Finite Group Databasebeta

← #10695434

Subgroups — #10695434

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
1#82199C211yesmaximal, characteristic
2C2×#4986C211yesmaximal, characteristic, complemented
3C2×#4735C211yesmaximal, characteristic
4C2×C3×#324C211yesmaximal, characteristic, complemented
5C2×#4470C211yesmaximal, characteristic
6#69925C211yesmaximal, characteristic, complemented
7#67675C211yesmaximal, characteristic, complemented
8#9921—31nomaximal, Sylow
9C22×C3×C4⋊C4C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C2×#929—21no—
12C22×C4⋊Dic3C2211yescharacteristic
13C2×C3×C22⋊C8C2211yescharacteristic
14C2×C3×C22⋊C8C2211yescharacteristic
15#5213—21no—
16#5212C2211yescharacteristic
17C2×C3⋊#95—21no—
18C2×C3⋊#95—21no—
19C2×C3⋊#95C2211yescharacteristic
20C2×#821—21no—
21C2×#817—21no—
22#4994—22no—
23#4991—21no—
24#4986—22no—
25#4986—22no—
26#4981—21no—
27#4735—22no—
28#4735—22no—
29C3×#324—22no—
30#4551—22no—
31#4470—22no—
32C2×#481—31no—
33#2618—31no—
34C2×#400—31no—
35#1824—31no—
36C2×#327—31no—
37C2×#324D311yescharacteristic, complemented
38#1294—31no—
39C23×C3×C4D411yescharacteristic
40C23×C3×C4D411yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C22×#67—21no—
43C23×Dic3—21no—
44C23×Dic3—21no—
45C22×D12D411yescharacteristic
46C22×C3×C8—21no—
47C22×C3×C8—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×#217—21no—
55#930—21no—
56#930—21no—
57#930—21no—
58#930—21no—
59#929—42no—
60#929—42no—
61C2×(C2×D3)⋊C4—22no—
62C2×(C2×D3)⋊C4—21no—
63C2×(C2×D3)⋊C4—22no—
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3—21no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3D411yescharacteristic
70C2×C4⋊Dic3D412yes—
71C2×C4⋊Dic3D412yes—
72C2×Dic3⋊C4—22no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×C22⋊C8D412yes—
77C3⋊#95—24no—
78C3⋊#95—24no—
79C3⋊#95—22no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82C3⋊#95D412yes—
83#821—24no—
84#817—24no—
85C2×#123—61no—
86C2×#121—31no—
87C22×C4⋊C4—31no—
88C22×C4⋊C4C2×D311yescharacteristic
89#481—62no—
90#481—62no—
91C2×#106—61no—
92C2×#102—61no—
93C2×C22⋊C8C2×D311yescharacteristic
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—22no—
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—21no—
118C22×C3×C4—21no—
119C22×C3×C4—42no—
120C22×C3×C4—42no—
121C22×C3×C4SD1612yes—
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4#4111yescharacteristic
128C2×#67—41no—
129C2×#67—41no—
130C2×#67—41no—
131C2×#67—41no—
132C2×#67—41no—
133C2×#67—41no—
134C22×Dic3—22no—
135C22×Dic3—22no—
136C22×Dic3—21no—
137C22×Dic3—22no—
138C22×Dic3—42no—
139C22×Dic3—42no—
140C22×Dic3—44no—
141C22×Dic3—41no—
142C22×Dic3—41no—
143C22×Dic3—41no—
144C22×Dic3—41no—
145C2×D12—21no—
146C2×D12—21no—
147C2×D12—42no—
148C2×C3×C8—22no—
149C2×C3×C8—42no—
150C2×C3×C8—42no—
151C2×C3×C8—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—22no—
164C4⋊Dic3—22no—
165C4⋊Dic3—21no—
166C4⋊Dic3—41no—
167C4⋊Dic3—41no—
168C4⋊Dic3—42no—
169Dic3⋊C4—44no—
170C22×D4—31no—
171C22×D4—61no—
172C23×C4—61no—
173C23×C4—61no—
174C23×C4D1211yescharacteristic
175C23×C4D1211yescharacteristic
176C23×C4C22×D311yescharacteristic
177C22×C8—21no—
178C22×C8—21no—
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—32no—
192C2×C4⋊C4—31no—
193C2×C4⋊C4—32no—
194C2×C4⋊C4—61no—
195C2×C4⋊C4—61no—
196C2×C4⋊C4—61no—
197C2×C4⋊C4—61no—
198C2×C4⋊C4—62no—
199C2×C4⋊C4—61no—
200C2×C22⋊C4—62no—
201C2×C22⋊C4—61no—
202C2×C22⋊C4—61no—
203C2×C22⋊C4—62no—
204#106—64no—
205#102—64no—
206C22⋊C8—21no—
207C22⋊C8—21no—
208C22⋊C8—21no—
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#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—41no—
