Finite Group Databasebeta

← #10705689

Subgroups — #10705689

Order 768.

622 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 87 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#82192C211yesmaximal, characteristic
2C3×#1463C211yesmaximal, characteristic, complemented
3C2×#4530C211yesmaximal, characteristic, complemented
4#67457C212yesmaximal, complemented
5#67456C212yesmaximal, complemented
6#11762—31nomaximal, Sylow
7C22×#235C411yescharacteristic
8C22×C3×C4⋊C4C2211yescharacteristic
9C2×#929—21no—
10C22×#208—21no—
11C3×#407—21no—
12#5213—21no—
13#5211C411yescharacteristic
14#4991C411yescharacteristic
15#4991C2211yescharacteristic
16#4989—21no—
17#4989—21no—
18#4979C411yescharacteristic
19#4979C2211yescharacteristic
20C3×#324—21no—
21C3×#324C2212yes—
22#4530—21no—
23#4530C2212yes—
24#2406—31no—
25C2×#343—31no—
26#1463D311yescharacteristic, complemented
27#1354—32no—
28#1345—32no—
29C23×C3×C4C2×C411yescharacteristic
30C23×C3×C4D411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C22×#67—21no—
33C2×#235—41no—
34C2×#235—41no—
35C2×#235—44no—
36C22×D12C2×C411yescharacteristic
37C22×C4×D3—21no—
38C22×Dic6C2×C411yescharacteristic
39C2×C3×M16—41no—
40C2×C3×C4⋊C4—41no—
41C2×C3×C4⋊C4—41no—
42C2×C3×C4⋊C4C2×C411yescharacteristic
43C2×C3×C4⋊C4D411yescharacteristic
44C2×C3×C4⋊C4D411yescharacteristic
45C2×C3×C4⋊C4C2311yescharacteristic
46C2×#217—21no—
47#929—44no—
48C2×(C2×D3)⋊C4—21no—
49C2×(C2×D3)⋊C4—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4—21no—
54C2×#208—24no—
55C2×#208—22no—
56C2×#208—41no—
57C2×#208—42no—
58C22×C3⋊C8—41no—
59C3×C22⋊C8—22no—
60C3×C22⋊C8—41no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95C2×C411yescharacteristic
65C3⋊#95C2×C411yescharacteristic
66C22×#41—31no—
67C22×M16—61no—
68C2×#123—61no—
69C22×C4⋊C4C2×D311yescharacteristic
70#407—21no—
71#395—31no—
72#395—31no—
73#394—31no—
74#394—31no—
75#386—61no—
76#385—61no—
77#385—61no—
78#381—31no—
79#343—32no—
80#343—61no—
81#324—21no—
82#324C2×D312yes—
83C24×C3C22⋊C411yescharacteristic
84C23×D3—21no—
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4—22no—
88C22×C3×C4—21no—
89C22×C3×C4—21no—
90C22×C3×C4—21no—
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4—42no—
95C22×C3×C4—42no—
96C22×C3×C4C22⋊C411yescharacteristic
97C22×C3×C4C22⋊C411yescharacteristic
98C22×C3×C4C22⋊C411yescharacteristic
99C22×C3×C4C22×C411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C2×#67—24no—
103C2×#67—42no—
104C2×#67—41no—
105C2×#67—41no—
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—21no—
109#235—44no—
110#235—44no—
111#235—44no—
112#235—44no—
113C2×D12—41no—
114C2×D12—42no—
115C2×C4×D3—42no—
116C2×C4×D3—42no—
117C2×C4×D3—42no—
118C2×Dic6—41no—
119C2×Dic6—42no—
120C3×M16—82no—
121C2×C3×C8—42no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126#217—42no—
127(C2×D3)⋊C4—42no—
128(C2×D3)⋊C4—42no—
129(C2×D3)⋊C4—42no—
130(C2×D3)⋊C4—41no—
131(C2×D3)⋊C4—41no—
132Dic3⋊C4—42no—
133Dic3⋊C4—42no—
134#208—24no—
135#208—44no—
136#208—42no—
137C2×C3⋊C8—44no—
138C2×C3⋊C8—42no—
139C2×#41—121no—
140C2×#41—121no—
141C2×#41—124no—
142C22×Q8—31no—
143C22×D4—31no—
144C22×D4—61no—
145C23×C4—61no—
146C23×C4C4×D311yescharacteristic
147C23×C4D1211yescharacteristic
148C23×C4#6711yescharacteristic
149C2×M16—41no—
150C2×M16—64no—
151C2×M16—62no—
152C2×M16—121no—
153C2×M16—122no—
154C22×C8—121no—
155#123—124no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—61no—
159C2×C4⋊C4—61no—
160C2×C4⋊C4C4×D311yescharacteristic
161C2×C4⋊C4D1211yescharacteristic
162C2×C4⋊C4#6711yescharacteristic
163C2×C4⋊C4C22×D311yescharacteristic
164C2×C22⋊C4—61no—
165C2×C22⋊C4—61no—
166C2×C22⋊C4—61no—
167C2×C22⋊C4—61no—
168C2×C22⋊C4—61no—
169C22⋊C8—22no—
170C22⋊C8—41no—
171#95—31no—
172#95—31no—
173#95—61no—
174#95—61no—
175#95—61no—
176C23×C3—22no—
177C23×C3—21no—
178C23×C3—21no—
179C23×C3—41no—
180C23×C3#9911yescharacteristic
181C23×C3#9911yescharacteristic
182C23×C3C2×C22⋊C411yescharacteristic
