Finite Group Databasebeta

← #10690546

Subgroups — #10690546

Order 768.

670 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#924C211yesmaximal, characteristic
2C2×C4×#208C211yesmaximal, characteristic
3C3×#1797C211yesmaximal, characteristic
4C2×#4484C212yesmaximal, complemented
5#67704C212yesmaximal, complemented
6#9077—31nomaximal, Sylow
7C22×C3×C42C2211yescharacteristic
8C2×#924C411yescharacteristic
9C22×#208—21no—
10C2×C4×Dic6C2211yescharacteristic
11C2×C3×C4⋊C8C2212yes—
12C2×C3×C22⋊C8—21no—
13#5212C411yescharacteristic
14#5210—21no—
15C4×#217—21no—
16C4×C4⋊Dic3C2211yescharacteristic
17#4933—21no—
18C4×Dic3⋊C4—21no—
19#4929C411yescharacteristic
20#4928C411yescharacteristic
21C4×#208—24no—
22C2×#803—21no—
23C2×C4×C3⋊C8C2212yes—
24C3×#336—21no—
25#4484—24no—
26C2×C4×M16—31no—
27C4×C22⋊Q8—31no—
28#1797D311yescharacteristic
29C2×Q8⋊C8—32no—
30#1279—32no—
31C23×C3×C4C2×C411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C23×C3×C4D411yescharacteristic
34C22×Dic6C2×C411yescharacteristic
35C22×C3×C8—22no—
36C2×C3×C42—22no—
37C2×C3×C42—22no—
38C2×C3×C42C2×C411yescharacteristic
39C2×C3×C42D411yescharacteristic
40C2×C3×C42D411yescharacteristic
41C2×C3×C42C2311yescharacteristic
42C2×#217—21no—
43C2×#217—21no—
44C2×C4⋊Dic3C2×C411yescharacteristic
45C2×C4⋊Dic3C2×C411yescharacteristic
46C2×C4⋊Dic3C2×C411yescharacteristic
47#924—22no—
48#924—22no—
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—21no—
52C2×C4×Dic3—21no—
53C2×C4×Dic3—21no—
54C2×#208—24no—
55C2×#208—24no—
56C2×#208—24no—
57C22×C3⋊C8—22no—
58C4×Dic6—22no—
59C4×Dic6—22no—
60C3×C4⋊C8—22no—
61C3×C4⋊C8—22no—
62C3×C22⋊C8—42no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68#803—24no—
69C4×C3⋊C8—22no—
70C4×C3⋊C8D414yes—
71C22×M16—61no—
72C2×C22⋊Q8—31no—
73C2×C4×Q8—31no—
74C22×C42C2×D311yescharacteristic
75C2×C4⋊C8C2×D312yes—
76C2×C22⋊C8—21no—
77C4×M16—64no—
78C2×#97—61no—
79C2×C4×C8—32no—
80#391—31no—
81#389—31no—
82#387—61no—
83#385—61no—
84#380—31no—
85C4×C4⋊C4—31no—
86C4×C4⋊C4—61no—
87C4×C22⋊C4—61no—
88#336—21no—
89Q8⋊C8—64no—
90C24×C3C22⋊C411yescharacteristic
91C22×C3×C4—22no—
92C22×C3×C4—22no—
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4—22no—
96C22×C3×C4—22no—
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4—22no—
100C22×C3×C4—21no—
101C22×C3×C4—21no—
102C22×C3×C4—22no—
103C22×C3×C4C22⋊C411yescharacteristic
104C22×C3×C4C22⋊C411yescharacteristic
105C22×C3×C4C22⋊C411yescharacteristic
106C22×C3×C4C22×C411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×Dic3—21no—
110C22×Dic3—21no—
111C22×Dic3—21no—
112C22×Dic3—21no—
113C2×Dic6—22no—
114C2×Dic6—22no—
115C2×Dic6—21no—
116C2×Dic6M1612yescomplemented
117C2×C3×C8—42no—
118C2×C3×C8—42no—
119C2×C3×C8—42no—
120C3×C42—22no—
121C3×C42—21no—
122C3×C42—22no—
123C3×C42—21no—
124C3×C42C2×D412yes—
125C3×C42#4112yes—
126#217—24no—
127#217—42no—
128C4⋊Dic3—21no—
129C4⋊Dic3—22no—
130C4⋊Dic3—22no—
131C4⋊Dic3M1612yescomplemented
132Dic3⋊C4—24no—
133Dic3⋊C4—42no—
134Dic3⋊C4—41no—
135Dic3⋊C4—41no—
136C4×Dic3—42no—
137C4×Dic3—41no—
138C4×Dic3—41no—
139#208—216no—
140C2×C3⋊C8—24no—
141C2×C3⋊C8—24no—
142C2×C3⋊C8—24no—
143C22×Q8—31no—
144C23×C4C4×D311yescharacteristic
145C23×C4D1211yescharacteristic
146C23×C4#6711yescharacteristic
147C2×M16—64no—
148C2×M16—64no—
149C2×M16—64no—
150C22×C8—22no—
151C22×C8—62no—
152C22⋊Q8—62no—
153C22⋊Q8—62no—
154C4×Q8—62no—
155C4×Q8—62no—
156C2×C4⋊C4—31no—
157C2×C4⋊C4—31no—
158C2×C4⋊C4—31no—
159C2×C4⋊C4—61no—
160C2×C4⋊C4—61no—
161C2×C4⋊C4—61no—
162C2×C22⋊C4—61no—
163C2×C22⋊C4—61no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—61no—
167C2×C42—61no—
168C2×C42C4×D311yescharacteristic
169C2×C42D1211yescharacteristic
170C2×C42#6711yescharacteristic
171C2×C42C22×D311yescharacteristic
172C4⋊C8—22no—
173C4⋊C8—22no—
174C22⋊C8—42no—
175#97—64no—
176C4×C8—34no—
177C4×C8—62no—
178#95—61no—
179#95—61no—
180#95—61no—
181#95—61no—
182#95—61no—
183C23×C3—21no—
184C23×C3—21no—
185C23×C3—21no—
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—21no—
189C23×C3#10412yes—
190C23×C3C2×C22⋊C411yescharacteristic
191C2×C3×C4—24no—
192C2×C3×C4—24no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—42no—
