Finite Group Databasebeta

← #10696408

Subgroups — #10696408

Order 768.

790 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#486C211yesmaximal, characteristic, complemented
2C2×#5087C211yesmaximal, characteristic, complemented
3#83188C211yesmaximal, characteristic
4#71778C211yesmaximal, characteristic, complemented
5C2×#4544C211yesmaximal, characteristic, complemented
6C2×#4459C211yesmaximal, characteristic, complemented
7#67655C211yesmaximal, characteristic, complemented
8#10070—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×#127C2211yescharacteristic
11C2×#940—21no—
12C2×#938—22no—
13C2×C4×D12C2211yescharacteristic
14C3×#486—22no—
15C3×#486—22no—
16C2×C3×#102—22no—
17C2×C3×C4×C8C2211yescharacteristic
18C2×#835—22no—
19#5223—22no—
20#5221—21no—
21#5087—22no—
22#5087—22no—
23C2×#809—21no—
24C2×#807—21no—
25C4×C4⋊Dic3C2211yescharacteristic
26C2×C4⋊(C3⋊C8)C2211yescharacteristic
27C2×C4⋊(C3⋊C8)C2211yescharacteristic
28#4544—22no—
29#4459—22no—
30C2×#486D311yescharacteristic, complemented
31C2×#474—31no—
32#2541—31no—
33#1981—31no—
34C2×#331—31no—
35C2×D4⋊C8—31no—
36#1279—31no—
37C22×C3×Q8—21no—
38C22×C3×D4—21no—
39C22×C3×D4—21no—
40C22×C3×D4D412yes—
41C22×#67—24no—
42C22×D12D411yescharacteristic
43C22×C4×D3—21no—
44C22×C3×C8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8—21no—
48C3×C4⋊Q8D412yescomplemented
49C3×#127—21no—
50C3×#127—21no—
51C3×#127—21no—
52C3×#127—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C42C2311yescharacteristic
56C2×#217—24no—
57#940—42no—
58#940—42no—
59#938—44no—
60#938—44no—
61C2×(C2×D3)⋊C4—21no—
62C2×C4⋊Dic3D411yescharacteristic
63C2×C4⋊Dic3D412yes—
64C2×C4×Dic3—21no—
65C22×C3⋊C8—21no—
66C22×C3⋊C8—21no—
67C4×D12—21no—
68C4×D12—21no—
69C4×D12—21no—
70C4×D12—21no—
71C3×#102—28no—
72C3×C4×C8—21no—
73C3×C4×C8D412yescomplemented
74#835—44no—
75C3⋊#95—21no—
76#809—24no—
77#807—24no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)D412yescomplemented
82C2×C4⋊Q8C2×D311yescharacteristic
83C2×#127—61no—
84C2×#127C2×D311yescharacteristic
85C2×#121—62no—
86C2×C4×D4—31no—
87#486—22no—
88#486—22no—
89#474—62no—
90#474—62no—
91C2×#107—61no—
92C2×C4⋊C8—31no—
93C2×C4⋊C8—31no—
94C2×#103—62no—
95C2×#102—22no—
96C2×#102—61no—
97C2×C4×C8C2×D311yescharacteristic
98#392—62no—
99#390—61no—
100C4×C4⋊C4—31no—
101#331—62no—
102D4⋊C8—62no—
103C24×C3—22no—
104C23×D3—21no—
105C2×C3×Q8—22no—
106C2×C3×Q8—41no—
107C2×C3×Q8—42no—
108C2×C3×Q8—42no—
109C2×C3×D4—22no—
110C2×C3×D4—22no—
111C2×C3×D4—44no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—42no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—42no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C22×C3×C4—41no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C2×#67—48no—
127C2×#67—48no—
128C2×#67—84no—
129C22×Dic3—22no—
130C22×Dic3—21no—
131C2×D12—21no—
132C2×D12—21no—
133C2×D12—21no—
134C2×D12—21no—
135C2×D12—21no—
136C2×D12SD1612yes—
137C2×C4×D3—41no—
138C2×C4×D3—41no—
139C2×C4×D3—42no—
140C2×C4×D3—42no—
141C2×C3×C8—22no—
142C2×C3×C8—42no—
143C3×C4⋊C4—22no—
144C3×C4⋊C4—22no—
145C3×C4⋊C4—41no—
146C3×C4⋊C4—41no—
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42#4111yescharacteristic
