Finite Group Databasebeta

← #10692254

Subgroups — #10692254

Order 768.

470 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 93 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#82616C211yesmaximal, characteristic
2#78584C211yesmaximal, characteristic, complemented
3C3×#1519C211yesmaximal, characteristic
4#71676C211yesmaximal, characteristic
5#69503C211yesmaximal, characteristic, complemented
6#67680C211yesmaximal, characteristic, complemented
7#67626C211yesmaximal, characteristic, complemented
8#9348—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×#926C2211yescharacteristic
11C2×C3×C22⋊C8C2211yescharacteristic
12C3×#380C2211yescharacteristic
13#5215C411yescharacteristic, complemented
14#5215C411yescharacteristic, complemented
15#5207C2211yescharacteristic
16#5207C2211yescharacteristic
17C2×#830C2211yescharacteristic
18#5102—21no—
19#5101—21no—
20#5035—21no—
21#5035—21no—
22#5034—21no—
23#5033—21no—
24#4964C411yescharacteristic, complemented
25#4964C411yescharacteristic, complemented
26#4720—22no—
27#4719—21no—
28#4718—21no—
29C3×#327—21no—
30C3×#327—21no—
31#4529—21no—
32#4529—21no—
33#4525—22no—
34#3094—31no—
35#2924—31no—
36#1825—31no—
37#1519D311yescharacteristic
38#1471—31no—
39#1305—31no—
40#1298—31no—
41C22×C3×D4—21no—
42C22×C3×D4C2×C411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C22×C3×C8—21no—
45C3×#121—21no—
46C3×#121—21no—
47C3×#121—21no—
48C3×#121—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4C2×C411yescharacteristic
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4—21no—
53C2×#217—21no—
54#926—21no—
55#926—21no—
56#926C2×C411yescharacteristic
57#926C2×C411yescharacteristic
58#926D411yescharacteristic
59#926D411yescharacteristic
60C2×C4⋊Dic3D411yescharacteristic
61C2×C4⋊Dic3D411yescharacteristic
62C2×C4×Dic3—21no—
63C2×#208—21no—
64C2×#208C2×C411yescharacteristic
65C2×#208C2×C411yescharacteristic
66C22×C3⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×C22⋊C8D411yescharacteristic
69C3×C22⋊C8D411yescharacteristic
70C3×#95—21no—
71#832—21no—
72#832—21no—
73#832—21no—
74#832—21no—
75#830—21no—
76#830—21no—
77#830—21no—
78#830—21no—
79#830—21no—
80#830D412yes—
81#815—22no—
82Dic3⋊C8—22no—
83Dic3⋊C8—22no—
84C8×Dic3—22no—
85#808—21no—
86#808—21no—
87#807—21no—
88#807—21no—
89C2×#121C2×D311yescharacteristic
90C2×#117—31no—
91#482—61no—
92#482—61no—
93#481—61no—
94#480—61no—
95#433—62no—
96#432—61no—
97#431—61no—
98#428—31no—
99#428—31no—
100#424—61no—
101#418—31no—
102#418—31no—
103#407—31no—
104#407—31no—
105C2×C22⋊C8—31no—
106C2×C22⋊C8C2×D311yescharacteristic
107#405—61no—
108#380C2×D311yescharacteristic
109#340—61no—
110#340—61no—
111#327—21no—
112#327—21no—
113#323—62no—
114C24×C3—21no—
115C24×C3#4111yescharacteristic
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—81no—
123C2×C3×D4—81no—
124C22×C3×C4—21no—
125C22×C3×C4—41no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4C22×C411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4#4111yescharacteristic
134C22×Dic3—21no—
135C22×Dic3—21no—
136C2×C3×C8—21no—
137C2×C3×C8—21no—
138C2×C3×C8—41no—
139C2×C3×C8—41no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C22⋊C4—41no—
146C3×C22⋊C4—81no—
147#217—21no—
148#217—21no—
149#217—21no—
150#217—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3C22⋊C411yescharacteristic
154C4⋊Dic3C22⋊C411yescharacteristic
155C4⋊Dic3C22⋊C411yescharacteristic
156C4⋊Dic3C22⋊C411yescharacteristic
157C4×Dic3—21no—
158C4×Dic3—21no—
159C4×Dic3—21no—
160C4×Dic3—21no—
161#208—41no—
162#208—41no—
163#208—41no—
164#208—41no—
165C2×C3⋊C8—21no—
166C2×C3⋊C8—21no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—41no—
169C2×C3⋊C8—41no—
170C2×C3⋊C8—41no—
171C22×D4—21no—
172C22×D4C4×D311yescharacteristic
173C23×C4C22×D311yescharacteristic
174C2×M16—31no—
175C2×M16—31no—
176C2×M16—61no—
177C22×C8—21no—
178C22×C8—61no—
179#121—21no—
180#121—21no—
181#121—21no—
182#121—21no—
183#117—31no—
184#117—31no—
185#117—31no—
186#117—31no—
187#117—61no—
188#117—61no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—31no—
191C2×C4⋊C4—31no—
192C2×C4⋊C4C4×D311yescharacteristic
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—61no—
196C2×C42—61no—
