Finite Group Databasebeta

← #10693078

Subgroups — #10693078

Order 768.

514 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 89 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#82418C211yesmaximal, characteristic
2#78518C211yesmaximal, characteristic, complemented
3C3×#1490C211yesmaximal, characteristic, complemented
4#71599C211yesmaximal, characteristic, complemented
5#70826C211yesmaximal, characteristic, complemented
6#67752C211yesmaximal, characteristic, complemented
7#67666C211yesmaximal, characteristic, complemented
8#9494—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×#872C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13#5223C411yescharacteristic
14#5222C411yescharacteristic
15#5213—21no—
16#5210—21no—
17#5126—21no—
18#5118—21no—
19#4993C411yescharacteristic
20#4985C411yescharacteristic
21#4956—21no—
22#4944—21no—
23#4937C2211yescharacteristic
24#4926C2211yescharacteristic
25C4×#208C2211yescharacteristic
26#4709—22no—
27#4690—22no—
28C3×D4⋊C8—22no—
29D4⋊(C3⋊C8)—22no—
30#4469—21no—
31#4468—21no—
32#2921—31no—
33#2598—31no—
34#1490—31no—
35#1490D311yescharacteristic, complemented
36#1464—31no—
37#1288—31no—
38#1288—31no—
39C22×C3×D4C2×C411yescharacteristic
40C23×C3×C4—21no—
41C22×#67—21no—
42C22×C4×D3D411yescharacteristic
43C2×C3×M16—21no—
44C3×C4×D4—21no—
45C3×C4×D4—21no—
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C2×C3×C4⋊C4C2×C411yescharacteristic
49C2×C3×C22⋊C4—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#217—21no—
52C2×#217—21no—
53C2×#217—21no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4C2×C411yescharacteristic
56C2×(C2×D3)⋊C4D411yescharacteristic
57C2×Dic3⋊C4—21no—
58C2×Dic3⋊C4C2×C411yescharacteristic
59C2×Dic3⋊C4D411yescharacteristic
60C2×C4×Dic3D411yescharacteristic
61C2×#208—21no—
62#872—21no—
63#872—21no—
64#872—21no—
65#872—21no—
66C3×C4⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×#97—21no—
69C3×#97—21no—
70C3×C4×C8D412yescomplemented
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95C2×C411yescharacteristic
76C3⋊#95C2×C411yescharacteristic
77#830—21no—
78(C2×D3)⋊C8—24no—
79Dic3⋊C8—24no—
80C4⋊(C3⋊C8)—21no—
81#803—21no—
82#803—21no—
83C4×C3⋊C8D412yescomplemented
84C2×C4×D4C2×D311yescharacteristic
85C2×#117—31no—
86#433—62no—
87#433—62no—
88#432—31no—
89#432C2×D311yescharacteristic
90#423—61no—
91#407—61no—
92C4×M16—31no—
93C4×M16C2×D311yescharacteristic
94#400—31no—
95#399—31no—
96#394—31no—
97#392—31no—
98#388—61no—
99#386—61no—
100#386—61no—
101#385—61no—
102#383—31no—
103D4⋊C8—22no—
104D4⋊C8—62no—
105#324—61no—
106#323—61no—
107C24×C3—21no—
108C23×D3C22⋊C411yescharacteristic
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—42no—
112C22×C3×C4—21no—
113C22×C3×C4—41no—
114C22×C3×C4—41no—
115C22×C3×C4—81no—
116C22×C3×C4—81no—
117C22×C3×C4C22×C411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C2×#67—42no—
121C2×#67—42no—
122C2×#67—81no—
123C22×Dic3—21no—
124C22×Dic3—21no—
125C22×Dic3C22⋊C411yescharacteristic
126C22×Dic3C22⋊C411yescharacteristic
127C22×Dic3C22⋊C411yescharacteristic
128C2×C4×D3—21no—
129C2×C4×D3—21no—
130C2×C4×D3—41no—
131C2×C4×D3—41no—
132C3×M16—41no—
133C3×M16—41no—
134C2×C3×C8—22no—
135C2×C3×C8—41no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C22⋊C4—41no—
139C3×C22⋊C4—41no—
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143C3×C42#4111yescharacteristic
144#217—42no—
145#217—42no—
146#217—81no—
147(C2×D3)⋊C4—21no—
148(C2×D3)⋊C4—21no—
149(C2×D3)⋊C4—21no—
150(C2×D3)⋊C4—21no—
151(C2×D3)⋊C4—81no—
152Dic3⋊C4—21no—
153Dic3⋊C4—21no—
154Dic3⋊C4—21no—
155Dic3⋊C4—21no—
156Dic3⋊C4—81no—
157C4×Dic3—21no—
158C4×Dic3—21no—
159#208—41no—
160#208—41no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—41no—
163C22×D4—61no—
164C22×D4C4×D311yescharacteristic
165C23×C4—21no—
166C23×C4—31no—
167C2×M16—21no—
168C2×M16—61no—
169C4×D4—21no—
170C4×D4—21no—
171C4×D4—21no—
172C4×D4—21no—
173#117—61no—
174#117—61no—
175#117—61no—
176#117—61no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4C4×D311yescharacteristic
181C2×C22⋊C4—21no—
182C2×C22⋊C4—31no—
183C2×C22⋊C4—31no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—61no—
186C2×C22⋊C4—61no—
187C2×C22⋊C4—61no—
188C2×C42—31no—
189C2×C42C22×D311yescharacteristic
190C4⋊C8—21no—
191C4⋊C8—64no—
192C4⋊C8—61no—
193C22⋊C8—21no—
194C22⋊C8—64no—
195C22⋊C8—61no—
196#97—21no—
197#97—21no—
198#97—61no—
199#97—61no—
200C4×C8—32no—
201C4×C8#6712yescomplemented
202#95—31no—
203#95—31no—
204#95—61no—
205#95—61no—
206#95—61no—
