Finite Group Databasebeta

← #10897668

Subgroups — #10897668

Order 768.

646 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 141 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#84000C211yesmaximal, characteristic, complemented
2#78649C211yesmaximal, characteristic, complemented
3C3×#2017C211yesmaximal, characteristic
4#72116C212yesmaximal, complemented
5#72115C211yesmaximal, characteristic, complemented
6#72114C212yesmaximal, complemented
7#70005C211yesmaximal, characteristic, complemented
8#70002C212yesmaximal, complemented
9#69998C212yesmaximal, complemented
10#68718C211yesmaximal, characteristic, complemented
11#68716C211yesmaximal, characteristic, complemented
12#29387—31nomaximal, Sylow
13C3×#575C2211yescharacteristic
14C3×#575C2212yes—
15Dic3×#41C2211yescharacteristic
16#5819C2212yes—
17C3×#408C2211yescharacteristic
18C3×#408C2212yes—
19#5240C2212yes—
20#5236C2212yes—
21C2×#837C2212yes—
22#5234C2211yescharacteristic
23#5233C2212yes—
24#5197—22no—
25#5196—22no—
26#5125C2211yescharacteristic
27#5119C2211yescharacteristic
28Dic3×M16C2211yescharacteristic
29#5102C2211yescharacteristic
30#5102C2212yes—
31C2×#815C2211yescharacteristic
32#4720—22no—
33#4719—22no—
34C3×#338C2211yescharacteristic
35#4559C2212yes—
36#4559C2212yes—
37#4558C2212yes—
38#4557C2211yescharacteristic
39#4555C2212yes—
40#2956—31no—
41#2923—31no—
42#2103—31no—
43#2098—31no—
44#2053—31no—
45#2017D311yescharacteristic
46#1938—32no—
47#1906—32no—
48#1881—32no—
49#1868—32no—
50#1841—31no—
51C2×C3×#41C2312yes—
52C2×C3×#41C2311yescharacteristic
53C3×#137—22no—
54C3×#137—22no—
55C3×#137—22no—
56C3×AGL(1,ℤ/8)—22no—
57C3×AGL(1,ℤ/8)—22no—
58C3×AGL(1,ℤ/8)—22no—
59C3×#135—22no—
60C3×#135—42no—
61C2×C3×M16C2312yes—
62C2×C3×M16C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64#948—24no—
65#948—42no—
66Q8×Dic3—21no—
67D4×Dic3—21no—
68D4×Dic3—22no—
69#926—21no—
70#926C2312yes—
71#926C2311yescharacteristic
72C2×C4×Dic3—21no—
73C2×C4×Dic3C2311yescharacteristic
74C2×#208—42no—
75C2×#208C2312yes—
76C2×#208C2311yescharacteristic
77C22×C3⋊C8C2311yescharacteristic
78C3×C22⋊C8—22no—
79C3×C22⋊C8—42no—
80#837—22no—
81#837—22no—
82#837—42no—
83#835—22no—
84#834—24no—
85#832—22no—
86#830—22no—
87#819—22no—
88#815—21no—
89#815—21no—
90#815D412yes—
91Dic3⋊C8—22no—
92Dic3⋊C8—22no—
93Dic3⋊C8—42no—
94C8×Dic3D412yes—
95#575C2×D311yescharacteristic
96#575C2×D312yes—
97#568—32no—
98C4×#41—31no—
99#444—62no—
100#443—62no—
101#433—62no—
102#432—62no—
103#430—31no—
104#424—31no—
105#424—31no—
106#424—32no—
107#421—32no—
108C2×#104—32no—
109#416—32no—
110#410—32no—
111#408—31no—
112#408C2×D311yescharacteristic
113#408C2×D312yes—
114C4×M16—31no—
