Finite Group Databasebeta

← #10900921

Subgroups — #10900921

Order 768.

1292 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#888C211yesmaximal, characteristic, complemented
2C2×#4967C211yesmaximal, characteristic
3C2×#4736C211yesmaximal, characteristic
4#73999C211yesmaximal, characteristic
5C3×#1823C211yesmaximal, characteristic, complemented
6#69910C211yesmaximal, characteristic, complemented
7#69501C211yesmaximal, characteristic, complemented
8#68898C214yesmaximal, complemented
9#68826C214yesmaximal, complemented
10#29638—31nomaximal, Sylow
11C23×Dic6—21no—
12C22×C3×C4⋊C4C2211yescharacteristic
13C2×#930C2211yescharacteristic
14C22×C4⋊Dic3C2211yescharacteristic
15C22×Dic3⋊C4—21no—
16C22×C4×Dic3C2211yescharacteristic
17C2×#888—24no—
18C2×#888—24no—
19C2×#888—22no—
20C2×#888C2214yes—
21C2×#886—21no—
22C2×C3×C22⋊C8C2211yescharacteristic
23C3×#380C2211yescharacteristic
24C2×#830C2211yescharacteristic
25C2×#821C2212yes—
26C2×#818C2212yes—
27#4988C2212yes—
28#4984C2212yes—
29#4967—24no—
30C2×#809C2212yes—
31C2×#807C2212yes—
32#4958—21no—
33#4736—24no—
34#4670—22no—
35C3×#340C2214yes—
36#4550C2214yes—
37#4529C2214yes—
38C22×C4⋊Q8—31no—
39C2×#480—31no—
40C2×#480—31no—
41#2854—31no—
42#2057—34no—
43#2023—34no—
44#1823—31no—
45#1823D311yescharacteristic, complemented
46#1817—31no—
47C23×C3×C4—21no—
48C23×C3×C4C2311yescharacteristic
49C22×#67—21no—
50C23×Dic3—21no—
51C23×Dic3D411yescharacteristic
52C23×Dic3D411yescharacteristic
53C22×D12C2311yescharacteristic
54C22×Dic6—28no—
55C22×Dic6—24no—
56C22×Dic6—44no—
57C22×Dic6—44no—
58C22×C3×C8C2312yes—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4D414yes—
62C2×C3×C4⋊C4C2312yes—
63C2×C3×C4⋊C4C2312yes—
64C2×C3×C22⋊C4—21no—
65#930—22no—
66#930—22no—
67C2×(C2×D3)⋊C4—21no—
68C2×(C2×D3)⋊C4—22no—
69C2×(C2×D3)⋊C4—22no—
70C2×C4⋊Dic3—22no—
71C2×C4⋊Dic3—21no—
72C2×C4⋊Dic3—22no—
73C2×C4⋊Dic3D414yes—
74C2×C4⋊Dic3C2311yescharacteristic
75C2×C4⋊Dic3C2312yes—
76C2×Dic3⋊C4—28no—
77C2×Dic3⋊C4—24no—
78C2×C4×Dic3—22no—
79C2×C4×Dic3—22no—
80C2×C4×Dic3D412yes—
81C2×C4×Dic3C2312yes—
82C22×C3⋊C8C2312yes—
83#888—216no—
84#888—28no—
85#888—28no—
86#886—44no—
87C3×C22⋊C8—22no—
88C3×#95—21no—
89C3⋊#95—22no—
90#830—22no—
91#821—24no—
92#818—24no—
93#809—24no—
94#807—24no—
95C23×Q8—61no—
96C2×C4⋊Q8—34no—
97C2×C4⋊Q8—62no—
98C2×C4⋊Q8—64no—
99C2×C4⋊Q8—64no—
100C2×#123—61no—
101C2×#121—31no—
102C22×C4⋊C4—31no—
103C22×C4⋊C4—61no—
104C22×C4⋊C4C2×D311yescharacteristic
105C22×C42—31no—
106#480—64no—
107#480—64no—
108C2×#107—32no—
109C2×#107—32no—
110C2×#102—32no—
111C2×#102—32no—
112C2×C22⋊C8—31no—
113C2×C22⋊C8C2×D311yescharacteristic
114#390—32no—
115#389—32no—
116#388—62no—
117#386—61no—
118#380C2×D311yescharacteristic
119#340—34no—
120#340C2×D314yes—
121#336—34no—
122C24×C3C2×D411yescharacteristic
123C23×D3—21no—
124C22×C3×C4—24no—
125C22×C3×C4—24no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—42no—
129C22×C3×C4—42no—
130C22×C3×C4C2×D412yes—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D412yes—
133C22×C3×C4C2411yescharacteristic
