(C2×D3)⋊C411yescharacteristic
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4(C2×D3)⋊C411yescharacteristic
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—61no—
298C22×C4—61no—
299C22×C4—81no—
300C22×C4—81no—
301C22×C4—81no—
302C22×C4C2×C4×D311yescharacteristic
303C22×C4C2×D1211yescharacteristic
304C22×C4D3×D411yescharacteristic
305C22×C4D3×D411yescharacteristic
306C22×C4#23711yescharacteristic
307C22×C4C2×#6711yescharacteristic
308Q16—241no—
309SD16—241no—
310M16—81no—
311M16—241no—
312C2×C8—41no—
313C2×C8—61no—
314C2×C8—61no—
315C2×C8—61no—
316C2×C8—61no—
317C2×C8—61no—
318C2×C8—61no—
319C2×C8—241no—
320C2×C8—241no—
321C2×C8—241no—
322C2×C8—241no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—61no—
326C4⋊C4—61no—
327C4⋊C4—121no—
328C4⋊C4—121no—
329C4⋊C4(C2×D3)⋊C411yescharacteristic
330C4⋊C4(C2×D3)⋊C411yescharacteristic
331C22⋊C4—21no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—121no—
337C22⋊C4—121no—
338C42—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—41no—
352C22×C3—41no—
353C22×C3#38611yescharacteristic
354C22×C3#44311yescharacteristic
355C22×C3#44411yescharacteristic
356C22×C3#47111yescharacteristic
357C22×C3#47311yescharacteristic
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—41no—
363C3×C4—41no—
364C3×C4—41no—
365C3×C4—82no—
366Dic3—81no—
367Dic3—81no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—41no—
374C23#67⋊C411yescharacteristic
375C23#88611yescharacteristic
376C23#93911yescharacteristic
377Q8—41no—
378Q8—121no—
379Q8—121no—
380Q8—241no—
381D4—41no—
382D4—41no—
383D4—41no—
384D4—41no—
385D4—41no—
386D4—81no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—81no—
400C2×C4—82no—
401C2×C4—81no—
402C2×C4—82no—
403C2×C4—121no—
404C2×C4—121no—
405C2×C4—121no—
406C2×C4—121no—
407C2×C4—241no—
408C2×C4#67⋊C411yescharacteristic
409C2×C4#88211yescharacteristic
410C2×C4C2×(C2×D3)⋊C411yescharacteristic
411C2×C4#93911yescharacteristic
412C8—81no—
413C8—121no—
414C8—121no—
415C8—121no—
416C8—121no—
417C8—121no—
418C8—121no—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—41no—
424C2×C3—81no—
425C2×C3#187611yescharacteristic
426C2×C3#190811yescharacteristic
427C2×C3#191311yescharacteristic
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22#482311yescharacteristic
443C22#482411yescharacteristic
444C22#495811yescharacteristic
445C22#520011yescharacteristic
446C22#520411yescharacteristic
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—41no—
452C4—41no—
453C4—41no—
454C4—82no—
455C4—241no—
456C4—241no—
457C3#926311yescharacteristic, complemented, Sylow
458C2—21no—
459C2—21no—
460C2—21no—
461C2—21no—
462C2—41no—
463C2—81no—
464C2#7002411yescharacteristic
465C2#7030411yescharacteristic
466C2#7037711yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1504C2
socleC22×C3#386
prime radicalC3×#525C22
cosocleC23×C3×C4C23
hypercenter#1504D3
hypocenterC3#9263
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1504

Complements

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

NH ≅ G/Ntype
4. #69972C2semidirect
5. C2×#4452C2semidirect
6. #67688C2semidirect
7. #67613C2semidirect
19. #5050C4semidirect
20. #5043C4semidirect
22. #5040C4semidirect
23. #5033C4semidirect
457. C3#9263semidirect

Maximal subgroups

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

subgroupindex
#829342
#829202
C3×#15042
#699722
C2×#44522
#676882
#676132
#92633