Finite Group Databasebeta

← #10697698

Subgroups — #10697698

Order 768.

678 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#459C211yesmaximal, characteristic
2C2×#5070C211yesmaximal, characteristic, complemented
3C2×#5019C211yesmaximal, characteristic, complemented
4C2×#4544C211yesmaximal, characteristic, complemented
5#71614C211yesmaximal, characteristic, complemented
6C2×#4482C211yesmaximal, characteristic, complemented
7#67703C211yesmaximal, characteristic, complemented
8#10261—31nomaximal, Sylow
9C22×C3×D8—21no—
10C2×C3×#127C2211yescharacteristic
11C2×C3×C4×D4C2211yescharacteristic
12C22×#214—21no—
13C2×#869C2211yescharacteristic
14C3×#459—22no—
15C3×#459—22no—
16C2×C3×C4⋊C8C2211yescharacteristic
17C2×C3×#102—21no—
18C2×#832—22no—
19C2×#830—22no—
20#5070—22no—
21#5070—22no—
22#5019—22no—
23#5019—22no—
24C2×#810—21no—
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26C2×C4⋊(C3⋊C8)C2211yescharacteristic
27C2×C4×C3⋊C8C2211yescharacteristic
28#4544—22no—
29#4482—22no—
30C2×#486—31no—
31C2×#461—31no—
32C2×#459D311yescharacteristic
33#1954—31no—
34C2×#332—31no—
35C2×#331—31no—
36#1277—31no—
37C22×C3×D4—21no—
38C22×C3×D4—21no—
39C22×C3×D4—21no—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4—21no—
42C22×Dic6—21no—
43C2×C3×D8—24no—
44C2×C3×D8—42no—
45C2×C3×D8—42no—
46C22×C3×C8—21no—
47C3×#127—21no—
48C3×#127—21no—
49C3×#127—21no—
50C3×#127D412yes—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C22⋊C4—21no—
57C2×C3×C42C2311yescharacteristic
58C2×#214—24no—
59C2×#214—42no—
60C2×#214—42no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—21no—
63C22×C3⋊C8—21no—
64C22×C3⋊C8—21no—
65C22×C3⋊C8D412yes—
66#869—21no—
67#869—21no—
68#869—21no—
69#869D412yes—
70C3×C4⋊C8—21no—
71C3×C4⋊C8D412yes—
72C3×#102—24no—
73#832—28no—
74#830—44no—
75#810—24no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)D412yes—
80C4×C3⋊C8—21no—
81C4×C3⋊C8—21no—
82C22×SD16—61no—
83C22×D8—21no—
84C2×C4⋊Q8—31no—
85C2×#127C2×D311yescharacteristic
86C2×C4×D4C2×D311yescharacteristic
87#486—62no—
88#486—62no—
89#461—62no—
90#461—62no—
91#459—22no—
92#459—22no—
93C2×C4⋊C8—31no—
94C2×C4⋊C8—31no—
95C2×C4⋊C8C2×D311yescharacteristic
96C2×#103—61no—
97C2×#102—21no—
98C2×#102—62no—
99C2×C22⋊C8—62no—
100C2×C4×C8—31no—
101#332—62no—
102#331—62no—
103C24×C3—21no—
104C24×C3—41no—
105C2×C3×D4—21no—
106C2×C3×D4—21no—
107C2×C3×D4—22no—
108C2×C3×D4—22no—
109C2×C3×D4—22no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—42no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—42no—
121C2×C3×D4—41no—
122C2×C3×D4—82no—
123C2×C3×D4D812yes—
124C22×C3×C4—21no—
125C22×C3×C4—41no—
126C22×C3×C4—41no—
127C22×C3×C4—42no—
128C22×C3×C4—42no—
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4#4111yescharacteristic
132C22×Dic3—41no—
133C2×Dic6—22no—
134C2×Dic6—41no—
135C2×Dic6—41no—
136C2×Dic6—41no—
137C2×Dic6—41no—
138C2×Dic6—41no—
139C3×D8—48no—
140C2×C3×C8—22no—
141C2×C3×C8—41no—
142C2×C3×C8—41no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C22⋊C4—41no—
