Finite Group Databasebeta

← #10695432

Subgroups — #10695432

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×#400C211yesmaximal, characteristic, complemented
2#82199C211yesmaximal, characteristic
3C2×#4968C211yesmaximal, characteristic
4#71554C211yesmaximal, characteristic, complemented
5C2×#4528C211yesmaximal, characteristic, complemented
6C2×#4451C211yesmaximal, characteristic
7#67612C211yesmaximal, characteristic, complemented
8#9921—31nomaximal, Sylow
9C22×C3×C4⋊C4C2211yescharacteristic
10C22×C3×C4⋊C4C2211yescharacteristic
11C2×#930C2211yescharacteristic
12C2×#929—21no—
13C3×#400—22no—
14C3×#400—22no—
15C2×C3×#95—21no—
16C2×C3×#95—21no—
17C2×C3×#95C2211yescharacteristic
18#5213—21no—
19#5212C2211yescharacteristic
20C2×#830C2211yescharacteristic
21C2×#830C2211yescharacteristic
22#4994—22no—
23#4991—21no—
24#4981—21no—
25#4968—22no—
26#4968—22no—
27C2×#809—21no—
28C2×#808—21no—
29#4529—22no—
30#4528—22no—
31#4451—22no—
32C2×#481—31no—
33#2618—31no—
34C2×#400D311yescharacteristic, complemented
35#1824—31no—
36C2×#327—31no—
37C2×#324—31no—
38#1294—31no—
39C23×C3×C4—21no—
40C23×C3×C4—21no—
41C23×C3×C4D411yescharacteristic
42C23×C3×C4D411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C22×#67—21no—
45C22×D12D411yescharacteristic
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—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⋊C4—21no—
57C2×C3×C4⋊C4D412yes—
58C2×#217—21no—
59#930—21no—
60#930—21no—
61#930—21no—
62#930—21no—
63#929—42no—
64#929—42no—
65C2×(C2×D3)⋊C4—22no—
66C2×(C2×D3)⋊C4—21no—
67C2×(C2×D3)⋊C4—22no—
68C2×C4⋊Dic3D411yescharacteristic
69C2×C4⋊Dic3D412yes—
70C2×Dic3⋊C4—22no—
71C22×C3⋊C8—21no—
72C22×C3⋊C8—21no—
73C3×#95—24no—
74C3×#95—24no—
75C3×#95—21no—
76C3×#95D412yes—
77C3⋊#95—22no—
78C3⋊#95—21no—
79#830—21no—
80#830—21no—
81#830—21no—
82#830D412yes—
83#809—24no—
84#808—24no—
85C2×#123—61no—
86C2×#121—31no—
87C22×C4⋊C4C2×D311yescharacteristic
88C22×C4⋊C4C2×D311yescharacteristic
89#481—62no—
90#481—62no—
91C2×#106—61no—
92C2×#102—61no—
93C2×C22⋊C8—31no—
94C2×C22⋊C8—31no—
95#400—22no—
96#400—22no—
97#398—61no—
98#396—62no—
99#394—61no—
100#386—61no—
101#380—31no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95C2×D311yescharacteristic
105#340—62no—
106#327—62no—
107#324—62no—
108C24×C3C2×D411yescharacteristic
109C23×D3—21no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4—22no—
118C22×C3×C4—21no—
119C22×C3×C4—22no—
120C22×C3×C4—21no—
121C22×C3×C4—42no—
122C22×C3×C4—42no—
123C22×C3×C4—42no—
124C22×C3×C4—42no—
125C22×C3×C4—44no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4—41no—
129C22×C3×C4—41no—
130C22×C3×C4SD1612yes—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4#4111yescharacteristic
137C2×#67—41no—
138C2×#67—41no—
139C2×#67—41no—
140C2×#67—41no—
141C2×#67—41no—
142C2×#67—41no—
143C22×Dic3—21no—
144C22×Dic3—22no—
145C2×D12—21no—
146C2×D12—21no—
147C2×D12—42no—
148C3×C4⋊C4—22no—
149C3×C4⋊C4—22no—
150C3×C4⋊C4—41no—
151C3×C4⋊C4—41no—
152C3×C4⋊C4—41no—
153C3×C4⋊C4—41no—
154C3×C4⋊C4—41no—
155C3×C4⋊C4—41no—
156#217—41no—
157#217—41no—
158(C2×D3)⋊C4—44no—
159(C2×D3)⋊C4—41no—
160(C2×D3)⋊C4—41no—
161(C2×D3)⋊C4—82no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3—42no—
165Dic3⋊C4—44no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—42no—
168C2×C3⋊C8—42no—
169C2×C3⋊C8—41no—
170C22×D4—31no—
171C22×D4—61no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4#6711yescharacteristic
175C23×C4#6711yescharacteristic
176C23×C4C22×D311yescharacteristic
177C22×C8—61no—
178C22×C8—61no—
179#123—122no—
180#123—122no—
181#121—61no—
182#121—61no—
183#121—61no—
184#121—61no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—31no—
197C2×C4⋊C4—32no—
198C2×C4⋊C4—62no—
199C2×C4⋊C4#6712yes—
200C2×C22⋊C4—62no—
201C2×C22⋊C4—61no—
202C2×C22⋊C4—61no—
203C2×C22⋊C4—62no—
204#106—64no—
205#102—64no—
206C22⋊C8—32no—
207C22⋊C8—61no—
208C22⋊C8—61no—
209C22⋊C8—61no—
210#95—24no—
211#95—24no—
212#95—21no—
213#95—62no—
214#95—61no—
215#95#6712yes—
216C23×C3—22no—
217C23×C3—22no—
218C23×C3#12011yescharacteristic
219C23×C3#12011yescharacteristic
220C23×C3#12111yescharacteristic
221C23×C3C2×SD1611yescharacteristic
222C23×C3#13511yescharacteristic
223C23×C3AGL(1,ℤ/8)11yescharacteristic