245C2×C3×C4—42no—
246C2×C3×C4—81no—
247C2×C3×C4#12011yescharacteristic
248C2×C3×C4#12111yescharacteristic
249C2×C3×C4#12111yescharacteristic
250C2×C3×C4#12711yescharacteristic
251#67—81no—
252#67—81no—
253#67—81no—
254#67—81no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—44no—
258C2×Dic3—42no—
259C2×Dic3—41no—
260C2×Dic3—42no—
261C2×Dic3—41no—
262C2×Dic3—42no—
263C2×Dic3—41no—
264C2×Dic3—42no—
265C2×Dic3—42no—
266C2×Dic3—82no—
267C2×Dic3—82no—
268C2×Dic3—82no—
269C2×Dic3—82no—
270D12—42no—
271D12—81no—
272C3×C8—42no—
273C3×C8—81no—
274C24—61no—
275C24C2×D1211yescharacteristic
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—21no—
293C22×C4—21no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—62no—
297C22×C4—62no—
298C22×C4—62no—
299C22×C4—61no—
300C22×C4—122no—
301C22×C4—122no—
302C22×C4—124no—
303C22×C4—121no—
304C22×C4—121no—
305C22×C4—121no—
306C22×C4—121no—
307C22×C4#20412yes—
308C22×C4#23511yescharacteristic
309C22×C4D3×D411yescharacteristic
310C22×C4D3×D411yescharacteristic
311C22×C4D3×D411yescharacteristic
312C22×C4D3×D411yescharacteristic
313C22×C4C2×#6711yescharacteristic
314C2×C8—22no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—62no—
323C4⋊C4—61no—
324C4⋊C4—62no—
325C4⋊C4—61no—
326C4⋊C4—122no—
327C4⋊C4—124no—
328C4⋊C4—121no—
329C4⋊C4—121no—
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—81no—
364Dic3—82no—
365Dic3—82no—
366Dic3—81no—
367Dic3—82no—
368C23—22no—
369C23—22no—
370C23—121no—
371C23—121no—
372C23—121no—
373C23—122no—
374C23—122no—
375C23#88211yescharacteristic
376C23#88211yescharacteristic
377C23C2×#20411yescharacteristic
378C23#90411yescharacteristic
379C23#90811yescharacteristic
380C23#90911yescharacteristic
381C23#93011yescharacteristic
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—41no—
403C2×C4—42no—
404C2×C4—81no—
405C2×C4—122no—
406C2×C4—122no—
407C2×C4—124no—
408C2×C4—121no—
409C2×C4—122no—
410C2×C4—122no—
411C2×C4—121no—
412C2×C4—121no—
413C2×C4—122no—
414C2×C4—122no—
415C2×C4—122no—
416C2×C4—242no—
417C2×C4—242no—
418C2×C4—242no—
419C2×C4—242no—
420C2×C4#88411yescharacteristic
421C2×C4#88411yescharacteristic
422C2×C4#93711yescharacteristic
423C2×C4#94011yescharacteristic
424C8—42no—
425C8—81no—
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—22no—
437C22—22no—
438C22—22no—
439C22—22no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—242no—
445C22—242no—
446C22—241no—
447C22—241no—
448C22—241no—
449C22#473111yescharacteristic
450C22#473311yescharacteristic
451C22#473411yescharacteristic
452C22#473811yescharacteristic
453C22#474212yes—
454C22#475312yes—
455C22#496011yescharacteristic
456C22#511411yescharacteristic
457C22#513411yescharacteristic
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—42no—
463C4—81no—
464C4—242no—
465C4—242no—
466C4—241no—
467C4—242no—
468C3#992111yescharacteristic, complemented, Sylow
469C2—21no—
470C2—21no—
471C2—21no—
472C2—21no—
473C2—481no—
474C2#6813612yes—
475C2#6813712yes—
476C2#6996211yescharacteristic
477C2#7003111yescharacteristic
478C2#7011611yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#324C2
socleC23×C3#120
prime radicalC2×#930C22
hypercenterC2×#324D3
hypocenterC3#9921
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. C2×#4986C2semidirect
4. C2×C3×#324C2semidirect
6. #69925C2semidirect
7. #67675C2semidirect
37. C2×#324D3semidirect
468. C3#9921semidirect

Maximal subgroups

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

subgroupindex
#821992
C2×#49862
C2×#47352
C2×C3×#3242
C2×#44702
#699252
#676752
#99213