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—32no—
209C22⋊C8—61no—
210C22⋊C8—61no—
211C22⋊C8—61no—
212C22⋊C8—61no—
213C22⋊C8—61no—
214C22⋊C8#6711yescharacteristic
215C22⋊C8#6711yescharacteristic
216#97—62no—
217C4×C8—62no—
218#95—21no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—42no—
224C23×C3—41no—
225C23×C3—81no—
226C23×C3—81no—
227C23×C3C4×D411yescharacteristic
228C23×C3#12111yescharacteristic
229C23×C3#12311yescharacteristic
230C3×D4—41no—
231C3×D4—41no—
232C3×D4—81no—
233C3×D4—82no—
234C3×D4—81no—
235C3×D4—81no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—81no—
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
254C2×Dic3—21no—
255C2×Dic3—21no—
256C2×Dic3—21no—
257C2×Dic3—21no—
258C2×Dic3—41no—
259C2×Dic3—41no—
260C2×Dic3—81no—
261C3×C8—41no—
262C3×C8—41no—
263C3⋊C8—42no—
264C3⋊C8—41no—
265C3⋊C8—41no—
266C3⋊C8—81no—
267C24—21no—
268C24#23711yescharacteristic
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—81no—
276C2×D4—81no—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—61no—
282C22×C4—61no—
283C22×C4C2×C4×D311yescharacteristic
284C22×C4#23511yescharacteristic
285C22×C4D3×D411yescharacteristic
286C22×C4D3×D411yescharacteristic
287C22×C4D3×D411yescharacteristic
288C22×C4C2×#6711yescharacteristic
289M16—121no—
290M16—121no—
291M16—121no—
292M16—121no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—61no—
298C2×C8—61no—
299C2×C8—62no—
300C2×C8—121no—
301C2×C8—121no—
302C2×C8—121no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—31no—
306C4⋊C4—31no—
307C4⋊C4—31no—
308C4⋊C4—31no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—61no—
312C4⋊C4—61no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—61no—
316C22⋊C4—61no—
317C22⋊C4—61no—
318C22⋊C4—61no—
319C22⋊C4—81no—
320C42—61no—
321C42—61no—
322C42—61no—
323C42—61no—
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—81no—
335C22×C3—81no—
336C22×C3—81no—
337C22×C3—162no—
338C22×C3#38611yescharacteristic
339C22×C3#47111yescharacteristic
340C22×C3#47211yescharacteristic
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—81no—
345C3×C4—161no—
346C3×C4#41011yescharacteristic
347C3×C4#41011yescharacteristic
348Dic3—41no—
349Dic3—41no—
350Dic3—41no—
351Dic3—41no—
352C23—21no—
353C23—21no—
354C23—41no—
355C23—41no—
356C23—42no—
357C23—41no—
358C23—81no—
359C23—81no—
360C23#67⋊C411yescharacteristic
361C23#88311yescharacteristic
362C23#93811yescharacteristic
363D4—41no—
364D4—41no—
365D4—81no—
366D4—82no—
367D4—81no—
368D4—81no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—61no—
380C2×C4—61no—
381C2×C4—61no—
382C2×C4—61no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—241no—
390C2×C4D3×C22⋊C411yescharacteristic
391C2×C4#88411yescharacteristic
392C2×C4C4×#6711yescharacteristic
393C2×C4#93711yescharacteristic
394C8—41no—
395C8—41no—
396C8—122no—
397C8—121no—
398C8—121no—
399C8—241no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—41no—
403C2×C3—41no—
404C2×C3—161no—
405C2×C3#188111yescharacteristic
406C2×C3#189911yescharacteristic
407C2×C3#190711yescharacteristic
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—162no—
422C22#495611yescharacteristic
423C22#516011yescharacteristic
424C22#517711yescharacteristic
425C4—21no—
426C4—21no—
427C4—21no—
428C4—81no—
429C4—121no—
430C4—121no—
431C4—121no—
432C4—121no—
433C4—161no—
434C4#474011yescharacteristic
435C4#474011yescharacteristic
436C3#934811yescharacteristic, complemented, Sylow
437C2—21no—
438C2—21no—
439C2—41no—
440C2—41no—
441C2—161no—
442C2#7002911yescharacteristic
443C2#7021811yescharacteristic
444C2#7041611yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1519C2
socleC22×C3#386
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenter#1519D3
hypocenterC3#9348
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1519

Complements

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

NH ≅ G/Ntype
2. #78584C2semidirect
5. #69503C2semidirect
6. #67680C2semidirect
7. #67626C2semidirect
13. #5215C4semidirect
14. #5215C4semidirect
24. #4964C4semidirect
25. #4964C4semidirect
436. C3#9348semidirect

Maximal subgroups

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

subgroupindex
#826162
#785842
C3×#15192
#716762
#695032
#676802
#676262
#93483