Finite Group Databasebeta

← #10900937

Subgroups — #10900937

Order 768.

1076 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#870C211yesmaximal, characteristic, complemented
2#82080C211yesmaximal, characteristic
3C2×#4735C211yesmaximal, characteristic
4C2×#4736C211yesmaximal, characteristic
5C3×#1817C211yesmaximal, characteristic, complemented
6#69925C211yesmaximal, characteristic, complemented
7#69924C211yesmaximal, characteristic, complemented
8#68679C214yesmaximal, complemented
9#68634C214yesmaximal, complemented
10#29640—31nomaximal, Sylow
11C22×C3×C42C2211yescharacteristic
12C2×#930C2211yescharacteristic
13C2×#929—21no—
14C22×C4⋊Dic3C2211yescharacteristic
15C22×C4⋊Dic3C2211yescharacteristic
16C22×Dic3⋊C4—21no—
17C22×Dic3⋊C4—21no—
18C2×#870—22no—
19C2×#870—24no—
20C2×#870—24no—
21C2×#870C2214yes—
22C2×C3×C22⋊C8C2211yescharacteristic
23C2×C3×C22⋊C8C2211yescharacteristic
24#5213—21no—
25#5212C2211yescharacteristic
26C2×#821C2212yes—
27C2×#821C2212yes—
28C2×#818C2212yes—
29C2×#817C2212yes—
30#4940—22no—
31#4939C2212yes—
32#4928C2212yes—
33#4736—24no—
34#4735—24no—
35C3×#336C2214yes—
36#4551C2214yes—
37#4551C2214yes—
38C22×#125—31no—
39C2×#481—31no—
40C2×#480—31no—
41#2854—31no—
42#2059—34no—
43#2025—34no—
44#1824—31no—
45#1823—31no—
46#1817D311yescharacteristic, complemented
47C23×C3×C4D411yescharacteristic
48C23×C3×C4D411yescharacteristic
49C23×C3×C4C2311yescharacteristic
50C22×#67—21no—
51C23×Dic3—21no—
52C23×Dic3—21no—
53C22×D12C2311yescharacteristic
54C22×C3×C8C2312yes—
55C22×C3×C8C2312yes—
56C2×C3×C42—22no—
57C2×C3×C42—22no—
58C2×C3×C42D412yes—
59C2×C3×C42C2312yes—
60C2×#217—21no—
61#930—22no—
62#930—22no—
63#929—44no—
64C2×(C2×D3)⋊C4—22no—
65C2×(C2×D3)⋊C4—22no—
66C2×(C2×D3)⋊C4—21no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3—22no—
70C2×C4⋊Dic3—22no—
71C2×C4⋊Dic3D414yes—
72C2×C4⋊Dic3D414yes—
73C2×C4⋊Dic3C2311yescharacteristic
74C2×C4⋊Dic3C2312yes—
75C2×C4⋊Dic3C2312yes—
76C2×C4⋊Dic3C2312yes—
77C2×Dic3⋊C4—28no—
78C2×Dic3⋊C4—28no—
79C2×Dic3⋊C4—24no—
80C2×Dic3⋊C4—24no—
81C2×Dic3⋊C4—22no—
82#870—216no—
83#870—28no—
84#870—28no—
85C3×C22⋊C8—22no—
86C3×C22⋊C8—22no—
87C3⋊#95—22no—
88C3⋊#95—21no—
89#821—24no—
90#821—24no—
91#818—24no—
92#817—24no—
93C2×#125—34no—
94C2×#125—64no—
95C2×#125—64no—
96C2×#125—62no—
97C2×#123—61no—
98C2×#121—31no—
99C22×C4⋊C4—31no—
100C22×C4⋊C4—31no—
101C22×C4⋊C4—61no—
102C22×C4⋊C4—61no—
103C22×C42C2×D311yescharacteristic
104#481—64no—
105#480—64no—
106C2×#107—32no—
107C2×#106—32no—
108C2×#102—32no—
109C2×#102—32no—
110C2×C22⋊C8C2×D311yescharacteristic
111C2×C22⋊C8C2×D311yescharacteristic
112#390—32no—
113#389—32no—
114#388—62no—
115#386—61no—
116#380—31no—
117#340—34no—
118#340—34no—
119#336C2×D314yes—
120C24×C3C2×D411yescharacteristic
121C23×D3—21no—
122C22×C3×C4—24no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—42no—
128C22×C3×C4C2×D412yes—
129C22×C3×C4C2×D412yes—
