Finite Group Databasebeta

← #10691467

Subgroups — #10691467

Order 768.

550 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 111 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#79298C211yesmaximal, characteristic
2#73617C211yesmaximal, characteristic
3#70514C211yesmaximal, characteristic, complemented
4#70496C211yesmaximal, characteristic
5#69471C211yesmaximal, characteristic, complemented
6C3×#1303C211yesmaximal, characteristic, complemented
7#67736C211yesmaximal, characteristic, complemented
8#9235—31nomaximal, Sylow
9C2×C4×D12C2211yescharacteristic
10C2×#872—21no—
11#5470C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15#4988—21no—
16#4972—21no—
17C4×(C2×D3)⋊C4—21no—
18C42×Dic3C2211yescharacteristic
19#4925C2211yescharacteristic
20#4842C412yescomplemented
21#4841C412yescomplemented
22#4731—21no—
23#4721—21no—
24C4×#204—22no—
25#4669—22no—
26#4555—21no—
27#4555—21no—
28#4483—22no—
29#4452—22no—
30#3101—31no—
31#2293—31no—
32#1994—31no—
33#1503—31no—
34#1486—31no—
35#1303D311yescharacteristic, complemented
36#1276—31no—
37C22×D12D411yescharacteristic
38C22×C4×D3—21no—
39C2×C3×M16—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C42—21no—
42C2×C3×C42C2311yescharacteristic
43C2×(C2×D3)⋊C4—21no—
44C2×(C2×D3)⋊C4—22no—
45C2×(C2×D3)⋊C4—22no—
46#926—21no—
47C2×C4⋊Dic3D411yescharacteristic
48#924—21no—
49#924—21no—
50C2×Dic3⋊C4—22no—
51C2×C4×Dic3—21no—
52C2×C4×Dic3—21no—
53C2×C4×Dic3D411yescharacteristic
54C2×C4×Dic3D411yescharacteristic
55C2×#204—22no—
56C4×D12—21no—
57C4×D12—21no—
58C4×D12C2×C412yes—
59C4×D12D412yes—
60#872—42no—
61#872—42no—
62#870—21no—
63#869C2×C411yescharacteristic
64#869C2×C411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8C2×C412yes—
70C3×C22⋊C8—21no—
71C3×C4×C8—21no—
72C3×#95D412yes—
73C3⋊#95—22no—
74#830—21no—
75#821—22no—
76#817—22no—
77#816—22no—
78C4⋊(C3⋊C8)—21no—
79C4×C3⋊C8—21no—
80#533—31no—
81C2×C4×D4—31no—
82C2×#117—61no—
83#483—61no—
84#476—32no—
85#461—32no—
86#450—61no—
87C4×SD16—62no—
88#431—31no—
89#431C2×D311yescharacteristic
90#423C2×D311yescharacteristic
91#390—61no—
92#389—61no—
93#388—62no—
94C4×C4⋊C4C2×D311yescharacteristic
95C4×C22⋊C4—61no—
96C43—31no—
97#339—61no—
98#339—61no—
99#328—62no—
100D4⋊C8—62no—
101C23×D3—21no—
102C22×C3×C4—22no—
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—21no—
109C2×D12—21no—
110C2×D12—42no—
111C2×D12C22⋊C412yes—
112C2×C4×D3—22no—
113C2×C4×D3—22no—
114C2×C4×D3—41no—
115C2×C4×D3—41no—
116C2×C4×D3—41no—
117C2×C4×D3—41no—
118C2×Dic6—21no—
119C2×Dic6—21no—
120C3×M16—41no—
121C3×M16—41no—
122C2×C3×C8—22no—
123C2×C3×C8—41no—
124C3×C4⋊C4—42no—
125C3×C42—41no—
126C3×C42—41no—
127C3×C42C22×C411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42#4111yescharacteristic
130C3×C42#4111yescharacteristic
131#217—41no—
132(C2×D3)⋊C4—22no—
133(C2×D3)⋊C4—22no—
134(C2×D3)⋊C4—44no—
135(C2×D3)⋊C4—44no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3C22⋊C412yes—
142Dic3⋊C4—44no—
143Dic3⋊C4—41no—
144Dic3⋊C4—41no—
145C4×Dic3—21no—
146C4×Dic3—21no—
147C4×Dic3—21no—
148C4×Dic3—42no—
149C4×Dic3—42no—
150C4×Dic3—41no—
151C4×Dic3SD1612yescomplemented
152C2×C3⋊C8—41no—
153#204—44no—
154C22×D4—31no—
155C23×C4—61no—
156C2×SD16—62no—
157C2×M16—21no—
158C4⋊Q8—31no—
159C4⋊Q8—31no—
160#125—61no—
161C22⋊Q8—61no—
162C22⋊Q8—61no—
163C4×Q8—61no—
164C4×Q8—61no—
165C4×D4—32no—
166C4×D4—32no—
167C4×D4—61no—
168C4×D4—61no—
169#117—61no—
170#117—122no—
171#117—122no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—31no—
174C2×C4⋊C4—62no—
175C2×C22⋊C4—61no—
176C2×C22⋊C4—62no—
177C2×C22⋊C4—62no—
178C2×C42—21no—
179C2×C42—31no—
180C2×C42—31no—
181C2×C42—61no—
182C2×C42—61no—
183C2×C42C22×D311yescharacteristic
184#106—62no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—61no—
188C4⋊C8C4×D312yes—
189#103—62no—
190#102—62no—
191C22⋊C8—21no—
192C22⋊C8—61no—
193C4×C8—21no—
194C4×C8—61no—
195#95—62no—
196#95D1212yes—
197C23×C3#12011yescharacteristic
198C22×D3—22no—
199C22×D3—42no—
200C22×D3—42no—
201C22×D3—41no—
202C22×D3—41no—
203C2×C3×C4—22no—
204C2×C3×C4—41no—
205C2×C3×C4—42no—
206C2×C3×C4—42no—
207C2×C3×C4—42no—
208C2×C3×C4C2×C22⋊C411yescharacteristic