207#95—61no—
208C23×C3—22no—
209C23×C3—41no—
210C23×C3—41no—
211C23×C3—41no—
212C23×C3—41no—
213C23×C3—41no—
214C23×C3—41no—
215C23×C3C2×C22⋊C411yescharacteristic
216C22×D3—21no—
217C22×D3—21no—
218C22×D3—21no—
219C22×D3—41no—
220C22×D3—41no—
221C3×D4—22no—
222C3×D4—22no—
223C3×D4—41no—
224C3×D4—41no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—82no—
233C2×C3×C4—82no—
234C2×C3×C4C4×D411yescharacteristic
235C2×C3×C4C4×D411yescharacteristic
236C2×C3×C4#12011yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12311yescharacteristic
240#67—84no—
241C2×Dic3—21no—
242C2×Dic3—21no—
243C2×Dic3—21no—
244C2×Dic3—21no—
245C2×Dic3—21no—
246C2×Dic3—41no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252C2×Dic3—41no—
253C2×Dic3—41no—
254C4×D3—42no—
255C4×D3—42no—
256C3×C8—42no—
257C3×C8—81no—
258C3⋊C8—42no—
259C3⋊C8—81no—
260C24—21no—
261C24—31no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—42no—
265C2×D4—122no—
266C2×D4—122no—
267C2×D4—241no—
268C22×C4—21no—
269C22×C4—31no—
270C22×C4—31no—
271C22×C4—31no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4—61no—
277C22×C4—61no—
278C22×C4—81no—
279C22×C4—81no—
280C22×C4—121no—
281C22×C4—121no—
282C22×C4C2×C4×D311yescharacteristic
283C22×C4D3×D411yescharacteristic
284C22×C4D3×D411yescharacteristic
285M16—41no—
286M16—41no—
287M16—121no—
288M16—121no—
289C2×C8—22no—
290C2×C8—41no—
291C2×C8—62no—
292C2×C8—121no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—61no—
296C4⋊C4—61no—
297C4⋊C4—61no—
298C4⋊C4—61no—
299C4⋊C4—241no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—61no—
303C22⋊C4—61no—
304C22⋊C4—61no—
305C22⋊C4—61no—
306C22⋊C4—122no—
307C22⋊C4—122no—
308C22⋊C4—241no—
309C22⋊C4—241no—
310C42—61no—
311C42—61no—
312C42#23511yescharacteristic
313C42D3×D411yescharacteristic
314C42#23711yescharacteristic
315C42C2×#6711yescharacteristic
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—42no—
319C22×C3—42no—
320C22×C3—42no—
321C22×C3—81no—
322C22×C3—81no—
323C22×C3—81no—
324C22×C3—81no—
325C22×C3#38611yescharacteristic
326C22×C3#42111yescharacteristic
327C22×C3#42111yescharacteristic
328C2×D3—42no—
329C2×D3—41no—
330C2×D3—41no—
331C2×D3—41no—
332C2×D3—81no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—81no—
340Dic3—41no—
341Dic3—41no—
342Dic3—41no—
343Dic3—81no—
344Dic3—81no—
345C23—22no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—61no—
353C23—61no—
354C23—61no—
355C23—121no—
356C23—121no—
357C23D3×C22⋊C411yescharacteristic
358D4—22no—
359D4—22no—
360D4—41no—
361D4—41no—
362D4—244no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—61no—
371C2×C4—61no—
372C2×C4—61no—
373C2×C4—61no—
374C2×C4—61no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—122no—
378C2×C4—122no—
379C2×C4—121no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4#67⋊C411yescharacteristic
388C2×C4#88311yescharacteristic
389C2×C4#88411yescharacteristic
390C2×C4C4×#6711yescharacteristic
391C2×C4#93711yescharacteristic
392C2×C4#93811yescharacteristic
393C8—42no—
394C8—81no—
395C8—122no—
396C8—241no—
397C2×C3—21no—
398C2×C3—21no—
399C2×C3—82no—
400C2×C3#189111yescharacteristic
401C2×C3#189711yescharacteristic
402C2×C3#191011yescharacteristic
403D3—81no—
404C22—21no—
405C22—21no—
406C22—42no—
407C22—42no—
408C22—42no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22—122no—
414C22—121no—
415C22—121no—
416C22—121no—
417C22—241no—
418C22#482111yescharacteristic
419C22#482111yescharacteristic
420C22#495611yescharacteristic
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—81no—
428C4—121no—
429C4—121no—
430C4—121no—
431C4—241no—
432C4—241no—
433C3#949411yescharacteristic, complemented, Sylow
434C2—21no—
435C2—21no—
436C2—82no—
437C2—241no—
438C2#6999211yescharacteristic
439C2#7033011yescharacteristic
440C2#7038311yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1490C2
socleC22×C3#386
prime radical#5223C4
cosocleC2×C3×C42C23
hypercenter#1490D3
hypocenterC3#9494
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1490

Complements

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

NH ≅ G/Ntype
2. #78518C2semidirect
3. C3×#1490C2semidirect
4. #71599C2semidirect
5. #70826C2semidirect
6. #67752C2semidirect
7. #67666C2semidirect
35. #1490D3semidirect
70. C3×C4×C8D4semidirect
83. C4×C3⋊C8D4semidirect
201. C4×C8#67semidirect
433. C3#9494semidirect

Maximal subgroups

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

subgroupindex
#824182
#785182
C3×#14902
#715992
#708262
#677522
#676662
#94943