Finite Group Databasebeta

← #10691422

Subgroups — #10691422

Order 768.

646 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5015C211yesmaximal, characteristic
2C2×#5012C211yesmaximal, characteristic, complemented
3C3×#1802C211yesmaximal, characteristic, complemented
4C2×#4485C211yesmaximal, characteristic, complemented
5#69580C211yesmaximal, characteristic, complemented
6#67703C211yesmaximal, characteristic, complemented
7#67634C211yesmaximal, characteristic, complemented
8#9229—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C22×#213—21no—
11C22×#208—21no—
12C2×#874C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×C3×#102—21no—
15C3×#384C2211yescharacteristic
16C2×#830—21no—
17#5015—22no—
18#5015C414yescomplemented
19#5012—22no—
20#5012—22no—
21C2×#809—21no—
22C2×#809—21no—
23C2×#809—21no—
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26C2×C4×C3⋊C8C2211yescharacteristic
27C3×#340—21no—
28#4485—22no—
29C2×#459—31no—
30C2×#436—31no—
31#1957—31no—
32#1802D311yescharacteristic, complemented
33C2×#331—31no—
34#1296—31no—
35#1277—31no—
36C22×C3×D4D411yescharacteristic
37C23×C3×C4—21no—
38C22×D12—21no—
39C22×D12—21no—
40C22×D12—21no—
41C22×C3×C8—21no—
42C3×C4×D4—21no—
43C3×C4×D4—21no—
44C3×C4×D4C2×C412yes—
45C3×C4×D4D412yes—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4D411yescharacteristic
50C2×C3×C22⋊C4—21no—
51C2×C3×C42C2311yescharacteristic
52C2×#213—24no—
53C2×#213—42no—
54C2×#213—42no—
55C2×#208—24no—
56C2×#208—42no—
57C2×#208—42no—
58C22×C3⋊C8—21no—
59C22×C3⋊C8—21no—
60C22×C3⋊C8D411yescharacteristic
61C22×C3⋊C8D411yescharacteristic
62#874—21no—
63#874—21no—
64#874—21no—
65#874C2×C412yes—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×#102—42no—
69#830—24no—
70#809—24no—
71#809—42no—
72#809—42no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)C2×C412yes—
76C4⋊(C3⋊C8)D412yes—
77C4×C3⋊C8—21no—
78C4×C3⋊C8—21no—
79C22×D8—61no—
80C22×M16—61no—
81C2×#127—31no—
82C2×C4×D4C2×D311yescharacteristic
83#459—34no—
84#459—62no—
85#436—62no—
86#436—62no—
87C2×C4⋊C8—31no—
88C2×C4⋊C8—31no—
89C2×C4⋊C8C2×D311yescharacteristic
90C2×#102—21no—
91C2×#102—61no—
92C2×#102—61no—
93C2×#102—61no—
94C2×C22⋊C8—61no—
95C2×C4×C8—31no—
96#384C2×D311yescharacteristic
97#340—21no—
98#331—62no—
99C24×C3—21no—
100C23×D3—41no—
101C2×C3×D4—21no—
102C2×C3×D4—21no—
103C2×C3×D4—21no—
104C2×C3×D4—21no—
105C2×C3×D4—21no—
106C2×C3×D4C22⋊C412yes—
107C22×C3×C4—21no—
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4—41no—
111C22×C3×C4—41no—
112C22×C3×C4—41no—
113C22×C3×C4—41no—
114C22×C3×C4—41no—
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4#4111yescharacteristic
118C2×D12—22no—
119C2×D12—22no—
120C2×D12—22no—
121C2×D12—41no—
122C2×D12—41no—
123C2×D12—41no—
124C2×D12—41no—
125C2×D12—41no—
126C2×D12—41no—
127C2×D12—41no—
128C2×D12—41no—
129C2×D12—41no—
130C2×D12—41no—
131C2×D12—41no—
132C2×D12—81no—
133C2×D12—81no—
134C2×C3×C8—41no—
135C2×C3×C8—41no—
136C2×C3×C8—41no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—81no—
143C3×C4⋊C4C22⋊C412yes—
144C3×C22⋊C4—22no—
145C3×C22⋊C4—22no—
146C3×C42C22×C411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42#4111yescharacteristic
150#213—48no—
151#208—48no—
152C2×C3⋊C8—21no—
153C2×C3⋊C8—21no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—21no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—21no—
158C2×C3⋊C8—21no—
159C2×C3⋊C8—41no—
