Finite Group Databasebeta

← #10697180

Subgroups — #10697180

Order 768.

530 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C3×#3037C211yesmaximal, characteristic
2#83317C211yesmaximal, characteristic
3C2×#5021C211yesmaximal, characteristic, complemented
4#71800C211yesmaximal, characteristic
5#71624C211yesmaximal, characteristic, complemented
6#70508C211yesmaximal, characteristic, complemented
7#67714C211yesmaximal, characteristic, complemented
8#10181—31nomaximal, Sylow
9C2×C3×#137—21no—
10C3×#532C2211yescharacteristic
11C2×C3×C4×Q8C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C3×#464—22no—
14C3×#462—21no—
15C3×#460—21no—
16C3×#423C2211yescharacteristic
17C3×#417—21no—
18C2×#835—21no—
19#5093—21no—
20#5076—21no—
21#5040—22no—
22#5021—22no—
23#5021—22no—
24#5019—22no—
25C2×#807—21no—
26#4925C2211yescharacteristic
27#4925C2211yescharacteristic
28C2×C4⋊(C3⋊C8)C2211yescharacteristic
29D4⋊(C3⋊C8)—22no—
30#4528—22no—
31#4486—22no—
32#3206—31no—
33C2×#477—31no—
34#3037D311yescharacteristic
35#1549—31no—
36#1480—31no—
37#1461—31no—
38#1301—31no—
39C2×C3×#41—21no—
40C22×C3×Q8D411yescharacteristic
41C22×Dic6—21no—
42C3×#137—42no—
43C3×#137—42no—
44C2×C3×Q16—22no—
45C2×C3×SD16—22no—
46C2×C3×M16—21no—
47C3×C4⋊Q8D411yescharacteristic
48C3×#127D411yescharacteristic
49C3×#124—21no—
50C3×#121—21no—
51C3×#121—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×Q8—21no—
55C3×C4×Q8—21no—
56C3×C4×D4—21no—
57C3×C4×D4—21no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C42—21no—
61C2×C3×C42C2311yescharacteristic
62C2×#214—24no—
63C2×C4⋊Dic3—21no—
64C2×C4⋊Dic3—21no—
65C22×C3⋊C8—21no—
66#869—21no—
67#869—21no—
68#869—21no—
69#869D412yes—
70C3×C4⋊C8—21no—
71C3×C4⋊C8D412yes—
72C3×#103—22no—
73C3×#102—22no—
74#835—24no—
75#830—21no—
76#830D412yes—
77#810—24no—
78#807—24no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)D412yes—
83C4×C3⋊C8—21no—
84C4×C3⋊C8—21no—
85C2×#137—21no—
86#532C2×D311yescharacteristic
87C2×C4⋊Q8—31no—
88C2×C4×Q8C2×D311yescharacteristic
89#492—61no—
90#487—61no—
91#477—62no—
92#477—62no—
93#464—22no—
94#462—21no—
95#461—62no—
96#460—21no—
97#448—62no—
98#431—31no—
99#431—31no—
100C2×#107—61no—
101#423C2×D311yescharacteristic
102C2×C4⋊C8—31no—
103#417—21no—
104C2×#103—61no—
105#332—62no—
106D4⋊C8—62no—
107#324—62no—
108C3×#41—41no—
109C3×#41—41no—
110C3×#41—41no—
111C3×#41—41no—
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×Q8—21no—
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8SD1612yescomplemented
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—41no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—41no—
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4#4111yescharacteristic
129C22×Dic3—41no—
130C2×Dic6—22no—
131C2×Dic6—41no—
132C2×Dic6—82no—
133C3×Q16—44no—
134C3×SD16—44no—
135C3×M16—41no—
136C3×M16—41no—
137C2×C3×C8—22no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
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×C42—41no—
147C3×C42—41no—
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152#214—84no—
153C4⋊Dic3—22no—
154C4⋊Dic3—22no—
155C4⋊Dic3—41no—
156C4⋊Dic3—41no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—41no—
160C2×C3⋊C8—41no—
161C2×C3⋊C8—41no—
162C2×#41—21no—
163C22×Q8—61no—
164C22×Q8#6711yescharacteristic
165#137—42no—
166#137—42no—
167C2×Q16—22no—
168C2×SD16—22no—
169C2×SD16—64no—
170C2×M16—21no—
171C22×C8—61no—
172C4⋊Q8—32no—
173C4⋊Q8—61no—
174C4⋊Q8—61no—
175C4⋊Q8—61no—
176C4⋊Q8#6711yescharacteristic
177#127#6711yescharacteristic
178#124—21no—
179#121—21no—
180#121—21no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×D4—21no—
186C4×D4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4#6711yescharacteristic
191C2×C42—21no—
192C2×C42C22×D311yescharacteristic
193#107—64no—
194C4⋊C8—21no—
195C4⋊C8—32no—
196C4⋊C8—61no—
197C4⋊C8—61no—
198C4⋊C8—61no—
199C4⋊C8#6712yes—
200#103—22no—
201#103—64no—
202#103—64no—
203#102—22no—
204C22⋊C8—32no—
205C22⋊C8—61no—
206C4×C8—61no—
207C4×C8—61no—
208C23×C3—41no—
209C23×C3#12111yescharacteristic