134C2×#67—42no—
135C2×#67—42no—
136C2×#67—42no—
137C22×Dic3—22no—
138C22×Dic3—22no—
139C22×Dic3—22no—
140C22×Dic3—24no—
141C22×Dic3—24no—
142C22×Dic3—22no—
143C22×Dic3—21no—
144C22×Dic3—42no—
145C22×Dic3—42no—
146C22×Dic3D814yes—
147C22×Dic3C2×D412yes—
148C22×Dic3C2×D412yes—
149C2×D12—21no—
150C2×D12—21no—
151C2×D12—24no—
152C2×Dic6—216no—
153C2×Dic6—48no—
154C2×Dic6—432no—
155C2×Dic6—48no—
156C2×C3×C8—22no—
157C2×C3×C8—24no—
158C3×C4⋊C4—24no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—24no—
161C3×C4⋊C4C2×D414yes—
162C3×C22⋊C4—42no—
163(C2×D3)⋊C4—42no—
164(C2×D3)⋊C4—44no—
165(C2×D3)⋊C4—44no—
166C4⋊Dic3—24no—
167C4⋊Dic3—22no—
168C4⋊Dic3—21no—
169C4⋊Dic3—21no—
170C4⋊Dic3—44no—
171C4⋊Dic3C2×D414yes—
172Dic3⋊C4—216no—
173C4×Dic3—24no—
174C4×Dic3—22no—
175C4×Dic3—22no—
176C2×C3⋊C8—22no—
177C2×C3⋊C8—24no—
178C22×Q8—64no—
179C22×Q8—68no—
180C22×Q8—124no—
181C22×Q8—124no—
182C22×D4—31no—
183C22×D4—61no—
184C23×C4—21no—
185C23×C4—31no—
186C23×C4—31no—
187C23×C4—61no—
188C23×C4C22×D311yescharacteristic
189C22×C8—32no—
190C22×C8C22×D312yes—
191C4⋊Q8—68no—
192C4⋊Q8—616no—
193C4⋊Q8—68no—
194#123—124no—
195#121—62no—
196#121—62no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—34no—
200C2×C4⋊C4—32no—
201C2×C4⋊C4—31no—
202C2×C4⋊C4—64no—
203C2×C4⋊C4—68no—
204C2×C4⋊C4—62no—
205C2×C4⋊C4—61no—
206C2×C4⋊C4—62no—
207C2×C4⋊C4D1214yes—
208C2×C4⋊C4C22×D312yes—
209C2×C4⋊C4C22×D312yes—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—62no—
212C2×C22⋊C4—62no—
213C2×C22⋊C4—61no—
214C2×C42—32no—
215C2×C42—32no—
216C2×C42—62no—
217C2×C42—62no—
218#107—64no—
219#107—64no—
220#102—64no—
221#102—64no—
222C22⋊C8—22no—
223C22⋊C8—62no—
224#95—21no—
225#95—62no—
226C23×C3—22no—
227C23×C3—22no—
228C23×C3#12012yes—
229C23×C3C2×D812yes—
230C23×C3AGL(1,ℤ/8)12yes—
231C23×C3C22×D411yescharacteristic
232C22×D3—41no—
233C22×D3—42no—
234C22×D3—41no—
235C22×D3—41no—
236C22×D3—42no—
237C2×C3×C4—22no—
238C2×C3×C4—24no—
239C2×C3×C4—24no—
240C2×C3×C4—24no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—48no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4#12012yes—
248C2×C3×C4C22×D411yescharacteristic
249C2×C3×C4C22×D411yescharacteristic
250C2×C3×C42−1+414yes—
251#67—84no—
252C2×Dic3—28no—
253C2×Dic3—22no—
254C2×Dic3—22no—
255C2×Dic3—22no—
256C2×Dic3—22no—
257C2×Dic3—28no—
258C2×Dic3—24no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261C2×Dic3—48no—
262C2×Dic3—41no—
263C2×Dic3—41no—
264C2×Dic3—41no—
265C2×Dic3C2×D814yes—
266C2×Dic3#13714yes—
267D12—42no—
268D12—42no—
269Dic6—432no—
270C3×C8—42no—
271C3⋊C8—42no—
272C24—61no—
273C24D3×D411yescharacteristic
274C2×Q8—616no—
275C2×Q8—128no—
276C2×Q8—128no—
277C2×Q8—1232no—
278C2×D4—61no—
279C2×D4—61no—
280C2×D4—64no—
281C2×D4—122no—
282C2×D4—122no—
283C2×D4—122no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—34no—
289C22×C4—32no—
290C22×C4—32no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—64no—
294C22×C4—62no—
295C22×C4—62no—
296C22×C4—62no—
297C22×C4—64no—
298C22×C4—62no—
299C22×C4—61no—
300C22×C4—122no—
