Finite Group Databasebeta

← #10698901

Subgroups — #10698901

Order 768.

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

normal subgroups only

This group has 406 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#82296C211yesmaximal, characteristic
2C2×C3×D4⋊C8C211yesmaximal, characteristic
3#71617C211yesmaximal, characteristic
4#70438C211yesmaximal, characteristic, complemented
5#69738C211yesmaximal, characteristic, complemented
6#69731C211yesmaximal, characteristic, complemented
7#67705C211yesmaximal, characteristic, complemented
8#10461—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×C3×C4⋊C8C2211yescharacteristic
12C2×C3×C4×C8C2211yescharacteristic
13C2×#833—21no—
14#5215—21no—
15#5009—22no—
16#5008—21no—
17#5007—21no—
18#4964—21no—
19#4924C2211yescharacteristic
20#4924C2211yescharacteristic
21#4924C2211yescharacteristic
22C3×D4⋊C8—21no—
23C3×D4⋊C8—21no—
24#4530—21no—
25#4482—21no—
26#4481—21no—
27#4458—22no—
28#4456—21no—
29#4456—21no—
30#3077—31no—
31#1800—31no—
32#1567—31no—
33#1550—31no—
34#1493—31no—
35C2×D4⋊C8D311yescharacteristic
36#1292—31no—
37C22×C3×D4D411yescharacteristic
38C23×C3×C4—21no—
39C22×C3×C8—21no—
40C22×C3×C8—21no—
41C3×C4×D4—21no—
42C3×C4×D4—21no—
43C3×C4×D4—21no—
44C3×C4×D4—21no—
45C2×C3×C4⋊C4D411yescharacteristic
46C2×C3×C22⋊C4—21no—
47C2×C3×C42C2311yescharacteristic
48#926—21no—
49#924—21no—
50#924—21no—
51C2×#208—21no—
52C2×#208—21no—
53C2×#208—21no—
54#870—21no—
55#869D411yescharacteristic
56#869D411yescharacteristic
57C4×Dic6—21no—
58C4×Dic6—21no—
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×C4×C8—21no—
62C3×C4×C8D411yescharacteristic
63C3×C4×C8D411yescharacteristic
64#833—42no—
65#832—22no—
66#832—22no—
67#808—22no—
68#807—22no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)D412yes—
75#533—31no—
76C2×C4×D4C2×D311yescharacteristic
77#478—62no—
78#476—61no—
79#474—61no—
80#428—61no—
81#423—31no—
82#423—31no—
83#423—31no—
84C2×C4⋊C8C2×D311yescharacteristic
85#418—61no—
86C2×#100—61no—
87C2×C4×C8C2×D311yescharacteristic
88#343—61no—
89#335—62no—
90#333—61no—
91#332—61no—
92Q8⋊C8—61no—
93Q8⋊C8—61no—
94D4⋊C8—21no—
95D4⋊C8—21no—
96C24×C3—21no—
97C2×C3×D4—21no—
98C2×C3×D4—21no—
99C2×C3×D4—21no—
100C2×C3×D4—21no—
101C2×C3×D4—21no—
102C2×C3×D4Q1612yes—
103C22×C3×C4—21no—
104C22×C3×C4—41no—
105C22×C3×C4—41no—
106C22×C3×C4—81no—
107C22×C3×C4—81no—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C22×C3×C4#4111yescharacteristic
111C2×Dic6—21no—
112C2×Dic6—21no—
113C2×C3×C8—21no—
114C2×C3×C8—21no—
115C2×C3×C8—42no—
116C2×C3×C8—41no—
117C2×C3×C8—81no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—21no—
120C3×C22⋊C4—41no—
121C3×C22⋊C4—41no—
122C3×C42C2×D411yescharacteristic
123C3×C42C2×D411yescharacteristic
124C3×C42C2×D411yescharacteristic
125C3×C42#4111yescharacteristic
126#217—41no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129C4⋊Dic3—21no—
130C4⋊Dic3—21no—
131Dic3⋊C4—41no—
132Dic3⋊C4—41no—
133C4×Dic3—41no—
134#208—41no—
135#208—41no—
136#208—41no—
137#208—41no—
138#208—41no—
139#208—41no—
140C2×C3⋊C8—22no—
141C2×C3⋊C8—41no—
142C2×C3⋊C8—41no—
143C22×D4D1211yescharacteristic
144C23×C4—21no—
145C2×M16—61no—
146C2×M16—61no—
147C2×M16—61no—
148C22×C8—21no—
149C22×C8—21no—
150C4⋊Q8—31no—
151C4⋊Q8—31no—
152#125—61no—
153C22⋊Q8—61no—
154C22⋊Q8—61no—
155C4×Q8—61no—
156C4×Q8—61no—
157C4×D4—21no—
158C4×D4—21no—
159C4×D4—21no—
160C4×D4—21no—
161#117—61no—
162C2×C4⋊C4D1211yescharacteristic
163C2×C22⋊C4—21no—
164C2×C42C22×D311yescharacteristic
165#107—62no—
166#106—62no—
167C4⋊C8—21no—
168C4⋊C8—21no—
169C4⋊C8—32no—
170C4⋊C8—61no—
171C4⋊C8—61no—
172C4⋊C8—61no—
173C4⋊C8—61no—
174C4⋊C8—61no—
175#102—62no—
176#102—62no—
177#100—122no—
178C4×C8—21no—
179C4×C8#6711yescharacteristic
180C4×C8#6711yescharacteristic
181C23×C3—22no—
182C23×C3—41no—
183C23×C3—41no—
184C23×C3—81no—
185C23×C3—81no—
186C23×C3#12311yescharacteristic
187C3×D4—22no—
188C3×D4—22no—
189C3×D4—41no—
190C3×D4—41no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—82no—
