Finite Group Databasebeta

← #10705754

Subgroups — #10705754

Order 768.

532 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 83 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#82380C211yesmaximal, characteristic
2C3×#1466C211yesmaximal, characteristic, complemented
3C4×#834C211yesmaximal, characteristic, complemented
4#67450C212yesmaximal, complemented
5#67448C212yesmaximal, complemented
6#11775—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#939—21no—
9C2×#930C411yescharacteristic, complemented
10C3×#432—21no—
11C2×#834—21no—
12#5213—21no—
13C4×#217—21no—
14C2×C3⋊#95C411yescharacteristic, complemented
15#4991C411yescharacteristic, complemented
16#4981C411yescharacteristic, complemented
17#4960C2211yescharacteristic
18#4958—21no—
19#4953C2211yescharacteristic
20#4940—21no—
21C3×#323—21no—
22C3×#323C2212yes—
23#4526C2212yes—
24#2543—31no—
25C4×#99—31no—
26#1466D311yescharacteristic, complemented
27#1344—32no—
28#1343—32no—
29C22×C3×D4D411yescharacteristic
30C23×C3×C4C2×C411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C22×#67—21no—
33C22×#67—21no—
34C23×Dic3—21no—
35C22×D12C2×C411yescharacteristic
36C3×C4×D4—41no—
37C3×C4×D4—41no—
38C2×C3×C4⋊C4C2×C411yescharacteristic
39C2×C3×C22⋊C4D411yescharacteristic
40C2×C3×C22⋊C4C2311yescharacteristic
41C2×C3×C42D411yescharacteristic
42C2×#217—22no—
43C2×#217—21no—
44#939—44no—
45#930—41no—
46#930—41no—
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—21no—
49C2×(C2×D3)⋊C4—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×C4⋊Dic3C2×C411yescharacteristic
52C2×Dic3⋊C4—21no—
53C2×C4×Dic3—41no—
54C3×C4⋊C8—41no—
55C3×C22⋊C8—22no—
56C3×#97—41no—
57#834—24no—
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—41no—
62C3⋊#95C2×C411yescharacteristic
63C3⋊#95C2×C411yescharacteristic
64C2×#121—31no—
65C2×#121—61no—
66C2×C4×D4C2×D311yescharacteristic
67#432—21no—
68C2×#99—61no—
69#398—31no—
70#397—31no—
71#394—31no—
72#392—31no—
73#388—61no—
74#386—61no—
75#386—61no—
76C4×C22⋊C4—61no—
77C2×#95—31no—
78#342—32no—
79#323—21no—
80#323C2×D312yes—
81C24×C3C22⋊C411yescharacteristic
82C24×C3C2×D411yescharacteristic
83C23×D3—21no—
84C2×C3×D4—22no—
85C2×C3×D4—22no—
86C2×C3×D4—41no—
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4—21no—
90C22×C3×C4—21no—
91C22×C3×C4—81no—
92C22×C3×C4—81no—
93C22×C3×C4C22⋊C411yescharacteristic
94C22×C3×C4C22⋊C411yescharacteristic
95C22×C3×C4C22⋊C411yescharacteristic
96C22×C3×C4C22×C411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C2×#67—42no—
99C2×#67—42no—
100C2×#67—42no—
101C2×#67—42no—
102C2×#67—42no—
103C2×#67—81no—
104C22×Dic3—21no—
105C22×Dic3—21no—
106C22×Dic3—21no—
107C22×Dic3—42no—
108C22×Dic3—42no—
109C22×Dic3—42no—
110C22×Dic3—81no—
111C2×D12—41no—
112C2×D12—81no—
113C2×C3×C8—42no—
114C3×C4⋊C4—41no—
115C3×C22⋊C4—41no—
116C3×C22⋊C4—41no—
117C3×C42—21no—
118C3×C42—21no—
119#217—24no—
120#217—42no—
121#217—42no—
122(C2×D3)⋊C4—42no—
123(C2×D3)⋊C4—42no—
124(C2×D3)⋊C4—41no—
125(C2×D3)⋊C4—41no—
126(C2×D3)⋊C4—81no—
127C4⋊Dic3—41no—
128Dic3⋊C4—42no—
129C4×Dic3—82no—
130C22×D4—31no—
131C22×D4—61no—
132C22×D4—61no—
133C22×D4D1211yescharacteristic
134C23×C4—61no—
135C23×C4C4×D311yescharacteristic
136C23×C4D1211yescharacteristic
137#121—124no—
138#121—121no—
139#121—121no—
140C4×D4—41no—
141C4×D4—41no—
142C2×C4⋊C4—31no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4C4×D311yescharacteristic
145C2×C22⋊C4—62no—
146C2×C22⋊C4—61no—
147C2×C22⋊C4—61no—
148C2×C22⋊C4—61no—
149C2×C22⋊C4—61no—
150C2×C22⋊C4—61no—
151C2×C22⋊C4#6711yescharacteristic
152C2×C22⋊C4C22×D311yescharacteristic
153C2×C42—121no—
154C2×C42#6711yescharacteristic
155C4⋊C8—41no—
156#99—64no—
157C22⋊C8—22no—
158#97—41no—
159#95—31no—
160#95—31no—
161#95—61no—
162#95—61no—
163#95—61no—
164#95—121no—
165C23×C3—21no—
166C23×C3—21no—
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—41no—
170C23×C3—41no—
171C23×C3—41no—
172C23×C3—41no—
173C23×C3#10412yes—
174C23×C3C2×C22⋊C411yescharacteristic
175C23×C3AGL(1,ℤ/8)12yes—
176C22×D3—41no—
177C22×D3—41no—
178C22×D3—41no—
179C22×D3—41no—
180C22×D3—41no—
181C22×D3—41no—