210C3×Q8—22no—
211C3×Q8—22no—
212C3×Q8—41no—
213C3×Q8—41no—
214C3×Q8—41no—
215C3×Q8—41no—
216C3×D4—42no—
217C3×D4—41no—
218C3×D4—41no—
219C3×D4—42no—
220C3×D4—81no—
221C3×D4—81no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—81no—
230C2×C3×C4—81no—
231C2×C3×C4—81no—
232C2×C3×C4#12011yescharacteristic
233C2×C3×C4#12011yescharacteristic
234C2×C3×C4#12011yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12711yescharacteristic
238C2×C3×C4C2×SD1611yescharacteristic
239C2×C3×C4#13511yescharacteristic
240C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
241C2×C3×C4#13711yescharacteristic
242C2×Dic3—42no—
243C2×Dic3—82no—
244Dic6—44no—
245Dic6—82no—
246C3×C8—42no—
247C3⋊C8—42no—
248C3⋊C8—42no—
249C3⋊C8—81no—
250#41—41no—
251#41—41no—
252#41—41no—
253#41—41no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—62no—
261C2×Q8—121no—
262C2×Q8—242no—
263C2×Q8#21412yescomplemented
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—41no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—41no—
271C22×C4—121no—
272C22×C4D3×D411yescharacteristic
273C22×C4#23711yescharacteristic
274C22×C4C2×#6711yescharacteristic
275Q16—44no—
276SD16—44no—
277SD16—244no—
278M16—41no—
279M16—41no—
280C2×C8—22no—
281C2×C8—62no—
282C2×C8—62no—
283C2×C8—121no—
284C2×C8—121no—
285C2×C8—121no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—62no—
293C4⋊C4—62no—
294C4⋊C4—121no—
295C4⋊C4—121no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C42—41no—
299C42—41no—
300C42D3×D411yescharacteristic
301C42D3×D411yescharacteristic
302C42C2×#6711yescharacteristic
303C42C2×#6711yescharacteristic
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—81no—
307C22×C3—81no—
308C22×C3—81no—
309C22×C3#39211yescharacteristic
310C22×C3#46511yescharacteristic
311C22×C3#46811yescharacteristic
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—42no—
316C3×C4—81no—
317C3×C4—81no—
318C3×C4#44811yescharacteristic
319C3×C4#44911yescharacteristic
320C3×C4#45011yescharacteristic
321C3×C4#45211yescharacteristic
322C3×C4#47312yes—
323Dic3—82no—
324C23—41no—
325C23#93911yescharacteristic
326Q8—22no—
327Q8—22no—
328Q8—41no—
329Q8—41no—
330Q8—41no—
331Q8—41no—
332Q8—124no—
333Q8—242no—
334D4—42no—
335D4—42no—
336D4—41no—
337D4—41no—
338D4—81no—
339D4—81no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—122no—
351C2×C4—242no—
352C2×C4#93211yescharacteristic
353C2×C4C2×#21411yescharacteristic
354C2×C4#93711yescharacteristic
355C2×C4#93711yescharacteristic
356C2×C4#93811yescharacteristic
357C2×C4#93911yescharacteristic
358C2×C4#94011yescharacteristic
359C2×C4#95011yescharacteristic
360C2×C4#95111yescharacteristic
361C2×C4#95311yescharacteristic
362C8—42no—
363C8—122no—
364C8—122no—
365C8—241no—
366C2×C3—21no—
367C2×C3—21no—
368C2×C3—161no—
369C2×C3#200011yescharacteristic
370C2×C3#200811yescharacteristic
371C2×C3#202011yescharacteristic
372C22—21no—
373C22—21no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22#504111yescharacteristic
378C22#504211yescharacteristic
379C22#522311yescharacteristic
380C4—21no—
381C4—21no—
382C4—21no—
383C4—42no—
384C4—81no—
385C4—81no—
386C4—242no—
387C4#503711yescharacteristic
388C4#504011yescharacteristic
389C4#520412yes—
390C4#521811yescharacteristic
391C4#523911yescharacteristic
392C3#1018111yescharacteristic, complemented, Sylow
393C2—21no—
394C2—21no—
395C2—161no—
396C2#7168511yescharacteristic
397C2#7210911yescharacteristic
398C2#7211911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3037C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3037D3
hypocenterC3#10181
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3037

Complements

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

NH ≅ G/Ntype
3. C2×#5021C2semidirect
5. #71624C2semidirect
6. #70508C2semidirect
7. #67714C2semidirect
118. C2×C3×Q8SD16semidirect
263. C2×Q8#214semidirect
392. C3#10181semidirect

Maximal subgroups

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

subgroupindex
C3×#30372
#833172
C2×#50212
#718002
#716242
#705082
#677142
#101813