196C2×C3×C4—81no—
197C2×C3×C4—81no—
198C2×C3×C4—81no—
199C2×C3×C4#12011yescharacteristic
200C2×C3×C4#12111yescharacteristic
201C2×C3×C4#12111yescharacteristic
202C2×C3×C4#12111yescharacteristic
203C2×C3×C4#12311yescharacteristic
204C2×C3×C4#12411yescharacteristic
205C2×C3×C4C2×Q1611yescharacteristic
206C2×C3×C4#13511yescharacteristic
207C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
208C2×C3×C4#13711yescharacteristic
209C2×Dic3—41no—
210C2×Dic3—41no—
211Dic6—41no—
212Dic6—41no—
213Dic6—81no—
214C3×C8—22no—
215C3×C8—41no—
216C3×C8—81no—
217C3⋊C8—42no—
218C3⋊C8—81no—
219C3⋊C8—81no—
220C24—21no—
221C2×Q8—61no—
222C2×Q8—61no—
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4Dic1212yes—
229C22×C4—21no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—81no—
233C22×C4—81no—
234C22×C4C2×D1211yescharacteristic
235C22×C4#23711yescharacteristic
236C22×C4#23711yescharacteristic
237M16—121no—
238M16—121no—
239M16—121no—
240M16—121no—
241M16—121no—
242M16—121no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—42no—
246C2×C8—41no—
247C2×C8—62no—
248C2×C8—81no—
249C2×C8—121no—
250C2×C8—121no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—121no—
258C4⋊C4—121no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—121no—
262C42—121no—
263C42#23511yescharacteristic
264C42D3×D411yescharacteristic
265C42D3×D411yescharacteristic
266C42C2×#6711yescharacteristic
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—42no—
270C22×C3—82no—
271C22×C3—82no—
272C22×C3—81no—
273C22×C3—81no—
274C22×C3#39411yescharacteristic
275C22×C3#48411yescharacteristic
276C22×C3#48511yescharacteristic
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—81no—
281C3×C4#44911yescharacteristic
282C3×C4#45111yescharacteristic
283C3×C4#46711yescharacteristic
284C3×C4#46811yescharacteristic
285C3×C4#47111yescharacteristic
286C3×C4#47211yescharacteristic
287Dic3—81no—
288Dic3—81no—
289C23—22no—
290C23—41no—
291C23—41no—
292C23—81no—
293C23—81no—
294C23#88611yescharacteristic
295Q8—121no—
296Q8—121no—
297Q8—241no—
298D4—22no—
299D4—22no—
300D4—41no—
301D4—41no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—82no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4#88211yescharacteristic
313C2×C4#88311yescharacteristic
314C2×C4#88511yescharacteristic
315C2×C4#90411yescharacteristic
316C2×C4C2×Dic1211yescharacteristic
317C2×C4#90811yescharacteristic
318C2×C4#90911yescharacteristic
319C2×C4#93011yescharacteristic
320C2×C4#93811yescharacteristic
321C2×C4#93911yescharacteristic
322C8—22no—
323C8—41no—
324C8—81no—
325C8—122no—
326C8—241no—
327C8—241no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3—82no—
331C2×C3#203611yescharacteristic
332C2×C3#205011yescharacteristic
333C2×C3#205111yescharacteristic
334C22—21no—
335C22—21no—
336C22—42no—
337C22—82no—
338C22—82no—
339C22—81no—
340C22—81no—
341C22#472211yescharacteristic
342C22#472311yescharacteristic
343C22#496111yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—81no—
348C4—241no—
349C4—241no—
350C4#473411yescharacteristic
351C4#473911yescharacteristic
352C4#511611yescharacteristic
353C4#513411yescharacteristic
354C4#516011yescharacteristic
355C4#517711yescharacteristic
356C3#1046111yescharacteristic, complemented, Sylow
357C2—21no—
358C2—21no—
359C2—82no—
360C2#6993711yescharacteristic
361C2#7019611yescharacteristic
362C2#7037911yescharacteristic
subgroupG/H
centerC22#4961
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×D4⋊C8C2
socleC22×C3#394
prime radicalC22×C3×D4D4
hypercenterC2×D4⋊C8D3
hypocenterC3#10461
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×D4⋊C8

Complements

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

NH ≅ G/Ntype
4. #70438C2semidirect
5. #69738C2semidirect
6. #69731C2semidirect
7. #67705C2semidirect
356. C3#10461semidirect

Maximal subgroups

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

subgroupindex
#822962
C2×C3×D4⋊C82
#716172
#704382
#697382
#697312
#677052
#104613