301C22×C4—122no—
302C22×C4C2×D1212yes—
303C22×C4D3×D411yescharacteristic
304C22×C4D3×D412yes—
305C22×C4C23×D311yescharacteristic
306C2×C8—22no—
307C2×C8—24no—
308C2×C8—62no—
309C2×C8—64no—
310C4⋊C4—24no—
311C4⋊C4—22no—
312C4⋊C4—24no—
313C4⋊C4—34no—
314C4⋊C4—616no—
315C4⋊C4—64no—
316C4⋊C4—62no—
317C4⋊C4—61no—
318C4⋊C4—61no—
319C4⋊C4—124no—
320C4⋊C4C2×D1214yes—
321C22⋊C4—42no—
322C22⋊C4—124no—
323C22⋊C4—124no—
324C22⋊C4—122no—
325C42—64no—
326C42—62no—
327C42—62no—
328C22×C3—22no—
329C22×C3—24no—
330C22×C3—24no—
331C22×C3—22no—
332C22×C3#44714yes—
333C22×C3C2×#12011yescharacteristic
334C22×C3#53612yes—
335C22×C3C22×D811yescharacteristic
336C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
337C22×C3C2×#13712yes—
338C2×D3—81no—
339C2×D3—82no—
340C2×D3—82no—
341C2×D3—81no—
342C2×D3—81no—
343C3×C4—22no—
344C3×C4—44no—
345C3×C4—82no—
346C3×C4#53612yes—
347C3×C4#53612yes—
348Dic3—24no—
349Dic3—24no—
350Dic3—44no—
351Dic3—81no—
352C23—22no—
353C23—22no—
354C23—122no—
355C23—121no—
356C23—121no—
357C23—121no—
358C23—122no—
359C23#88212yes—
360C23D3×D812yes—
361C23#91412yes—
362C23C2×D3×D411yescharacteristic
363Q8—1232no—
364D4—122no—
365D4—122no—
366D4—244no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—34no—
372C2×C4—34no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—48no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—68no—
380C2×C4—62no—
381C2×C4—62no—
382C2×C4—62no—
383C2×C4—62no—
384C2×C4—68no—
385C2×C4—64no—
386C2×C4—122no—
387C2×C4—122no—
388C2×C4—128no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4—121no—
392C2×C4#88212yes—
393C2×C4C22×D1211yescharacteristic
394C2×C4C2×D3×D411yescharacteristic
395C2×C4#101014yes—
396C8—42no—
397C8—122no—
398C2×C3—24no—
399C2×C3#261411yescharacteristic
400C2×C3C2×#44711yescharacteristic
401C2×C3#300411yescharacteristic
402C2×C3#322612yes—
403C2×C3#323812yes—
404D3—161no—
405C22—24no—
406C22—22no—
407C22—24no—
408C22—22no—
409C22—242no—
410C22—241no—
411C22—242no—
412C22—241no—
413C22—241no—
414C22#477314yes—
415C22C2×#88211yescharacteristic
416C22#564812yes—
417C22C2×D3×D811yescharacteristic
418C22C2×#91411yescharacteristic
419C22D3×#13712yes—
420C4—22no—
421C4—44no—
422C4—64no—
423C4—64no—
424C4—82no—
425C4—124no—
426C4—241no—
427C4#552612yes—
428C4#564812yes—
429C3#2963811yescharacteristic, complemented, Sylow
430C2—24no—
431C2—481no—
432C2#7413511yescharacteristic
433C2C2×#477311yescharacteristic
434C2#7895811yescharacteristic
435C2#8127412yes—
436C2#8142112yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1823C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1823D3
hypocenterC3#29638
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
1. C22×#888C2semidirect
5. C3×#1823C2semidirect
6. #69910C2semidirect
7. #69501C2semidirect
8. #68898C2semidirect
9. #68826C2semidirect
45. #1823D3semidirect
429. C3#29638semidirect

Maximal subgroups

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

subgroupindex
C22×#8882
C2×#49672
C2×#47362
#739992
C3×#18232
#699102
#695012
#688982
#688262
#296383