Finite Group Databasebeta

← #10691351

Subgroups — #10691351

Order 768.

594 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5087C211yesmaximal, characteristic
2C2×#4834C211yesmaximal, characteristic
3C2×C3×#332C211yesmaximal, characteristic, complemented
4#71785C211yesmaximal, characteristic, complemented
5#70479C211yesmaximal, characteristic
6#67703C211yesmaximal, characteristic, complemented
7#67702C211yesmaximal, characteristic, complemented
8#9219—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C22×D3×C8—21no—
11C2×C4×D12C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×#835—21no—
15C2×#835—21no—
16C2×#821—21no—
17C2×(C2×D3)⋊C8—21no—
18#5087—22no—
19#5087C414yescomplemented
20C2×#809—21no—
21C2×#807—21no—
22#4935C2211yescharacteristic
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24C2×C4×C3⋊C8C2211yescharacteristic
25#4834—22no—
26#4834—22no—
27C3×#332—22no—
28#4551—21no—
29C2×#474—31no—
30C2×C8×D4—31no—
31#1956—31no—
32#1802—31no—
33C2×#332D311yescharacteristic, complemented
34#1296—31no—
35#1277—31no—
36C22×C3×Q8—21no—
37C22×D12D411yescharacteristic
38C22×C4×D3—21no—
39C22×C3×C8—21no—
40C22×C3×C8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8C2×C412yes—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C42C2311yescharacteristic
48C2×(C2×D3)⋊C4—21no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3D411yescharacteristic
53C22×C3⋊C8—21no—
54C22×C3⋊C8D411yescharacteristic
55C22×C3⋊C8D411yescharacteristic
56C2×D3×C8—24no—
57C2×D3×C8—42no—
58C2×D3×C8—42no—
59C4×D12—21no—
60C4×D12—21no—
61C4×D12C2×C412yes—
62C4×D12D412yes—
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8D412yes—
67#835—42no—
68#835—42no—
69#821—42no—
70(C2×D3)⋊C8—24no—
71#809—24no—
72#807—24no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)C2×C412yes—
75C4×C3⋊C8—21no—
76C4×C3⋊C8—21no—
77C23×C8—61no—
78C2×C4⋊Q8C2×D311yescharacteristic
79C2×C4×D4—31no—
80#474—34no—
81#474—62no—
82C8×D4—62no—
83C8×D4—62no—
84C2×#107—61no—
85C2×C4⋊C8—31no—
86C2×C4⋊C8C2×D311yescharacteristic
87C2×C4⋊C8C2×D311yescharacteristic
88C2×#103—61no—
89C2×#103—61no—
90C2×#102—61no—
91C2×#102—61no—
92C2×C22⋊C8—61no—
93C2×C4×C8—31no—
94#384—31no—
95#340—61no—
96#332—22no—
97C23×D3—21no—
98C2×C3×Q8—22no—
99C2×C3×Q8—41no—
100C2×C3×Q8—41no—
101C2×C3×Q8—41no—
102C2×C3×Q8—41no—
103C2×C3×Q8—41no—
104C22×C3×C4—41no—
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4#4111yescharacteristic
108C22×Dic3—21no—
109C22×Dic3—41no—
110C2×D12—21no—
111C2×D12—21no—
112C2×D12—21no—
113C2×D12C22⋊C412yes—
114C2×D12SD1612yes—
115C2×D12Q1612yes—
116C2×C4×D3—22no—
117C2×C4×D3—22no—
118C2×C4×D3—41no—
119C2×C4×D3—41no—
120C2×C4×D3—41no—
121C2×C4×D3—41no—
122C2×C3×C8—22no—
123C2×C3×C8—41no—
124C2×C3×C8—41no—
125C2×C3×C8—41no—
126C2×C3×C8—41no—
127C2×C3×C8—41no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C42C22×C411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42#4111yescharacteristic
136(C2×D3)⋊C4—22no—
137(C2×D3)⋊C4—22no—
138C4⋊Dic3—21no—
139C4⋊Dic3—41no—
140C4⋊Dic3—41no—
141C4⋊Dic3—41no—
142C4⋊Dic3—41no—
143C4⋊Dic3—81no—
144C4⋊Dic3C22⋊C412yes—
145C2×C3⋊C8—21no—
146C2×C3⋊C8—21no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8—21no—