151#217—48no—
152(C2×D3)⋊C4—41no—
153(C2×D3)⋊C4—41no—
154C4⋊Dic3—22no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3D814yes—
158C4×Dic3—41no—
159C4×Dic3—41no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—41no—
162C2×C3⋊C8—41no—
163C2×C3⋊C8—41no—
164C2×C3⋊C8—42no—
165C22×Q8—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—31no—
169C22×D4—64no—
170C22×D4#6712yes—
171C23×C4—61no—
172C22×C8—21no—
173C22×C8—61no—
174C22×C8—61no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8#6712yescomplemented
179#127—21no—
180#127—21no—
181#127—21no—
182#127—21no—
183#127—122no—
184#127—122no—
185#121—124no—
186#121—124no—
187C4×D4—61no—
188C4×D4—61no—
189C4×D4—61no—
190C4×D4—61no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—32no—
194C2×C4⋊C4—31no—
195C2×C22⋊C4—64no—
196C2×C22⋊C4—61no—
197C2×C42—61no—
198C2×C42C22×D311yescharacteristic
199#107—64no—
200C4⋊C8—32no—
201C4⋊C8—61no—
202C4⋊C8—61no—
203C4⋊C8—61no—
204#103—124no—
205#102—28no—
206#102—64no—
207C4×C8—21no—
208C4×C8#6712yescomplemented
209#95—61no—
210C23×C3—42no—
211C23×C3—44no—
212C23×C3—42no—
213C23×C3—42no—
214C23×C3—44no—
215C23×C3#12011yescharacteristic
216C22×D3—22no—
217C22×D3—41no—
218C22×D3—42no—
219C22×D3—41no—
220C22×D3—42no—
221C3×Q8—44no—
222C3×Q8—81no—
223C3×Q8—81no—
224C3×D4—44no—
225C3×D4—82no—
226C3×D4—81no—
227C3×D4—81no—
228C3×D4—82no—
229C3×D4—81no—
230C3×D4—81no—
231C3×D4—81no—
232C3×D4—81no—
233C2×C3×C4—22no—
234C2×C3×C4—42no—
235C2×C3×C4—82no—
236C2×C3×C4#12011yescharacteristic
237C2×C3×C4#12011yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12111yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12711yescharacteristic
242C2×C3×C4C2×D812yes—
243C2×C3×C4C2×SD1611yescharacteristic
244C2×C3×C4#13511yescharacteristic
245C2×C3×C4AGL(1,ℤ/8)12yes—
246C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
247C2×C3×C4#13711yescharacteristic
248#67—816no—
249C2×Dic3—24no—
250C2×Dic3—42no—
251C2×Dic3—42no—
252C2×Dic3—41no—
253C2×Dic3—42no—
254D12—22no—
255D12—22no—
256D12—41no—
257D12—41no—
258C4×D3—82no—
259C4×D3—81no—
260C4×D3—81no—
261C3×C8—42no—
262C3⋊C8—42no—
263C3⋊C8—81no—
264C24—22no—
265C24—61no—
266C2×Q8—22no—
267C2×Q8—41no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—32no—
273C2×D4—44no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—61no—
285C2×D4—61no—
286C2×D4—61no—
287C2×D4—61no—
288C2×D4—61no—
289C2×D4—128no—
290C2×D4—128no—
291C2×D4—244no—
292C22×C4—41no—
293C22×C4—62no—
294C22×C4—61no—
295C22×C4—122no—
296C22×C4—122no—
297C22×C4—121no—
298C22×C4—121no—
299C22×C4D3×D411yescharacteristic
300C22×C4D3×D411yescharacteristic
301C22×C4C2×#6711yescharacteristic
302C2×C8—22no—
303C2×C8—42no—
304C2×C8—62no—
305C2×C8—121no—
306C2×C8—121no—
307C2×C8—122no—
308C2×C8—121no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—34no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—62no—
315C4⋊C4—61no—
316C4⋊C4—61no—
317C22⋊C4—128no—
318C22⋊C4—121no—
319C22⋊C4—121no—
320C42—121no—
321C42—121no—
322C42D3×D411yescharacteristic
323C42#23711yescharacteristic
324C42C2×#6711yescharacteristic
325C42C2×#6711yescharacteristic
326C22×C3—84no—