115C2×#97—31no—
116#344—32no—
117#344—32no—
118#341—32no—
119#339—32no—
120#338C2×D311yescharacteristic
121H(3,ℤ/4)—31no—
122C3×#41—22no—
123C3×#41—22no—
124C3×#41—22no—
125C3×#41—22no—
126C3×#41—22no—
127C3×#41—22no—
128C2×C3×Q8#4112yes—
129C2×C3×Q8#4111yescharacteristic
130C2×C3×D4—22no—
131C2×C3×D4—21no—
132C2×C3×D4#4112yes—
133C2×C3×D4#4111yescharacteristic
134C22×C3×C4—21no—
135C22×C3×C4—22no—
136C22×C3×C4C2411yescharacteristic
137C22×Dic3—22no—
138C22×Dic3—21no—
139C3×Q16—22no—
140C3×Q16—42no—
141C3×SD16—24no—
142C3×SD16—42no—
143C3×SD16—42no—
144C3×D8—22no—
145C3×D8—42no—
146C3×M16—22no—
147C3×M16—22no—
148C3×M16—22no—
149C2×C3×C8—21no—
150C2×C3×C8—22no—
151C2×C3×C8—22no—
152C2×C3×C8C2×D412yes—
153#217—22no—
154#217—22no—
155#217#4112yes—
156C4⋊Dic3—22no—
157C4⋊Dic3—21no—
158C4⋊Dic3—22no—
159C4×Dic3—22no—
160C4×Dic3—21no—
161C4×Dic3—21no—
162C4×Dic3—22no—
163C4×Dic3C2×D411yescharacteristic
164C4×Dic3C2×D411yescharacteristic
165#208—22no—
166#208—22no—
167#208—22no—
168#208—44no—
169C2×C3⋊C8—21no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—22no—
172C2×C3⋊C8—44no—
173C2×C3⋊C8C2×D412yes—
174C2×#41C22×D311yescharacteristic
175C2×#41C22×D312yes—
176#137—22no—
177#137—22no—
178#137—22no—
179AGL(1,ℤ/8)—22no—
180AGL(1,ℤ/8)—22no—
181AGL(1,ℤ/8)—22no—
182#135—22no—
183#135—42no—
184#131—64no—
185#131—122no—
186C2×M16—31no—
187C2×M16—32no—
188C2×M16—122no—
189C2×M16C22×D312yes—
190C2×M16C22×D311yescharacteristic
191C22×C8—31no—
192C22×C8C22×D311yescharacteristic
193C4×Q8—61no—
194C4×D4—62no—
195C4×D4—61no—
196#117—31no—
197#117—32no—
198#117—61no—
199C2×C42—31no—
200C2×C42—61no—
201#108—62no—
202C4⋊C8—62no—
203C4⋊C8—62no—
204C4⋊C8—122no—
205#104—62no—
206#104—62no—
207#104—122no—
208#103—62no—
209#102—62no—
210#99—64no—
211C22⋊C8—22no—
212C22⋊C8—42no—
213C22⋊C8—62no—
214#97—32no—
215#97—61no—
216#97—61no—
217C4×C8—32no—
218C23×C3—21no—
219C23×C3—22no—
220C23×C3C2×#4111yescharacteristic
221C3×Q8—22no—
222C3×Q8—22no—
223C3×Q8—22no—
224C3×D4—22no—
225C3×D4—22no—
226C3×D4—22no—
227C3×D4—42no—
228C3×D4—42no—
229C3×D4—42no—
230C2×C3×C4—21no—
231C2×C3×C4—22no—
232C2×C3×C4—41no—
233C2×C3×C4—42no—
234C2×C3×C4—41no—
235C2×C3×C4—42no—
236C2×C3×C4C22×D411yescharacteristic
237C2×C3×C4C2×#4112yes—
238C2×C3×C4C2×#4111yescharacteristic
239C2×C3×C4H(5,𝔽2)12yes—
240C2×Dic3—21no—
241C2×Dic3—22no—
242C2×Dic3—21no—
243C2×Dic3—42no—
244C2×Dic3—81no—
245C2×Dic3—82no—
246C3×C8—22no—
247C3×C8—41no—
248C3×C8—42no—
249C3⋊C8—22no—
250C3⋊C8—41no—
251C3⋊C8—42no—
252#41—22no—
253#41—22no—
254#41—22no—
255#41—22no—
256#41—22no—
257#41—22no—
258C2×Q8#23712yes—
259C2×Q8#23711yescharacteristic
260C2×D4—22no—
261C2×D4—21no—