160C2×C3⋊C8—41no—
161C2×C3⋊C8—41no—
162C2×C3⋊C8—41no—
163C2×C3⋊C8D812yescomplemented
164C22×D4—61no—
165C22×D4—61no—
166C22×D4—61no—
167C22×D4#6711yescharacteristic
168C23×C4—21no—
169C2×D8—64no—
170C2×D8—122no—
171C2×D8—122no—
172C2×M16—64no—
173C2×M16—122no—
174C2×M16—122no—
175C22×C8—21no—
176C22×C8—31no—
177C22×C8—31no—
178C22×C8—61no—
179C22×C8—61no—
180#127—32no—
181#127—61no—
182#127—61no—
183#127—61no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4C4×D312yes—
187C4×D4D1212yes—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4#6711yescharacteristic
192C2×C22⋊C4—21no—
193C2×C42C22×D311yescharacteristic
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—32no—
197C4⋊C8—32no—
198C4⋊C8—61no—
199C4⋊C8—61no—
200#102—42no—
201#102—64no—
202#102—122no—
203#102—122no—
204C22⋊C8—64no—
205C4×C8—61no—
206C4×C8—61no—
207C23×C3—22no—
208C23×C3—41no—
209C23×C3—41no—
210C23×C3—41no—
211C23×C3—41no—
212C23×C3—41no—
213C23×C3—41no—
214C23×C3#12111yescharacteristic
215C22×D3—42no—
216C22×D3—81no—
217C22×D3—81no—
218C22×D3—81no—
219C22×D3—81no—
220C22×D3—81no—
221C22×D3—81no—
222C3×D4—22no—
223C3×D4—22no—
224C3×D4—41no—
225C3×D4—41no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—22no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4—81no—
236C2×C3×C4—81no—
237C2×C3×C4—81no—
238C2×C3×C4C2×C22⋊C411yescharacteristic
239C2×C3×C4C4×D411yescharacteristic
240C2×C3×C4C4×D411yescharacteristic
241C2×C3×C4#12011yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12311yescharacteristic
244C2×C3×C4C2×D811yescharacteristic
245C2×C3×C4#13511yescharacteristic
246C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
247C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
248C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
249C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
250C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
251C2×C3×C4#13711yescharacteristic
252D12—42no—
253D12—42no—
254D12—42no—
255D12—42no—
256D12—82no—
257D12—82no—
258D12—81no—
259D12—81no—
260D12—81no—
261D12—81no—
262C3×C8—81no—
263C3⋊C8—22no—
264C3⋊C8—42no—
265C3⋊C8—42no—
266C3⋊C8—41no—
267C24—21no—
268C24—121no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—62no—
275C2×D4—62no—
276C2×D4—62no—
277C2×D4—121no—
278C2×D4—121no—
279C2×D4—121no—
280C2×D4—121no—
281C2×D4—121no—
282C2×D4—121no—
283C2×D4—121no—
284C2×D4—121no—
285C2×D4—121no—
286C2×D4—121no—
287C2×D4—121no—
288C2×D4—241no—
289C2×D4—241no—
290C2×D4(C2×D3)⋊C412yes—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4D3×D411yescharacteristic
300C22×C4#23711yescharacteristic
301C22×C4C2×#6711yescharacteristic
302D8—128no—
303M16—128no—
304C2×C8—32no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—61no—
309C2×C8—61no—
310C2×C8—61no—
311C2×C8—62no—
312C2×C8—61no—
313C2×C8—61no—
314C2×C8—62no—
315C2×C8—121no—
316C2×C8—121no—
317C2×C8—121no—
318C2×C8—121no—
319C4⋊C4—21no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—81no—
325C4⋊C4(C2×D3)⋊C412yes—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C42C2×C4×D311yescharacteristic
329C42C2×D1211yescharacteristic
330C42D3×D411yescharacteristic
331C42#23711yescharacteristic
332C22×C3—42no—
333C22×C3—42no—
334C22×C3—42no—
335C22×C3—81no—
336C22×C3—81no—
337C22×C3—81no—
338C22×C3—81no—
339C22×C3#38611yescharacteristic
340C22×C3#45911yescharacteristic
341C22×C3#46311yescharacteristic