146C3×C22⋊C4—41no—
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151#214—48no—
152C4⋊Dic3—22no—
153C4⋊Dic3—22no—
154C4⋊Dic3—41no—
155C4⋊Dic3—41no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—41no—
161C2×C3⋊C8—41no—
162C2×C3⋊C8—41no—
163C2×C3⋊C8—42no—
164C22×Q8—61no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4#6711yescharacteristic
169C23×C4—21no—
170C2×SD16—64no—
171C2×SD16—122no—
172C2×SD16—122no—
173C2×D8—24no—
174C2×D8—42no—
175C2×D8—42no—
176C22×C8—21no—
177C22×C8—32no—
178C22×C8—61no—
179C22×C8—61no—
180C4⋊Q8—32no—
181C4⋊Q8—61no—
182C4⋊Q8—61no—
183C4⋊Q8—61no—
184#127—21no—
185#127—21no—
186#127—21no—
187#127#6712yes—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—21no—
192C2×C4⋊C4—61no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4#6711yescharacteristic
195C2×C22⋊C4—21no—
196C2×C42C22×D311yescharacteristic
197C4⋊C8—21no—
198C4⋊C8—32no—
199C4⋊C8—61no—
200C4⋊C8—61no—
201C4⋊C8—61no—
202C4⋊C8#6712yes—
203#103—64no—
204#102—24no—
205#102—68no—
206C22⋊C8—124no—
207C4×C8—61no—
208C4×C8—61no—
209C23×C3—22no—
210C23×C3—41no—
211C23×C3—42no—
212C23×C3—41no—
213C23×C3—42no—
214C23×C3—42no—
215C23×C3—82no—
216C23×C3—81no—
217C23×C3—81no—
218C23×C3—82no—
219C23×C3#12111yescharacteristic
220C3×D4—22no—
221C3×D4—22no—
222C3×D4—42no—
223C3×D4—41no—
224C3×D4—41no—
225C3×D4—44no—
226C3×D4—42no—
227C3×D4—82no—
228C3×D4—82no—
229C3×D4—81no—
230C3×D4—81no—
231C3×D4—81no—
232C3×D4—81no—
233C2×C3×C4—22no—
234C2×C3×C4—41no—
235C2×C3×C4—42no—
236C2×C3×C4—82no—
237C2×C3×C4—81no—
238C2×C3×C4—81no—
239C2×C3×C4#12011yescharacteristic
240C2×C3×C4#12011yescharacteristic
241C2×C3×C4#12011yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12111yescharacteristic
244C2×C3×C4#12711yescharacteristic
245C2×C3×C4C2×D811yescharacteristic
246C2×C3×C4#13512yes—
247C2×C3×C4#13511yescharacteristic
248C2×C3×C4AGL(1,ℤ/8)12yes—
249C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
250C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
251C2×Dic3—42no—
252C2×Dic3—81no—
253C2×Dic3—81no—
254Dic6—42no—
255Dic6—42no—
256Dic6—81no—
257Dic6—81no—
258C3×C8—42no—
259C3⋊C8—42no—
260C3⋊C8—42no—
261C3⋊C8—81no—
262C24—21no—
263C24—41no—
264C2×Q8—62no—
265C2×Q8—121no—
266C2×Q8—121no—
267C2×Q8—121no—
268C2×Q8—121no—
269C2×Q8—121no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—22no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—82no—
288C2×D4#21312yes—
289C22×C4—21no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—121no—
295C22×C4D3×D411yescharacteristic
296C22×C4#23711yescharacteristic
297C22×C4C2×#6711yescharacteristic
298SD16—128no—
299D8—48no—
300C2×C8—22no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—62no—
304C2×C8—62no—
305C2×C8—62no—
306C2×C8—62no—
307C2×C8—121no—
308C2×C8—121no—
309C2×C8—122no—
310C2×C8—121no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—62no—
314C4⋊C4—62no—
315C4⋊C4—121no—
316C4⋊C4—121no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C42D3×D411yescharacteristic