149C2×C3⋊C8—21no—
150C2×C3⋊C8—21no—
151C2×C3⋊C8—41no—
152C2×C3⋊C8—41no—
153C2×C3⋊C8D812yes—
154D3×C8—48no—
155C22×Q8—21no—
156C22×D4—31no—
157C23×C4—61no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—31no—
161C22×C8—31no—
162C22×C8—64no—
163C22×C8—61no—
164C22×C8—122no—
165C22×C8—122no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8C4×D312yes—
170C4×D4—32no—
171C4×D4—32no—
172C4×D4—61no—
173C4×D4—61no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—31no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—61no—
180C2×C22⋊C4—61no—
181C2×C42C22×D311yescharacteristic
182#107—64no—
183C4⋊C8—21no—
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8—32no—
187C4⋊C8—61no—
188C4⋊C8#6712yes—
189#103—122no—
190#103—122no—
191#102—64no—
192#102—122no—
193C22⋊C8—64no—
194C4×C8—61no—
195C4×C8—61no—
196C23×C3#12111yescharacteristic
197C22×D3—22no—
198C22×D3—22no—
199C22×D3—22no—
200C22×D3—41no—
201C22×D3—41no—
202C22×D3—41no—
203C22×D3—41no—
204C3×Q8—42no—
205C3×Q8—42no—
206C3×Q8—81no—
207C3×Q8—81no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—42no—
211C2×C3×C4—81no—
212C2×C3×C4—81no—
213C2×C3×C4C2×C22⋊C411yescharacteristic
214C2×C3×C4C4×D411yescharacteristic
215C2×C3×C4C4×D411yescharacteristic
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12311yescharacteristic
219C2×C3×C4C2×D811yescharacteristic
220C2×C3×C4C2×SD1611yescharacteristic
221C2×C3×C4C2×Q1611yescharacteristic
222C2×C3×C4#13511yescharacteristic
223C2×C3×C4#13511yescharacteristic
224C2×C3×C4#13511yescharacteristic
225C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
226C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
227C2×Dic3—22no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—81no—
231C2×Dic3—81no—
232C2×Dic3—81no—
233D12—22no—
234D12—22no—
235D12—22no—
236D12#10314yescomplemented
237C4×D3—42no—
238C4×D3—42no—
239C4×D3—42no—
240C4×D3—42no—
241C3×C8—42no—
242C3×C8—81no—
243C3⋊C8—22no—
244C3⋊C8—42no—
245C3⋊C8—41no—
246C24—61no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×D4—32no—
254C2×D4—32no—
255C2×D4—32no—
256C2×D4—61no—
257C2×D4—61no—
258C2×D4—61no—
259C22×C4—41no—
260C22×C4—61no—
261C22×C4—62no—
262C22×C4—62no—
263C22×C4—121no—
264C22×C4—121no—
265C22×C4—121no—
266C22×C4—121no—
267C22×C4—121no—
268C22×C4#23511yescharacteristic
269C22×C4D3×D411yescharacteristic
270C22×C4D3×D411yescharacteristic
271C2×C8—22no—
272C2×C8—32no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—61no—
279C2×C8—61no—
280C2×C8—62no—
281C2×C8—61no—
282C2×C8—61no—
283C2×C8—61no—
284C2×C8—128no—
285C2×C8—121no—
286C2×C8—121no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—32no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—61no—
293C4⋊C4—121no—
294C4⋊C4—121no—
295C4⋊C4—121no—
296C4⋊C4—121no—
297C4⋊C4—241no—
298C22⋊C4—62no—
299C22⋊C4—62no—
300C42C2×C4×D311yescharacteristic
301C42D3×D411yescharacteristic
302C42#23711yescharacteristic
303C42C2×#6711yescharacteristic
304C22×C3#38611yescharacteristic
305C22×C3#45911yescharacteristic
306C22×C3#46311yescharacteristic
307C22×C3#46511yescharacteristic
308C22×C3#46711yescharacteristic
309C22×C3#47111yescharacteristic
310C22×C3#47311yescharacteristic
311C2×D3—24no—