204C2×C3×C4—42no—
205C2×C3×C4—42no—
206C2×C3×C4—42no—
207C2×C3×C4#10412yes—
208C2×C3×C4C4×D412yes—
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12112yes—
211C2×C3×C4#12311yescharacteristic
212C2×C3×C4C2×M1611yescharacteristic
213C2×C3×C4C2×M1611yescharacteristic
214C2×C3×C4#13112yes—
215C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
216C2×C3×C4#13711yescharacteristic
217C2×C3×C4#13712yes—
218C2×Dic3—22no—
219C2×Dic3—22no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230Dic6—24no—
231Dic6—42no—
232C3×C8—82no—
233C3⋊C8—28no—
234C24(C2×D3)⋊C411yescharacteristic
235C2×Q8—32no—
236C2×Q8—61no—
237C2×Q8—62no—
238C2×Q8—62no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—61no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4—61no—
255C22×C4(C2×D3)⋊C411yescharacteristic
256C22×C4(C2×D3)⋊C411yescharacteristic
257C22×C4(C2×D3)⋊C411yescharacteristic
258C22×C4C2×C4×D311yescharacteristic
259C22×C4C2×D1211yescharacteristic
260C22×C4C2×#6711yescharacteristic
261M16—616no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—64no—
266C2×C8—64no—
267C2×C8—64no—
268C4⋊C4—32no—
269C4⋊C4—64no—
270C4⋊C4—61no—
271C4⋊C4—62no—
272C4⋊C4—62no—
273C4⋊C4—122no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C22⋊C4—64no—
277C22⋊C4—122no—
278C42—22no—
279C42—21no—
280C42—21no—
281C42—22no—
282C42—121no—
283C42—122no—
284C42—121no—
285C42D3×D412yes—
286C42#23712yes—
287C22×C3—22no—
288C22×C3—22no—
289C22×C3—22no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3#38611yescharacteristic
295C22×C3#40711yescharacteristic
296C22×C3#40811yescharacteristic
297C22×C3#41711yescharacteristic
298C22×C3#41911yescharacteristic
299C22×C3C2×#10411yescharacteristic
300C22×C3C2×#10411yescharacteristic
301C3×C4—21no—
302C3×C4—22no—
303C3×C4—21no—
304C3×C4—41no—
305C3×C4—41no—
306C3×C4#43512yes—
307C3×C4#43612yes—
308C3×C4#46812yes—
309C3×C4#47012yes—
310Dic3—42no—
311Dic3—42no—
312Dic3—81no—
313Dic3—81no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23#82512yes—
321C23C2×(C2×D3)⋊C411yescharacteristic
322Q8—64no—
323Q8—122no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—62no—
341C2×C4—62no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4—121no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4#82512yes—
353C2×C4#67⋊C412yes—
354C2×C4#88211yescharacteristic
355C2×C4#88611yescharacteristic
356C2×C4#90112yes—
357C2×C4D3×M1611yescharacteristic
358C2×C4D3×M1611yescharacteristic
359C2×C4#93211yescharacteristic
360C2×C4#93912yes—
361C2×C4#94211yescharacteristic
362C2×C4#95112yes—
363C8—68no—
364C8—82no—
365C2×C3—42no—
366C2×C3#148311yescharacteristic
367C2×C3#148511yescharacteristic
368C2×C3#148911yescharacteristic
369C2×C3#149111yescharacteristic
370C2×C3#187011yescharacteristic
371C2×C3#189411yescharacteristic
372C2×C3#189611yescharacteristic
373C22—22no—
374C22—22no—
375C22—22no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22#495811yescharacteristic
381C22#496611yescharacteristic
382C22#500411yescharacteristic
383C22#511011yescharacteristic
384C22#512611yescharacteristic
385C22C2×#82511yescharacteristic
386C22C2×#82511yescharacteristic
387C4—21no—
388C4—22no—
389C4—21no—
390C4—41no—
391C4—41no—
392C4—122no—
393C4—122no—
394C4—241no—
395C4—241no—
396C4#472812yes—
397C4#472912yes—
398C4#504212yes—
399C4#505212yes—
400C3#907711yescharacteristic, complemented, Sylow
401C2—42no—
402C2#6951611yescharacteristic
403C2#6995311yescharacteristic
404C2#6996511yescharacteristic
405C2#7042711yescharacteristic
406C2#7049711yescharacteristic
407C2#7076911yescharacteristic
408C2#7080211yescharacteristic
subgroupG/H
centerC22×C4C2×#67
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1797C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1797D3
hypocenterC3#9077
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
4. C2×#4484C2semidirect
5. #67704C2semidirect
116. C2×Dic6M16semidirect
131. C4⋊Dic3M16semidirect
400. C3#9077semidirect

Maximal subgroups

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

subgroupindex
C4×#9242
C2×C4×#2082
C3×#17972
C2×#44842
#677042
#90773