320C42D3×D411yescharacteristic
321C42C2×#6711yescharacteristic
322C42C2×#6711yescharacteristic
323C22×C3—44no—
324C22×C3—42no—
325C22×C3—82no—
326C22×C3—81no—
327C22×C3—81no—
328C22×C3—82no—
329C22×C3—82no—
330C22×C3—82no—
331C22×C3—162no—
332C22×C3—162no—
333C22×C3#39211yescharacteristic
334C22×C3#46312yes—
335C22×C3#46611yescharacteristic
336C22×C3#46911yescharacteristic
337C22×C3#47112yes—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—81no—
341C3×C4#44711yescharacteristic
342C3×C4#44711yescharacteristic
343C3×C4#44912yes—
344C3×C4#44911yescharacteristic
345C3×C4#44911yescharacteristic
346C3×C4#49511yescharacteristic
347C3×C4#49611yescharacteristic
348Dic3—82no—
349C23—22no—
350C23—42no—
351C23—42no—
352C23—41no—
353C23—41no—
354C23—42no—
355C23—82no—
356C23—82no—
357C23—81no—
358C23—81no—
359C23#93911yescharacteristic
360Q8—122no—
361Q8—122no—
362Q8—241no—
363Q8—241no—
364D4—22no—
365D4—22no—
366D4—44no—
367D4—42no—
368D4—41no—
369D4—42no—
370D4—41no—
371D4—82no—
372D4—82no—
373D4—81no—
374D4—81no—
375D4—81no—
376D4—81no—
377C2×C4—22no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—82no—
383C2×C4—122no—
384C2×C4—241no—
385C2×C4—241no—
386C2×C4#91112yes—
387C2×C4#91212yes—
388C2×C4C2×#21311yescharacteristic
389C2×C4#93211yescharacteristic
390C2×C4#93711yescharacteristic
391C2×C4#93711yescharacteristic
392C2×C4#93811yescharacteristic
393C2×C4#93911yescharacteristic
394C2×C4#94011yescharacteristic
395C2×C4#94911yescharacteristic
396C2×C4#95011yescharacteristic
397C2×C4#95311yescharacteristic
398C8—42no—
399C8—122no—
400C8—122no—
401C8—241no—
402C2×C3—82no—
403C2×C3—162no—
404C2×C3#168112yes—
405C2×C3#170112yes—
406C2×C3#199911yescharacteristic
407C2×C3#201211yescharacteristic
408C2×C3#201911yescharacteristic
409C22—42no—
410C22—44no—
411C22—82no—
412C22—81no—
413C22—81no—
414C22—82no—
415C22—82no—
416C22—82no—
417C22—162no—
418C22—162no—
419C22#503811yescharacteristic
420C22#503911yescharacteristic
421C22#515312yes—
422C22#520012yes—
423C22#522311yescharacteristic
424C4—21no—
425C4—21no—
426C4—81no—
427C4—242no—
428C4#503611yescharacteristic
429C4#503711yescharacteristic
430C4#515411yescharacteristic
431C4#515511yescharacteristic
432C4#515812yes—
433C4#521711yescharacteristic
434C4#523811yescharacteristic
435C3#1026111yescharacteristic, complemented, Sylow
436C2—82no—
437C2—162no—
438C2#7100712yes—
439C2#7101112yes—
440C2#7168411yescharacteristic
441C2#7204511yescharacteristic
442C2#7209711yescharacteristic
subgroupG/H
centerC23#939
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC2×C3×#459C2
socleC23×C3#121
hypercenterC2×#459D3
hypocenterC3#10261
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
2. C2×#5070C2semidirect
3. C2×#5019C2semidirect
4. C2×#4544C2semidirect
5. #71614C2semidirect
6. C2×#4482C2semidirect
7. #67703C2semidirect
435. C3#10261semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4592
C2×#50702
C2×#50192
C2×#45442
#716142
C2×#44822
#677032
#102613