312C2×D3—42no—
313C2×D3—42no—
314C2×D3—42no—
315C2×D3—42no—
316C2×D3—42no—
317C2×D3—42no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—82no—
321C3×C4C2×#10311yescharacteristic
322C3×C4#41611yescharacteristic
323C3×C4C4×D811yescharacteristic
324C3×C4#44211yescharacteristic
325C3×C4#44711yescharacteristic
326C3×C4#44911yescharacteristic
327C3×C4#46511yescharacteristic
328C3×C4#46711yescharacteristic
329Dic3—42no—
330Dic3—161no—
331C23—62no—
332C23—62no—
333C23—62no—
334C23—121no—
335C23—121no—
336C23—121no—
337C23—121no—
338C23#88411yescharacteristic
339Q8—42no—
340Q8—42no—
341Q8—81no—
342Q8—81no—
343D4—34no—
344D4—62no—
345D4—62no—
346D4—62no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—42no—
350C2×C4—62no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—121no—
354C2×C4—121no—
355C2×C4—122no—
356C2×C4—122no—
357C2×C4—122no—
358C2×C4—122no—
359C2×C4—241no—
360C2×C4—241no—
361C2×C4—241no—
362C2×C4D3×C22⋊C411yescharacteristic
363C2×C4#67⋊C411yescharacteristic
364C2×C4#88311yescharacteristic
365C2×C4D3×D811yescharacteristic
366C2×C4#91111yescharacteristic
367C2×C4D3×SD1611yescharacteristic
368C2×C4#91411yescharacteristic
369C2×C4#91611yescharacteristic
370C2×C4#91611yescharacteristic
371C2×C4D3×Q1611yescharacteristic
372C2×C4#91911yescharacteristic
373C2×C4C4×#6711yescharacteristic
374C2×C4#93711yescharacteristic
375C2×C4#93811yescharacteristic
376C8—42no—
377C8—62no—
378C8—81no—
379C8—121no—
380C8—122no—
381C2×C3#166511yescharacteristic
382C2×C3#166911yescharacteristic
383C2×C3#167911yescharacteristic
384C2×C3#168311yescharacteristic
385C2×C3#187911yescharacteristic
386C2×C3#189211yescharacteristic
387C2×C3#190811yescharacteristic
388D3—44no—
389C22—64no—
390C22—122no—
391C22—122no—
392C22—122no—
393C22—122no—
394C22—122no—
395C22—122no—
396C22#475411yescharacteristic
397C22#475511yescharacteristic
398C22#478611yescharacteristic
399C22#478711yescharacteristic
400C22#480711yescharacteristic
401C22#481311yescharacteristic
402C22#495611yescharacteristic
403C4—21no—
404C4—21no—
405C4—82no—
406C4—122no—
407C4—481no—
408C4#475611yescharacteristic
409C4#478511yescharacteristic
410C4D3×#10311yescharacteristic
411C4#480411yescharacteristic
412C4#515611yescharacteristic
413C4#517111yescharacteristic
414C4#517211yescharacteristic
415C4#518811yescharacteristic
416C3#921911yescharacteristic, complemented, Sylow
417C2—124no—
418C2#6844011yescharacteristic
419C2#6844111yescharacteristic
420C2#6844411yescharacteristic
421C2#6844711yescharacteristic
422C2#7012311yescharacteristic
423C2#7020811yescharacteristic
424C2#7027011yescharacteristic
subgroupG/H
centerC23#884
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×#332C2
socleC23×C3#121
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenterC2×#332D3
hypocenterC3#9219
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
3. C2×C3×#332C2semidirect
4. #71785C2semidirect
6. #67703C2semidirect
7. #67702C2semidirect
19. #5087C4semidirect
33. C2×#332D3semidirect
236. D12#103semidirect
416. C3#9219semidirect

Maximal subgroups

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

subgroupindex
C2×#50872
C2×#48342
C2×C3×#3322
#717852
#704792
#677032
#677022
#92193