Finite Group Databasebeta

← #10695429

Subgroups — #10695429

Order 768.

672 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#481C211yesmaximal, characteristic
2#82675C211yesmaximal, characteristic
3C2×#4985C211yesmaximal, characteristic, complemented
4C2×#4537C211yesmaximal, characteristic
5#71554C211yesmaximal, characteristic, complemented
6C2×#4469C211yesmaximal, characteristic, complemented
7#67675C211yesmaximal, characteristic, complemented
8#9921—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C22×C3×C4⋊C4C2211yescharacteristic
11C2×#935—21no—
12C22×Dic3⋊C4C2211yescharacteristic
13C3×#481—22no—
14C3×#481—22no—
15C2×C3×#106—21no—
16C2×C3×#102—21no—
17C2×C3×C22⋊C8C2211yescharacteristic
18#5222—21no—
19#5220—21no—
20#5212C2211yescharacteristic
21C2×C3⋊#95—21no—
22C2×C3⋊#95—21no—
23C2×C3⋊#95C2211yescharacteristic
24C2×#830C2211yescharacteristic
25#4985—22no—
26#4985—22no—
27#4983—21no—
28#4948—22no—
29#4537—22no—
30#4529—22no—
31#4469—22no—
32C2×#481D311yescharacteristic
33#2618—31no—
34C2×#400—31no—
35#1824—31no—
36C2×#327—31no—
37C2×#324—31no—
38#1294—31no—
39C22×C3×D4—21no—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4—21no—
42C23×C3×C4C2311yescharacteristic
43C23×Dic3—21no—
44C23×Dic3D411yescharacteristic
45C23×Dic3D411yescharacteristic
46C22×C3×C8—21no—
47C3×#121—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×#121—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C4⋊C4D412yes—
58C2×C3×C22⋊C4—21no—
59C2×#217—22no—
60C2×#217—21no—
61C2×#217—22no—
62#935—42no—
63#935—42no—
64C2×C4⋊Dic3D412yes—
65C2×Dic3⋊C4—22no—
66C2×Dic3⋊C4—21no—
67C2×Dic3⋊C4—21no—
68C2×Dic3⋊C4—21no—
69C2×Dic3⋊C4—21no—
70C2×Dic3⋊C4—21no—
71C2×Dic3⋊C4—21no—
72C22×C3⋊C8—21no—
73C3×#106—24no—
74C3×#102—24no—
75C3×C22⋊C8—21no—
76C3×C22⋊C8D412yes—
77C3⋊#95—22no—
78C3⋊#95—24no—
79C3⋊#95—24no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82C3⋊#95D412yes—
83#830—21no—
84#830—21no—
85C2×#123—61no—
86C2×#121C2×D311yescharacteristic
87C22×C4⋊C4—31no—
88C22×C4⋊C4C2×D311yescharacteristic
89#481—22no—
90#481—22no—
91C2×#106—21no—
92C2×#102—21no—
93C2×C22⋊C8—31no—
94C2×C22⋊C8C2×D311yescharacteristic
95#400—62no—
96#400—62no—
97#398—61no—
98#396—62no—
99#394—61no—
100#386—61no—
101#380—31no—
102C2×#95—31no—
103C2×#95—61no—
104C2×#95—61no—
105#340—62no—
106#327—62no—
107#324—62no—
108C24×C3—21no—
109C24×C3C2×D411yescharacteristic
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—42no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C22×C3×C4—22no—
120C22×C3×C4—21no—
121C22×C3×C4—42no—
122C22×C3×C4—44no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4#4111yescharacteristic
131C22×Dic3—21no—
132C22×Dic3—21no—
133C22×Dic3—21no—
134C22×Dic3—21no—
135C22×Dic3—21no—
136C22×Dic3—21no—
137C22×Dic3—22no—
138C22×Dic3—21no—
139C22×Dic3—21no—
140C22×Dic3—22no—
141C22×Dic3—21no—
142C22×Dic3—42no—
143C22×Dic3—42no—
144C22×Dic3—42no—
145C22×Dic3—41no—
146C22×Dic3—41no—
147C22×Dic3SD1612yes—
148C2×C3×C8—22no—
149C2×C3×C8—42no—
150C3×C4⋊C4—22no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—41no—
155C3×C4⋊C4—41no—
156C3×C22⋊C4—41no—
157C3×C22⋊C4—41no—
158#217—44no—