182C22×D3—41no—
183C3×D4—44no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—41no—
191C2×C3×C4—42no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—81no—
196C2×C3×C4—81no—
197C2×C3×C4—81no—
198C2×C3×C4#9911yescharacteristic
199C2×C3×C4#9911yescharacteristic
200#67—84no—
201#67—84no—
202C2×Dic3—22no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—44no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—84no—
213C2×Dic3—82no—
214C2×Dic3—82no—
215D12—82no—
216C3×C8—82no—
217C24—61no—
218C24(C2×D3)⋊C411yescharacteristic
219C24C2×D1211yescharacteristic
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—41no—
223C2×D4—121no—
224C2×D4—122no—
225C2×D4—122no—
226C2×D4—122no—
227C2×D4—122no—
228C2×D4—122no—
229C2×D4—241no—
230C2×D4—241no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—61no—
236C22×C4—61no—
237C22×C4—61no—
238C22×C4—81no—
239C22×C4—81no—
240C22×C4—122no—
241C22×C4—122no—
242C22×C4—122no—
243C22×C4—241no—
244C22×C4(C2×D3)⋊C411yescharacteristic
245C22×C4(C2×D3)⋊C411yescharacteristic
246C22×C4(C2×D3)⋊C411yescharacteristic
247C22×C4C2×C4×D311yescharacteristic
248C22×C4C2×#6711yescharacteristic
249C2×C8—42no—
250C4⋊C4—41no—
251C4⋊C4—121no—
252C4⋊C4—122no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—64no—
256C22⋊C4—122no—
257C22⋊C4—122no—
258C22⋊C4—122no—
259C22⋊C4—122no—
260C22⋊C4—121no—
261C22⋊C4—121no—
262C22⋊C4—241no—
263C42—21no—
264C42—21no—
265C42—242no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—22no—
269C22×C3—22no—
270C22×C3—42no—
271C22×C3—42no—
272C22×C3—42no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—81no—
278C22×C3C2×#9911yescharacteristic
279C22×C3#41811yescharacteristic
280C22×C3C2×#10411yescharacteristic
281C2×D3—81no—
282C2×D3—81no—
283C2×D3—81no—
284C2×D3—81no—
285C2×D3—81no—
286C2×D3—81no—
287C2×D3—81no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—41no—
291C3×C4—41no—
292C3×C4—41no—
293C3×C4—41no—
294Dic3—42no—
295Dic3—81no—
296Dic3—84no—
297Dic3—81no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—121no—
307C23—121no—
308C23—121no—
309C23—121no—
310C23—121no—
311C23—121no—
312C23—121no—
313C23#82512yes—
314C23#90812yes—
315C23C2×(C2×D3)⋊C411yescharacteristic
316D4—44no—
317D4—242no—
318D4—244no—
319D4—244no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—62no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—121no—
336C2×C4—121no—
337C2×C4—124no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—244no—
345C2×C4—242no—
346C2×C4—242no—
347C2×C4#80611yescharacteristic
348C2×C4#80611yescharacteristic
349C8—82no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—42no—
353C2×C3—42no—
354C2×C3#150711yescharacteristic
355C2×C3#212111yescharacteristic
356C2×C3#212611yescharacteristic
357D3—161no—
358C22—21no—
359C22—21no—
360C22—22no—
361C22—22no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—81no—
370C22—241no—
371C22—241no—
372C22—241no—
373C22—241no—
374C22—241no—
375C22—241no—
376C22—241no—
377C22C2×#80611yescharacteristic
378C22#513111yescharacteristic
379C22C2×#82511yescharacteristic
380C4—21no—
381C4—21no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—122no—
387C4—241no—
388C4—244no—
389C4—241no—
390C3#1177511yescharacteristic, complemented, Sylow
391C2—21no—
392C2—21no—
393C2—42no—
394C2—42no—
395C2—481no—
396C2#6997711yescharacteristic
397C2#7006011yescharacteristic
398C2#7009411yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C22⋊C4C22×D3
FittingC3×#1466C2
socleC22×C3C2×#99
prime radical#82380C2
cosocleC2×C3×C22⋊C4C23
hypercenter#1466D3
hypocenterC3#11775
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1466

Complements

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

NH ≅ G/Ntype
2. C3×#1466C2semidirect
3. C4×#834C2semidirect
4. #67450C2semidirect
5. #67448C2semidirect
9. C2×#930C4semidirect
14. C2×C3⋊#95C4semidirect
15. #4991C4semidirect
16. #4981C4semidirect
26. #1466D3semidirect
390. C3#11775semidirect

Maximal subgroups

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

subgroupindex
#823802
C3×#14662
C4×#8342
#674502
#674482
#117753