262C2×D4#23712yes—
263C2×D4#23711yescharacteristic
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—62no—
267C22×C4—61no—
268C22×C4C23×D311yescharacteristic
269Q16—22no—
270Q16—42no—
271SD16—24no—
272SD16—42no—
273SD16—42no—
274D8—22no—
275D8—42no—
276M16—22no—
277M16—22no—
278M16—22no—
279M16—62no—
280M16—62no—
281M16—62no—
282M16—124no—
283C2×C8—21no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—32no—
287C2×C8—61no—
288C2×C8—62no—
289C2×C8—62no—
290C2×C8—124no—
291C2×C8D3×D412yes—
292C4⋊C4—62no—
293C4⋊C4—61no—
294C4⋊C4—62no—
295C22⋊C4—32no—
296C22⋊C4—62no—
297C22⋊C4—62no—
298C42—31no—
299C42—31no—
300C42—62no—
301C42—61no—
302C42—61no—
303C42—62no—
304C22×C3—41no—
305C22×C3—42no—
306C22×C3—41no—
307C22×C3—42no—
308C22×C3—82no—
309C22×C3#52511yescharacteristic
310C22×C3#55312yes—
311C3×C4—41no—
312C3×C4—42no—
313C3×C4#52511yescharacteristic
314C3×C4#53511yescharacteristic
315C3×C4#55112yes—
316Dic3—42no—
317Dic3—41no—
318Dic3—41no—
319Dic3—82no—
320C23—21no—
321C23—22no—
322C23D3×#4111yescharacteristic
323Q8—22no—
324Q8—22no—
325Q8—22no—
326D4—22no—
327D4—22no—
328D4—22no—
329D4—42no—
330D4—42no—
331D4—42no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—62no—
341C2×C4—122no—
342C2×C4—241no—
343C2×C4—242no—
344C2×C4C2×D3×D411yescharacteristic
345C2×C4C2×#23712yes—
346C2×C4C2×#23711yescharacteristic
347C2×C4#100412yes—
348C8—22no—
349C8—41no—
350C8—42no—
351C8—62no—
352C8—122no—
353C8—121no—
354C2×C3—21no—
355C2×C3—22no—
356C2×C3—81no—
357C2×C3—82no—
358C2×C3#259811yescharacteristic
359C22—41no—
360C22—42no—
361C22—41no—
362C22—42no—
363C22—82no—
364C22#560112yes—
365C22#560611yescharacteristic
366C4—42no—
367C4—41no—
368C4—121no—
369C4—121no—
370C4—122no—
371C4—242no—
372C4#558611yescharacteristic
373C4#558712yes—
374C4#571511yescharacteristic
375C3#2938711yescharacteristic, complemented, Sylow
376C2—21no—
377C2—22no—
378C2—81no—
379C2—82no—
380C2#7669211yescharacteristic
subgroupG/H
centerC4#5715
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#2017C2
socleC2×C3#2598
hypercenter#2017D3
hypocenterC3#29387
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2017

Complements

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

NH ≅ G/Ntype
1. #84000C2semidirect
2. #78649C2semidirect
4. #72116C2semidirect
5. #72115C2semidirect
6. #72114C2semidirect
7. #70005C2semidirect
8. #70002C2semidirect
9. #69998C2semidirect
10. #68718C2semidirect
11. #68716C2semidirect
375. C3#29387semidirect

Maximal subgroups

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

subgroupindex
#840002
#786492
C3×#20172
#721162
#721152
#721142
#700052
#700022
#699982
#687182
#687162
#293873