159#217—41no—
160#217—41no—
161#217—82no—
162C4⋊Dic3—42no—
163Dic3⋊C4—44no—
164Dic3⋊C4—41no—
165Dic3⋊C4—41no—
166Dic3⋊C4—41no—
167Dic3⋊C4—41no—
168C2×C3⋊C8—42no—
169C2×C3⋊C8—41no—
170C22×D4—21no—
171C22×D4#6711yescharacteristic
172C23×C4—21no—
173C23×C4—31no—
174C23×C4—31no—
175C23×C4—61no—
176C23×C4C22×D311yescharacteristic
177C22×C8—21no—
178C22×C8—61no—
179#123—122no—
180#123—122no—
181#121—21no—
182#121—21no—
183#121—21no—
184#121—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—32no—
191C2×C4⋊C4—62no—
192C2×C4⋊C4—61no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4—61no—
195C2×C4⋊C4—61no—
196C2×C4⋊C4—61no—
197C2×C4⋊C4—61no—
198C2×C4⋊C4#6712yes—
199C2×C4⋊C4#6711yescharacteristic
200C2×C22⋊C4—21no—
201C2×C22⋊C4—62no—
202C2×C22⋊C4—61no—
203C2×C22⋊C4—62no—
204#106—24no—
205#102—24no—
206C22⋊C8—21no—
207C22⋊C8—61no—
208C22⋊C8—61no—
209C22⋊C8#6712yes—
210#95—32no—
211#95—64no—
212#95—64no—
213#95—62no—
214#95—61no—
215#95—61no—
216C23×C3—22no—
217C23×C3—22no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3—42no—
222C23×C3—42no—
223C23×C3#12011yescharacteristic
224C23×C3#12011yescharacteristic
225C23×C3#12111yescharacteristic
226C23×C3C2×SD1611yescharacteristic
227C23×C3#13511yescharacteristic
228C23×C3AGL(1,ℤ/8)11yescharacteristic
229C23×C3#13711yescharacteristic
230C3×D4—42no—
231C3×D4—81no—
232C3×D4—81no—
233C3×D4—81no—
234C3×D4—81no—
235C3×D4—81no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—44no—
243C2×C3×C4—41no—
244C2×C3×C4—42no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—82no—
248C2×C3×C4—82no—
249C2×C3×C4#12011yescharacteristic
250C2×C3×C4#12111yescharacteristic
251C2×C3×C4#12111yescharacteristic
252C2×C3×C4#12711yescharacteristic
253C2×Dic3—22no—
254C2×Dic3—22no—
255C2×Dic3—22no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258C2×Dic3—41no—
259C2×Dic3—41no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262C2×Dic3—42no—
263C2×Dic3—41no—
264C2×Dic3—42no—
265C2×Dic3—42no—
266C2×Dic3—42no—
267C2×Dic3—42no—
268C2×Dic3—42no—
269C2×Dic3—82no—
270C2×Dic3—82no—
271C2×Dic3—81no—
272C3×C8—42no—
273C3⋊C8—81no—
274C24—21no—
275C24D3×D411yescharacteristic
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—42no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—32no—
288C22×C4—42no—
289C22×C4—44no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—61no—
293C22×C4—61no—
294C22×C4—61no—
295C22×C4—61no—
296C22×C4—61no—
297C22×C4—61no—
298C22×C4—62no—
299C22×C4—61no—
300C22×C4—62no—
301C22×C4—61no—
302C22×C4—61no—
303C22×C4—122no—
304C22×C4—122no—
305C22×C4—122no—
306C22×C4—121no—
307C22×C4—121no—
308C22×C4D3×D411yescharacteristic
309C22×C4D3×D411yescharacteristic
310C22×C4#23711yescharacteristic
311C22×C4C2×#6711yescharacteristic
312C22×C4C2×#6711yescharacteristic
313C22×C4C2×#6711yescharacteristic
314C2×C8—22no—
315C2×C8—42no—
316C2×C8—122no—
317C2×C8—121no—
318C4⋊C4—22no—
319C4⋊C4—21no—
320C4⋊C4—22no—
321C4⋊C4—21no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—122no—
325C4⋊C4—124no—
326C4⋊C4—121no—
327C4⋊C4—121no—
328C4⋊C4—121no—
329C4⋊C4—121no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—124no—