183C22×D3—22no—
184C22×D3—41no—
185C22×D3—41no—
186C22×D3—41no—
187C22×D3—41no—
188C22×D3—41no—
189C22×D3—41no—
190C2×C3×C4—22no—
191C2×C3×C4—22no—
192C2×C3×C4—22no—
193C2×C3×C4—22no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—42no—
203C2×C3×C4—42no—
204C2×C3×C4—42no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—81no—
210C2×C3×C4#10412yes—
211C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
212C2×C3×C4#13711yescharacteristic
213#67—44no—
214#67—44no—
215C2×Dic3—22no—
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224D12—42no—
225D12—42no—
226C4×D3—48no—
227Dic6—42no—
228Dic6—42no—
229C3×C8—82no—
230C3⋊C8—44no—
231C24—61no—
232C24(C2×D3)⋊C411yescharacteristic
233#41—124no—
234#41—124no—
235#41—124no—
236#41—124no—
237C2×Q8—121no—
238C2×Q8—122no—
239C2×D4—64no—
240C2×D4—121no—
241C2×D4—122no—
242C2×D4—122no—
243C2×D4—121no—
244C2×D4—121no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—61no—
257C22×C4—61no—
258C22×C4—61no—
259C22×C4—122no—
260C22×C4—122no—
261C22×C4—122no—
262C22×C4(C2×D3)⋊C411yescharacteristic
263C22×C4(C2×D3)⋊C411yescharacteristic
264C22×C4(C2×D3)⋊C411yescharacteristic
265C22×C4C2×C4×D311yescharacteristic
266C22×C4C2×D1211yescharacteristic
267C22×C4C2×#6711yescharacteristic
268M16—64no—
269M16—82no—
270M16—124no—
271M16—122no—
272C2×C8—42no—
273C2×C8—124no—
274C2×C8—122no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—122no—
280C4⋊C4—122no—
281C22⋊C4—122no—
282C22⋊C4—122no—
283C22⋊C4—122no—
284C22⋊C4—122no—
285C22⋊C4—121no—
286C22⋊C4—121no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—42no—
295C22×C3—42no—
296C22×C3AΣL(1,𝔽16)12yes—
297C22×C3C2×#9911yescharacteristic
298C22×C3#41711yescharacteristic
299C22×C3C2×#10411yescharacteristic
300C2×D3—42no—
301C2×D3—42no—
302C2×D3—42no—
303C2×D3—81no—
304C2×D3—81no—
305C2×D3—81no—
306C2×D3—81no—
307C3×C4—21no—
308C3×C4—22no—
309C3×C4—21no—
310C3×C4—41no—
311C3×C4—41no—
312C3×C4—41no—
313C3×C4—41no—
314Dic3—42no—
315Dic3—81no—
316Dic3—81no—
317C23—21no—
318C23—21no—
319C23—22no—
320C23—41no—
321C23—62no—
322C23—121no—
323C23—121no—
324C23—121no—
325C23—121no—
326C23—121no—
327C23—121no—
328C23#80611yescharacteristic
329C23#80611yescharacteristic
330C23C2×(C2×D3)⋊C411yescharacteristic
331Q8—122no—
332Q8—122no—
333D4—122no—
334D4—122no—
335D4—124no—
336D4—124no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—62no—
357C2×C4—81no—
358C2×C4—128no—
359C2×C4—121no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4—121no—
366C2×C4—121no—
367C2×C4#82512yes—
368C2×C4#90811yescharacteristic
369C2×C4#90911yescharacteristic
370C8—82no—
371C8—124no—
372C2×C3—21no—
373C2×C3—21no—
374C2×C3—22no—
375C2×C3—41no—
376C2×C3#150411yescharacteristic
377C2×C3C2×AΣL(1,𝔽16)11yescharacteristic
378C2×C3#213311yescharacteristic
379D3—82no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—42no—
388C22—42no—
389C22—122no—
390C22—122no—
391C22—122no—
392C22—241no—
393C22—241no—
394C22—241no—
395C22—241no—
396C22#447112yes—
397C22C2×#80611yescharacteristic
398C22#513311yescharacteristic
399C22C2×#82511yescharacteristic
400C4—21no—
401C4—22no—
402C4—21no—
403C4—41no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—122no—
408C4—241no—
409C4—241no—
410C3#1176211yescharacteristic, complemented, Sylow
411C2—21no—
412C2—21no—
413C2—22no—
414C2—41no—
415C2—242no—
416C2#6997611yescharacteristic
417C2C2×#447111yescharacteristic
418C2#7012811yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1463C2
socleC22×C3C2×#99
prime radicalC22×#235C4
cosocleC2×C3×C4⋊C4C23
hypercenter#1463D3
hypocenterC3#11762
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
2. C3×#1463C2semidirect
3. C2×#4530C2semidirect
4. #67457C2semidirect
5. #67456C2semidirect
26. #1463D3semidirect
410. C3#11762semidirect

Maximal subgroups

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

subgroupindex
#821922
C3×#14632
C2×#45302
#674572
#674562
#117623