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D412yes—
132C22×C3×C4C2×D412yes—
133C22×C3×C4C2411yescharacteristic
134C2×#67—42no—
135C2×#67—42no—
136C2×#67—42no—
137C22×Dic3—24no—
138C22×Dic3—24no—
139C22×Dic3—22no—
140C22×Dic3—22no—
141C22×Dic3—21no—
142C22×Dic3—22no—
143C22×Dic3—44no—
144C22×Dic3—42no—
145C22×Dic3—42no—
146C2×D12—21no—
147C2×D12—21no—
148C2×D12—24no—
149C2×C3×C8—22no—
150C2×C3×C8—22no—
151C2×C3×C8—24no—
152C2×C3×C8—24no—
153C3×C42—24no—
154C3×C42—22no—
155C3×C42—22no—
156#217—42no—
157(C2×D3)⋊C4—44no—
158(C2×D3)⋊C4—44no—
159(C2×D3)⋊C4—42no—
160C4⋊Dic3—24no—
161C4⋊Dic3—24no—
162C4⋊Dic3—22no—
163C4⋊Dic3—22no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166C4⋊Dic3—24no—
167C4⋊Dic3C2×D414yes—
168C4⋊Dic3C2×D414yes—
169Dic3⋊C4—216no—
170Dic3⋊C4—216no—
171Dic3⋊C4—44no—
172C22×D4—31no—
173C22×D4—61no—
174C23×C4—61no—
175C23×C4—61no—
176C23×C4D1211yescharacteristic
177C23×C4D1211yescharacteristic
178C23×C4C22×D311yescharacteristic
179C22×C8C22×D312yes—
180C22×C8C22×D312yes—
181#125—68no—
182#125—68no—
183#125—616no—
184#123—124no—
185#121—62no—
186#121—62no—
187C2×C4⋊C4—34no—
188C2×C4⋊C4—34no—
189C2×C4⋊C4—32no—
190C2×C4⋊C4—32no—
191C2×C4⋊C4—31no—
192C2×C4⋊C4—32no—
193C2×C4⋊C4—68no—
194C2×C4⋊C4—68no—
195C2×C4⋊C4—61no—
196C2×C4⋊C4—61no—
197C2×C4⋊C4—64no—
198C2×C4⋊C4—64no—
199C2×C4⋊C4—62no—
200C2×C4⋊C4—62no—
201C2×C4⋊C4—62no—
202C2×C22⋊C4—61no—
203C2×C22⋊C4—61no—
204C2×C22⋊C4—62no—
205C2×C22⋊C4—62no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42D1212yes—
209C2×C42C22×D312yes—
210#107—64no—
211#106—64no—
212#102—64no—
213#102—64no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216#95—61no—
217#95—62no—
218C23×C3—22no—
219C23×C3—22no—
220C23×C3#12012yes—
221C23×C3#13512yes—
222C23×C3AGL(1,ℤ/8)12yes—
223C23×C3C22×D411yescharacteristic
224C22×D3—41no—
225C22×D3—42no—
226C22×D3—41no—
227C22×D3—41no—
228C22×D3—42no—
229C2×C3×C4—22no—
230C2×C3×C4—24no—
231C2×C3×C4—24no—
232C2×C3×C4—22no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—44no—
239C2×C3×C4—44no—
240C2×C3×C4#12012yes—
241C2×C3×C4C22×D411yescharacteristic
242C2×C3×C4C22×D411yescharacteristic
243C2×C3×C42−1+414yes—
244#67—84no—
245C2×Dic3—24no—
246C2×Dic3—24no—
247C2×Dic3—48no—
248C2×Dic3—42no—
249C2×Dic3—42no—
250C2×Dic3—44no—
251C2×Dic3—48no—
252C2×Dic3—41no—
253C2×Dic3—42no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256C2×Dic3—42no—
257C2×Dic3—42no—
258D12—42no—
259D12—42no—
260C3×C8—42no—
261C3×C8—42no—
262C24—61no—
263C24C2×D1211yescharacteristic
264C2×D4—61no—
265C2×D4—61no—
266C2×D4—64no—
267C2×D4—122no—
268C2×D4—122no—
269C2×D4—122no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—42no—
276C22×C4—64no—
277C22×C4—64no—
278C22×C4—62no—
279C22×C4—62no—
280C22×C4—62no—
281C22×C4—61no—
282C22×C4—124no—
283C22×C4—122no—
284C22×C4—122no—
285C22×C4C2×D1212yes—
286C22×C4C2×D1212yes—