327C22×C3—84no—
328C22×C3—82no—
329C22×C3—82no—
330C22×C3—82no—
331C22×C3#39211yescharacteristic
332C22×C3#44712yes—
333C22×C3#45011yescharacteristic
334C22×C3#45211yescharacteristic
335C22×C3#45512yes—
336C2×D3—44no—
337C2×D3—42no—
338C2×D3—82no—
339C2×D3—81no—
340C2×D3—81no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—82no—
344C3×C4#44811yescharacteristic
345C3×C4#44911yescharacteristic
346C3×C4#45912yes—
347C3×C4#46511yescharacteristic
348C3×C4#46811yescharacteristic
349C3×C4#49311yescharacteristic
350C3×C4#49611yescharacteristic
351Dic3—44no—
352Dic3—81no—
353C23—44no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—44no—
358C23—62no—
359C23—122no—
360C23—121no—
361C23—121no—
362C23—122no—
363C23#93711yescharacteristic
364Q8—44no—
365Q8—81no—
366Q8—81no—
367D4—44no—
368D4—62no—
369D4—62no—
370D4—82no—
371D4—82no—
372D4—81no—
373D4—81no—
374D4—81no—
375D4—81no—
376D4—81no—
377D4—81no—
378D4—121no—
379D4—121no—
380D4—2416no—
381C2×C4—22no—
382C2×C4—42no—
383C2×C4—64no—
384C2×C4—82no—
385C2×C4—122no—
386C2×C4—122no—
387C2×C4—122no—
388C2×C4—121no—
389C2×C4—242no—
390C2×C4—241no—
391C2×C4—241no—
392C2×C4D3×D812yes—
393C2×C4#91112yes—
394C2×C4D3×SD1611yescharacteristic
395C2×C4#91411yescharacteristic
396C2×C4#91511yescharacteristic
397C2×C4#91611yescharacteristic
398C2×C4#93711yescharacteristic
399C2×C4#93811yescharacteristic
400C2×C4#93911yescharacteristic
401C2×C4#93911yescharacteristic
402C2×C4#94011yescharacteristic
403C2×C4#95311yescharacteristic
404C8—42no—
405C8—122no—
406C8—241no—
407C2×C3—162no—
408C2×C3#163912yes—
409C2×C3#164112yes—
410C2×C3#200011yescharacteristic
411C2×C3#200411yescharacteristic
412C2×C3#201111yescharacteristic
413D3—82no—
414C22—84no—
415C22—84no—
416C22—82no—
417C22—82no—
418C22—82no—
419C22—124no—
420C22—122no—
421C22—242no—
422C22—241no—
423C22—241no—
424C22#506112yes—
425C22#515612yes—
426C22#516911yescharacteristic
427C22#517311yescharacteristic
428C22#522311yescharacteristic
429C4—21no—
430C4—21no—
431C4—82no—
432C4—124no—
433C4—241no—
434C4#515012yes—
435C4#515111yescharacteristic
436C4#515411yescharacteristic
437C4#517011yescharacteristic
438C4#517111yescharacteristic
439C4#517211yescharacteristic
440C4#517411yescharacteristic
441C3#1007011yescharacteristic, complemented, Sylow
442C2—162no—
443C2—242no—
444C2#7119812yes—
445C2#7120712yes—
446C2#7173111yescharacteristic
447C2#7204411yescharacteristic
448C2#7205411yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#486C2
socleC23×C3#120
prime radical#83188C2
hypercenterC2×#486D3
hypocenterC3#10070
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#486

Complements

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

NH ≅ G/Ntype
1. C2×C3×#486C2semidirect
2. C2×#5087C2semidirect
4. #71778C2semidirect
5. C2×#4544C2semidirect
6. C2×#4459C2semidirect
7. #67655C2semidirect
30. C2×#486D3semidirect
48. C3×C4⋊Q8D4semidirect
73. C3×C4×C8D4semidirect
81. C4⋊(C3⋊C8)D4semidirect
178. C4⋊Q8#67semidirect
208. C4×C8#67semidirect
441. C3#10070semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4862
C2×#50872
#831882
#717782
C2×#45442
C2×#44592
#676552
#100703