Finite Group Databasebeta

← #10695430

Subgroups — #10695430

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
1C3×#2618C211yesmaximal, characteristic
2C2×#5034C211yesmaximal, characteristic
3C2×#4986C211yesmaximal, characteristic, complemented
4C2×#4537C211yesmaximal, characteristic
5C2×#4528C211yesmaximal, characteristic, complemented
6#69502C211yesmaximal, characteristic, complemented
7#67624C211yesmaximal, characteristic, complemented
8#9921—31nomaximal, Sylow
9C2×C3×#123—21no—
10C2×C3×#121C2211yescharacteristic
11C22×C3×C4⋊C4C2211yescharacteristic
12C22×C4⋊Dic3C2211yescharacteristic
13C3×#398—21no—
14C3×#396—22no—
15C3×#394—21no—
16C3×#386—21no—
17C3×#380C2211yescharacteristic
18C2×#832—21no—
19C2×C3⋊#95—21no—
20C2×C3⋊#95—21no—
21C2×C3⋊#95C2211yescharacteristic
22C2×#830C2211yescharacteristic
23C2×#830C2211yescharacteristic
24#5034—22no—
25#5034—22no—
26#4986—22no—
27#4986—22no—
28C2×#808—21no—
29#4537—22no—
30#4529—22no—
31#4528—22no—
32C2×#481—31no—
33#2618D311yescharacteristic
34C2×#400—31no—
35#1824—31no—
36C2×#327—31no—
37C2×#324—31no—
38#1294—31no—
39C22×C3×D4—21no—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4D411yescharacteristic
42C23×C3×C4D411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C23×Dic3—21no—
45C23×Dic3—21no—
46C3×#123—42no—
47C3×#123—42no—
48C3×#121—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×#121—21no—
52C2×C3×C4⋊C4—22no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C4⋊C4D412yes—
61C2×C3×C22⋊C4—22no—
62C2×C3×C22⋊C4—21no—
63C2×C3×C22⋊C4—21no—
64C2×C3×C22⋊C4—22no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3—21no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3—21no—
70C2×C4⋊Dic3D412yes—
71C22×C3⋊C8—21no—
72C22×C3⋊C8—21no—
73C3×#95—22no—
74C3×#95—21no—
75#832—24no—
76C3⋊#95—24no—
77C3⋊#95—24no—
78C3⋊#95—21no—
79C3⋊#95D412yes—
80#830—21no—
81#830—21no—
82#830—21no—
83#830D412yes—
84#808—24no—
85C2×#123—21no—
86C2×#121C2×D311yescharacteristic
87C22×C4⋊C4—31no—
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—62no—
96#400—62no—
97#398—21no—
98#396—22no—
99#394—21no—
100#386—21no—
101#380C2×D311yescharacteristic
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—21no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—22no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—42no—
131C22×C3×C4—42no—
132C22×C3×C4SD1612yes—
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4#4111yescharacteristic
139C22×Dic3—22no—
140C22×Dic3—22no—
141C22×Dic3—42no—
142C22×Dic3—42no—
143C22×Dic3—44no—
144C22×Dic3—41no—
145C22×Dic3—41no—
146C22×Dic3—41no—
147C22×Dic3—41no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—42no—
151C3×C4⋊C4—44no—
152C3×C4⋊C4—41no—
153C3×C4⋊C4—41no—
154C3×C4⋊C4—41no—
155C3×C4⋊C4—41no—
156C3×C22⋊C4—44no—
157C3×C22⋊C4—41no—
158C3×C22⋊C4—41no—
159C3×C22⋊C4—41no—
160C3×C22⋊C4—41no—
161C3×C22⋊C4—82no—
162C4⋊Dic3—22no—
163C4⋊Dic3—22no—
164C4⋊Dic3—41no—
165C4⋊Dic3—41no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—42no—
168C2×C3⋊C8—42no—
169C2×C3⋊C8—41no—
170C22×D4—21no—
171C22×D4#6711yescharacteristic
172C23×C4—61no—
173C23×C4—61no—
174C23×C4#6711yescharacteristic
175C23×C4#6711yescharacteristic
176C23×C4C22×D311yescharacteristic
177C22×C8—61no—
178C22×C8—61no—
179#123—42no—
180#123—42no—
181#121—21no—
182#121—21no—
183#121—21no—
184#121—21no—
185C2×C4⋊C4—22no—
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—32no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4—61no—
195C2×C4⋊C4—61no—
196C2×C4⋊C4—61no—
197C2×C4⋊C4—61no—
198C2×C4⋊C4#6711yescharacteristic
199C2×C4⋊C4#6712yes—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204#106—64no—