287C22×C4C2×D1211yescharacteristic
288C22×C4D3×D412yes—
289C22×C4D3×D412yes—
290C22×C4C23×D311yescharacteristic
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—24no—
294C2×C8—24no—
295C4⋊C4—34no—
296C4⋊C4—34no—
297C4⋊C4—616no—
298C4⋊C4—616no—
299C4⋊C4—64no—
300C4⋊C4—64no—
301C4⋊C4—62no—
302C4⋊C4—62no—
303C4⋊C4—61no—
304C4⋊C4—61no—
305C4⋊C4—64no—
306C4⋊C4—124no—
307C22⋊C4—124no—
308C22⋊C4—124no—
309C22⋊C4—122no—
310C22⋊C4—122no—
311C42—24no—
312C42—22no—
313C42—22no—
314C22×C3—22no—
315C22×C3—24no—
316C22×C3—24no—
317C22×C3—22no—
318C22×C3#44914yes—
319C22×C3C2×#12011yescharacteristic
320C22×C3#53612yes—
321C22×C3C2×#13511yescharacteristic
322C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
323C22×C3#57512yes—
324C2×D3—81no—
325C2×D3—82no—
326C2×D3—82no—
327C2×D3—81no—
328C2×D3—81no—
329C3×C4—22no—
330C3×C4—42no—
331C3×C4—42no—
332C3×C4#53612yes—
333C3×C4#53612yes—
334Dic3—44no—
335Dic3—44no—
336Dic3—82no—
337Dic3—81no—
338C23—22no—
339C23—22no—
340C23—121no—
341C23—121no—
342C23—121no—
343C23—122no—
344C23—122no—
345C23#88212yes—
346C23#90412yes—
347C23#90812yes—
348C23C22×D1211yescharacteristic
349D4—122no—
350D4—122no—
351D4—244no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—64no—
364C2×C4—64no—
365C2×C4—128no—
366C2×C4—122no—
367C2×C4—122no—
368C2×C4—124no—
369C2×C4—128no—
370C2×C4—121no—
371C2×C4—121no—
372C2×C4—122no—
373C2×C4—121no—
374C2×C4—122no—
375C2×C4—122no—
376C2×C4#88212yes—
377C2×C4C2×D3×D411yescharacteristic
378C2×C4C2×D3×D411yescharacteristic
379C2×C4#101014yes—
380C8—42no—
381C8—42no—
382C2×C3—24no—
383C2×C3#261411yescharacteristic
384C2×C3C2×#44911yescharacteristic
385C2×C3#300511yescharacteristic
386C2×C3#323212yes—
387C2×C3#323412yes—
388D3—161no—
389C22—22no—
390C22—22no—
391C22—24no—
392C22—24no—
393C22—241no—
394C22—241no—
395C22—241no—
396C22—242no—
397C22—242no—
398C22#473414yes—
399C22C2×#88211yescharacteristic
400C22#552612yes—
401C22C2×#90411yescharacteristic
402C22C2×#90811yescharacteristic
403C22#574812yes—
404C4—22no—
405C4—42no—
406C4—42no—
407C4—124no—
408C4—124no—
409C4—241no—
410C4—242no—
411C4#564812yes—
412C4#564812yes—
413C3#2964011yescharacteristic, complemented, Sylow
414C2—24no—
415C2—481no—
416C2#7413511yescharacteristic
417C2C2×#473411yescharacteristic
418C2#7863811yescharacteristic
419C2#7927412yes—
420C2#7931812yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1817C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1817D3
hypocenterC3#29640
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
1. C22×#870C2semidirect
5. C3×#1817C2semidirect
6. #69925C2semidirect
7. #69924C2semidirect
8. #68679C2semidirect
9. #68634C2semidirect
46. #1817D3semidirect
413. C3#29640semidirect

Maximal subgroups

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

subgroupindex
C22×#8702
#820802
C2×#47352
C2×#47362
C3×#18172
#699252
#699242
#686792
#686342
#296403