224C23×C3#13711yescharacteristic
225C22×D3—41no—
226C22×D3—41no—
227C22×D3—41no—
228C22×D3—42no—
229C22×D3—42no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—22no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4—44no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4—41no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4—82no—
252C2×C3×C4—82no—
253C2×C3×C4—82no—
254C2×C3×C4—82no—
255C2×C3×C4—81no—
256C2×C3×C4#12011yescharacteristic
257C2×C3×C4#12111yescharacteristic
258C2×C3×C4#12111yescharacteristic
259C2×C3×C4#12711yescharacteristic
260#67—81no—
261#67—81no—
262#67—81no—
263#67—81no—
264C2×Dic3—41no—
265C2×Dic3—41no—
266C2×Dic3—42no—
267C2×Dic3—41no—
268C2×Dic3—42no—
269C2×Dic3—42no—
270D12—42no—
271D12—81no—
272C3⋊C8—42no—
273C3⋊C8—81no—
274C24—61no—
275C24C2×#6711yescharacteristic
276C2×D4—61no—
277C2×D4—61no—
278C2×D4—122no—
279C2×D4—121no—
280C2×D4—121no—
281C2×D4—121no—
282C2×D4—121no—
283C2×D4—121no—
284C2×D4—121no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—44no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—62no—
306C22×C4—61no—
307C22×C4#21512yes—
308C22×C4D3×D411yescharacteristic
309C22×C4D3×D411yescharacteristic
310C22×C4D3×D411yescharacteristic
311C22×C4#23711yescharacteristic
312C22×C4C2×#6711yescharacteristic
313C22×C4C2×#6711yescharacteristic
314C2×C8—62no—
315C2×C8—122no—
316C2×C8—122no—
317C2×C8—121no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—61no—
327C4⋊C4—61no—
328C4⋊C4—122no—
329C4⋊C4—124no—
330C22⋊C4—124no—
331C22⋊C4—121no—
332C22⋊C4—121no—
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#39211yescharacteristic
345C22×C3#44811yescharacteristic
346C22×C3#44911yescharacteristic
347C22×C3#45011yescharacteristic
348C22×C3#45211yescharacteristic
349C22×C3#45412yes—
350C22×C3#45812yes—
351C22×C3#46511yescharacteristic
352C22×C3#46811yescharacteristic
353C2×D3—82no—
354C2×D3—82no—
355C2×D3—81no—
356C2×D3—81no—
357C2×D3—81no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—42no—
363C3×C4—82no—
364C3×C4—82no—
365C3×C4—81no—
366Dic3—82no—
367Dic3—81no—
368C23—22no—
369C23—22no—
370C23—121no—
371C23—121no—
372C23—121no—
373C23—122no—
374C23—122no—
375C23#93711yescharacteristic
376C23#93911yescharacteristic
377C23C2×#21511yescharacteristic
378C23#94211yescharacteristic
379C23#94911yescharacteristic
380C23#95011yescharacteristic
381C23#95311yescharacteristic
382D4—122no—
383D4—241no—
384D4—241no—
385D4—241no—
386D4—241no—
387D4—241no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—81no—
414C2×C4—121no—
415C2×C4—122no—
416C2×C4—121no—
417C2×C4—121no—
418C2×C4—122no—
419C2×C4—122no—
420C2×C4#93711yescharacteristic
421C2×C4#93811yescharacteristic
422C2×C4#93911yescharacteristic
423C2×C4#94011yescharacteristic
424C8—122no—
425C8—241no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—21no—
429C2×C3—21no—
430C2×C3#161512yes—
431C2×C3#161612yes—
432C2×C3#200011yescharacteristic
433C2×C3#200311yescharacteristic
434C2×C3#202211yescharacteristic
435D3—161no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—241no—
445C22—241no—
446C22—241no—
447C22—242no—
448C22—242no—
449C22#504611yescharacteristic
450C22#504711yescharacteristic
451C22#504811yescharacteristic
452C22#504911yescharacteristic
453C22#505312yes—
454C22#506212yes—
455C22#522311yescharacteristic
456C22#522711yescharacteristic
457C22#523811yescharacteristic
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—42no—
463C4—82no—
464C4—81no—
465C4—82no—
466C4—241no—
467C4—242no—
468C3#992111yescharacteristic, complemented, Sylow
469C2—21no—
470C2—21no—
471C2—21no—
472C2—21no—
473C2—481no—
474C2#6971712yes—
475C2#6971812yes—
476C2#7169911yescharacteristic
477C2#7170611yescharacteristic
478C2#7173011yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#400C2
socleC23×C3#120
prime radicalC2×#930C22
hypercenterC2×#400D3
hypocenterC3#9921
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#400

Complements

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

NH ≅ G/Ntype
1. C2×C3×#400C2semidirect
4. #71554C2semidirect
5. C2×#4528C2semidirect
7. #67612C2semidirect
34. C2×#400D3semidirect
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×#4002
#821992
C2×#49682
#715542
C2×#45282
C2×#44512
#676122
#99213