205#102—64no—
206C22⋊C8—32no—
207C22⋊C8—61no—
208C22⋊C8—61no—
209C22⋊C8—61no—
210#95—22no—
211#95—21no—
212#95—32no—
213#95—64no—
214#95—64no—
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—22no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—42no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—42no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—81no—
259C2×C3×C4#12011yescharacteristic
260C2×C3×C4#12111yescharacteristic
261C2×C3×C4#12111yescharacteristic
262C2×C3×C4#12711yescharacteristic
263C2×Dic3—42no—
264C2×Dic3—42no—
265C2×Dic3—44no—
266C2×Dic3—42no—
267C2×Dic3—42no—
268C2×Dic3—82no—
269C2×Dic3—82no—
270C2×Dic3—82no—
271C2×Dic3—82no—
272C3⋊C8—42no—
273C3⋊C8—81no—
274C24—21no—
275C24C2×#6711yescharacteristic
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—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—21no—
295C22×C4—21no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—62no—
299C22×C4—62no—
300C22×C4—122no—
301C22×C4—122no—
302C22×C4—124no—
303C22×C4—121no—
304C22×C4—121no—
305C22×C4—121no—
306C22×C4—121no—
307C22×C4#21412yes—
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—21no—
319C4⋊C4—21no—
320C4⋊C4—42no—
321C4⋊C4—44no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—62no—
327C4⋊C4—62no—
328C4⋊C4—121no—
329C4⋊C4—121no—
330C22⋊C4—44no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—82no—
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—42no—
363C3×C4—82no—
364C3×C4—81no—
365C3×C4—81no—
366Dic3—82no—
367Dic3—82no—
368C23—22no—
369C23—22no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—42no—
374C23—42no—
375C23#93211yescharacteristic
376C23C2×#21411yescharacteristic
377C23#93711yescharacteristic
378C23#93911yescharacteristic
379C23#95011yescharacteristic
380C23#95111yescharacteristic
381C23#95311yescharacteristic
382D4—42no—
383D4—81no—
384D4—81no—
385D4—81no—
386D4—81no—
387D4—81no—
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—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—81no—
411C2×C4—122no—
412C2×C4—122no—
413C2×C4—124no—
414C2×C4—122no—
415C2×C4—122no—
416C2×C4—242no—
417C2×C4—242no—
418C2×C4—242no—
419C2×C4—242no—
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—161no—
431C2×C3#161512yes—
432C2×C3#161612yes—
433C2×C3#200011yescharacteristic
434C2×C3#200311yescharacteristic
435C2×C3#202211yescharacteristic
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22—82no—
448C22—82no—
449C22#503711yescharacteristic
450C22#504011yescharacteristic
451C22#504111yescharacteristic
452C22#504211yescharacteristic
453C22#521811yescharacteristic
454C22#522311yescharacteristic
455C22#523911yescharacteristic
456C22#524312yes—
457C22#524512yes—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—42no—
463C4—81no—
464C4—81no—
465C4—82no—
466C4—242no—
467C4—242no—
468C3#992111yescharacteristic, complemented, Sylow
469C2—21no—
470C2—21no—
471C2—21no—
472C2—21no—
473C2—161no—
474C2#7168511yescharacteristic
475C2#7184911yescharacteristic
476C2#7186711yescharacteristic
477C2#7187512yes—
478C2#7188712yes—
subgroupG/H
centerC23#953
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2618C2
socleC23×C3#120
prime radicalC2×C3×#121C22
hypercenter#2618D3
hypocenterC3#9921
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2618

Complements

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

NH ≅ G/Ntype
3. C2×#4986C2semidirect
5. C2×#4528C2semidirect
6. #69502C2semidirect
7. #67624C2semidirect
468. C3#9921semidirect

Maximal subgroups

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

subgroupindex
C3×#26182
C2×#50342
C2×#49862
C2×#45372
C2×#45282
#695022
#676242
#99213