333C22⋊C4—121no—
334C22⋊C4—121no—
335C22⋊C4—242no—
336C22×C3—22no—
337C22×C3—22no—
338C22×C3—22no—
339C22×C3—22no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—82no—
345C22×C3—82no—
346C22×C3—81no—
347C22×C3—81no—
348C22×C3—81no—
349C22×C3#39211yescharacteristic
350C22×C3#44811yescharacteristic
351C22×C3#44911yescharacteristic
352C22×C3#45011yescharacteristic
353C22×C3#45211yescharacteristic
354C22×C3#45412yes—
355C22×C3#45812yes—
356C22×C3#46511yescharacteristic
357C22×C3#46811yescharacteristic
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—81no—
363C3×C4—82no—
364Dic3—42no—
365Dic3—82no—
366Dic3—82no—
367Dic3—81no—
368C23—22no—
369C23—22no—
370C23—41no—
371C23—41no—
372C23—42no—
373C23—41no—
374C23—42no—
375C23D3×SD1611yescharacteristic
376C23#91411yescharacteristic
377C23#91511yescharacteristic
378C23#91611yescharacteristic
379C23#93711yescharacteristic
380C23#93711yescharacteristic
381C23#93811yescharacteristic
382D4—42no—
383D4—81no—
384D4—81no—
385D4—81no—
386D4—81no—
387D4—81no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—44no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—62no—
400C2×C4—62no—
401C2×C4—62no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—121no—
405C2×C4—121no—
406C2×C4—121no—
407C2×C4—121no—
408C2×C4—122no—
409C2×C4—122no—
410C2×C4—122no—
411C2×C4—122no—
412C2×C4—122no—
413C2×C4—122no—
414C2×C4—121no—
415C2×C4—122no—
416C2×C4—122no—
417C2×C4—242no—
418C2×C4—242no—
419C2×C4—241no—
420C2×C4#93911yescharacteristic
421C2×C4#93911yescharacteristic
422C2×C4#94011yescharacteristic
423C2×C4#95311yescharacteristic
424C8—42no—
425C8—241no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—21no—
429C2×C3—21no—
430C2×C3—161no—
431C2×C3#161512yes—
432C2×C3#161612yes—
433C2×C3#200011yescharacteristic
434C2×C3#200311yescharacteristic
435C2×C3#202211yescharacteristic
436C22—22no—
437C22—22no—
438C22—22no—
439C22—22no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—82no—
445C22—82no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22#505312yes—
450C22#516911yescharacteristic
451C22#517011yescharacteristic
452C22#517111yescharacteristic
453C22#517211yescharacteristic
454C22#517311yescharacteristic
455C22#517411yescharacteristic
456C22#520112yes—
457C22#522311yescharacteristic
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—81no—
463C4—82no—
464C4—122no—
465C4—242no—
466C4—242no—
467C4—241no—
468C3#992111yescharacteristic, complemented, Sylow
469C2—21no—
470C2—21no—
471C2—21no—
472C2—21no—
473C2—161no—
474C2#7117012yes—
475C2#7117112yes—
476C2#7170611yescharacteristic
477C2#7205411yescharacteristic
478C2#7209611yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#481C2
socleC23×C3#120
prime radicalC2×C3×#121C22
hypercenterC2×#481D3
hypocenterC3#9921
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#481

Complements

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

NH ≅ G/Ntype
3. C2×#4985C2semidirect
5. #71554C2semidirect
6. C2×#4469C2semidirect
7. #67675C2semidirect
468. C3#9921semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4812
#826752
C2×#49852
C2×#45372
#715542
C2×#44692
#676752
#99213