209C2×C3×C4C4×D411yescharacteristic
210C2×C3×C4C4×D411yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12311yescharacteristic
214C2×C3×C4C2×SD1611yescharacteristic
215C2×C3×C4#13511yescharacteristic
216C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
217C2×C3×C4#13711yescharacteristic
218C2×Dic3—22no—
219C2×Dic3—22no—
220C2×Dic3—22no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229D12—22no—
230D12—22no—
231D12—42no—
232C4×D3—42no—
233C4×D3—42no—
234C4×D3—42no—
235C4×D3—42no—
236Dic6—42no—
237Dic6—81no—
238C3×C8—42no—
239C3×C8—81no—
240C3⋊C8—81no—
241C24—61no—
242C2×Q8—61no—
243C2×Q8—61no—
244C2×D4—32no—
245C2×D4—61no—
246C2×D4—122no—
247C22×C4—22no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4—62no—
252C22×C4—62no—
253C22×C4—121no—
254C22×C4—121no—
255C22×C4—121no—
256C22×C4—121no—
257C22×C4C2×D1211yescharacteristic
258C22×C4D3×D411yescharacteristic
259C22×C4D3×D411yescharacteristic
260SD16—124no—
261M16—41no—
262M16—41no—
263C2×C8—22no—
264C2×C8—41no—
265C2×C8—121no—
266C4⋊C4—32no—
267C4⋊C4—42no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—124no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C22⋊C4—62no—
277C22⋊C4—62no—
278C22⋊C4—124no—
279C22⋊C4—124no—
280C22⋊C4—121no—
281C42—32no—
282C42—41no—
283C42—41no—
284C42—61no—
285C42—61no—
286C42—61no—
287C42—122no—
288C42—122no—
289C42—121no—
290C42C2×C4×D311yescharacteristic
291C42#23511yescharacteristic
292C42D3×D411yescharacteristic
293C42#23911yescharacteristic
294C22×C3—21no—
295C22×C3—21no—
296C22×C3#38611yescharacteristic
297C22×C3#45011yescharacteristic
298C22×C3#45211yescharacteristic
299C2×D3—42no—
300C2×D3—42no—
301C2×D3—42no—
302C2×D3—82no—
303C2×D3—82no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—82no—
308C3×C4#42112yes—
309C3×C4C4×SD1611yescharacteristic
310C3×C4#44011yescharacteristic
311C3×C4#46111yescharacteristic
312C3×C4#46311yescharacteristic
313Dic3—42no—
314Dic3—42no—
315Dic3—42no—
316Dic3—81no—
317Dic3—81no—
318C23—62no—
319C23—122no—
320C23—122no—
321C23—121no—
322C23—121no—
323C23#88211yescharacteristic
324Q8—122no—
325Q8—241no—
326D4—62no—
327D4—62no—
328D4—122no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—62no—
335C2×C4—62no—
336C2×C4—62no—
337C2×C4—121no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—122no—
344C2×C4—122no—
345C2×C4—122no—
346C2×C4—122no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4C4×D1211yescharacteristic
350C2×C4D3×C22⋊C411yescharacteristic
351C2×C4#88411yescharacteristic
352C2×C4#89311yescharacteristic
353C2×C4#89411yescharacteristic
354C2×C4#89511yescharacteristic
355C2×C4#91111yescharacteristic
356C2×C4D3×SD1611yescharacteristic
357C2×C4#91611yescharacteristic
358C2×C4#91811yescharacteristic
359C8—42no—
360C8—81no—
361C8—241no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3#187711yescharacteristic
365C2×C3#189711yescharacteristic
366C2×C3#190611yescharacteristic
367D3—82no—
368C22—21no—
369C22—21no—
370C22—122no—
371C22—122no—
372C22—122no—
373C22—242no—
374C22—242no—
375C22#466511yescharacteristic
376C22#477611yescharacteristic
377C22#480811yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—82no—
382C4—122no—
383C4—122no—
384C4—122no—
385C4—241no—
386C4—241no—
387C4#478711yescharacteristic
388C4#481811yescharacteristic
389C4#482112yes—
390C4#485211yescharacteristic
391C4#488911yescharacteristic
392C3#923511yescharacteristic, complemented, Sylow
393C2—21no—
394C2—21no—
395C2—242no—
396C2#6875711yescharacteristic
397C2#6889611yescharacteristic
398C2#6903911yescharacteristic
subgroupG/H
centerC2×C4#884
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1303C2
socleC22×C3#386
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenter#1303D3
hypocenterC3#9235
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
3. #70514C2semidirect
5. #69471C2semidirect
6. C3×#1303C2semidirect
7. #67736C2semidirect
20. #4842C4semidirect
21. #4841C4semidirect
35. #1303D3semidirect
151. C4×Dic3SD16semidirect
392. C3#9235semidirect

Maximal subgroups

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

subgroupindex
#792982
#736172
#705142
#704962
#694712
C3×#13032
#677362
#92353