342C22×C3#46811yescharacteristic
343C22×C3#46911yescharacteristic
344C22×C3#47111yescharacteristic
345C22×C3#47211yescharacteristic
346C2×D3—82no—
347C2×D3—82no—
348C2×D3—82no—
349C2×D3—161no—
350C2×D3—161no—
351C2×D3—161no—
352C2×D3—161no—
353C3×C4—21no—
354C3×C4—21no—
355C3×C4—42no—
356C3×C4—161no—
357C3×C4#41711yescharacteristic
358C3×C4#41811yescharacteristic
359C3×C4C4×D811yescharacteristic
360C3×C4#44211yescharacteristic
361C3×C4#44711yescharacteristic
362C3×C4#44911yescharacteristic
363C3×C4#46811yescharacteristic
364C3×C4#46911yescharacteristic
365C23—22no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—122no—
373C23—241no—
374C23—241no—
375C23—241no—
376C23—241no—
377C23—241no—
378C23—241no—
379C23#93911yescharacteristic
380D4—22no—
381D4—22no—
382D4—41no—
383D4—41no—
384D4—122no—
385D4—122no—
386D4—122no—
387D4—122no—
388D4—242no—
389D4—242no—
390D4—241no—
391D4—241no—
392D4—241no—
393D4—241no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4#67⋊C411yescharacteristic
407C2×C4#67⋊C411yescharacteristic
408C2×C4#88211yescharacteristic
409C2×C4#88611yescharacteristic
410C2×C4D3×D811yescharacteristic
411C2×C4#91111yescharacteristic
412C2×C4#91411yescharacteristic
413C2×C4#91611yescharacteristic
414C2×C4C2×(C2×D3)⋊C411yescharacteristic
415C2×C4#93211yescharacteristic
416C2×C4#93911yescharacteristic
417C2×C4#94211yescharacteristic
418C2×C4#94911yescharacteristic
419C2×C4#94911yescharacteristic
420C8—62no—
421C8—81no—
422C8—122no—
423C8—122no—
424C8—121no—
425C2×C3—82no—
426C2×C3#168511yescharacteristic
427C2×C3#168711yescharacteristic
428C2×C3#169711yescharacteristic
429C2×C3#170111yescharacteristic
430C2×C3#187911yescharacteristic
431C2×C3#189511yescharacteristic
432C2×C3#190711yescharacteristic
433D3—162no—
434C22—42no—
435C22—42no—
436C22—42no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—242no—
442C22—242no—
443C22—242no—
444C22—481no—
445C22—481no—
446C22—481no—
447C22—481no—
448C22#495811yescharacteristic
449C22#503911yescharacteristic
450C22#504911yescharacteristic
451C22#515011yescharacteristic
452C22#516511yescharacteristic
453C22#519911yescharacteristic
454C22#520311yescharacteristic
455C4—21no—
456C4—21no—
457C4—42no—
458C4—161no—
459C4#475611yescharacteristic
460C4#477311yescharacteristic
461C4#478111yescharacteristic
462C4#478511yescharacteristic
463C4#496611yescharacteristic
464C4#500111yescharacteristic
465C4#503911yescharacteristic
466C4#504911yescharacteristic
467C3#922911yescharacteristic, complemented, Sylow
468C2—82no—
469C2—482no—
470C2#6951711yescharacteristic
471C2#7017211yescharacteristic
472C2#7030311yescharacteristic
473C2#7100911yescharacteristic
474C2#7104311yescharacteristic
475C2#7110411yescharacteristic
476C2#7113811yescharacteristic
subgroupG/H
centerC23#939
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1802C2
socleC23×C3#121
prime radicalC2×#5015C2
cosocleC2×C3×C42C23
hypercenter#1802D3
hypocenterC3#9229
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. C2×#5012C2semidirect
3. C3×#1802C2semidirect
4. C2×#4485C2semidirect
5. #69580C2semidirect
6. #67703C2semidirect
7. #67634C2semidirect
18. #5015C4semidirect
32. #1802D3semidirect
163. C2×C3⋊C8D8semidirect
467. C3#9229semidirect

Maximal subgroups

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

subgroupindex
C2×#50152
C2×#50122
C3×#18022
C2×#44852
#695802
#677032
#676342
#92293