Finite Group Databasebeta

← #10697702

Subgroups — #10697702

Order 768.

742 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#460C211yesmaximal, characteristic, complemented
2C2×#5073C211yesmaximal, characteristic
3C2×#5027C211yesmaximal, characteristic, complemented
4C2×#4544C211yesmaximal, characteristic, complemented
5#71634C211yesmaximal, characteristic, complemented
6C2×#4485C211yesmaximal, characteristic, complemented
7#67703C211yesmaximal, characteristic, complemented
8#10262—31nomaximal, Sylow
9C22×C3×SD16—21no—
10C2×C3×#127C2211yescharacteristic
11C2×C3×C4×Q8C2211yescharacteristic
12C22×#215—21no—
13C22×#213—22no—
14C2×#874C2211yescharacteristic
15C3×#460—22no—
16C3×#460—22no—
17C2×C3×C4⋊C8C2211yescharacteristic
18C2×C3×#102—21no—
19#5073—22no—
20#5073—22no—
21#5027—22no—
22#5027—22no—
23C2×#809—21no—
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26C2×C4×C3⋊C8C2211yescharacteristic
27#4544—22no—
28#4524—22no—
29#4485—22no—
30C2×#493—31no—
31C2×#460—31no—
32C2×#460D311yescharacteristic, complemented
33#1984—31no—
34C2×#331—31no—
35C2×#331—31no—
36#1277—31no—
37C22×C3×Q8D411yescharacteristic
38C22×C3×D4—21no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×D12—21no—
42C22×D12—21no—
43C22×D12—21no—
44C2×C3×SD16—24no—
45C2×C3×SD16—42no—
46C2×C3×SD16—42no—
47C22×C3×C8—21no—
48C3×#127—21no—
49C3×#127—21no—
50C3×#127—21no—
51C3×#127D412yes—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×Q8—21no—
55C3×C4×Q8—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C42—21no—
59C2×C3×C42C2311yescharacteristic
60C2×#215—24no—
61C2×#215—42no—
62C2×#215—42no—
63C2×#213—28no—
64C2×#213—44no—
65C2×#213—44no—
66C22×C3⋊C8—21no—
67C22×C3⋊C8—21no—
68C22×C3⋊C8D412yes—
69#874—21no—
70#874—21no—
71#874—21no—
72#874D412yes—
73C3×C4⋊C8—21no—
74C3×C4⋊C8D412yes—
75C3×#102—24no—
76#809—24no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)D412yes—
81C4×C3⋊C8—21no—
82C4×C3⋊C8—21no—
83C22×SD16—21no—
84C22×SD16—61no—
85C22×D8—62no—
86C2×#127—31no—
87C2×#127C2×D311yescharacteristic
88C2×C4×Q8C2×D311yescharacteristic
89#493—62no—
90#493—62no—
91#460—22no—
92#460—22no—
93#460—62no—
94#460—62no—
95C2×C4⋊C8—31no—
96C2×C4⋊C8—31no—
97C2×C4⋊C8C2×D311yescharacteristic
98C2×#102—21no—
99C2×#102—61no—
100C2×C4×C8—31no—
101#336—62no—
102#331—62no—
103#331—62no—
104C24×C3—41no—
105C23×D3—41no—
106C2×C3×Q8—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8D812yescomplemented
112C2×C3×D4—22no—
113C2×C3×D4—22no—
114C2×C3×D4—22no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—42no—
120C2×C3×D4—41no—
121C2×C3×D4—42no—
122C2×C3×D4—41no—
123C2×C3×D4—41no—
124C2×C3×D4—82no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4#4111yescharacteristic
130C2×D12—22no—
131C2×D12—22no—
132C2×D12—22no—
133C2×D12—41no—
134C2×D12—42no—
135C2×D12—41no—
136C2×D12—42no—
137C2×D12—41no—
138C2×D12—41no—
139C2×D12—41no—
140C2×D12—41no—
141C2×D12—41no—
142C2×D12—82no—
143C3×SD16—48no—
144C2×C3×C8—22no—
145C2×C3×C8—41no—
146C2×C3×C8—41no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—41no—
150C3×C4⋊C4—41no—
151C3×C42—41no—
152C3×C42—41no—
153C3×C42C2×D411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157#215—48no—
158#213—416no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—41no—
163C2×C3⋊C8—41no—
164C2×C3⋊C8—41no—
165C2×C3⋊C8—42no—
166C2×C3⋊C8SD1614yescomplemented
167C22×Q8#6711yescharacteristic
168C22×D4—21no—
169C22×D4—21no—
170C22×D4—21no—
171C22×D4—61no—
172C22×D4—61no—
173C22×D4—61no—
174C2×SD16—24no—
175C2×SD16—42no—
176C2×SD16—42no—
177C2×SD16—64no—
178C2×SD16—122no—
179C2×SD16—122no—
180C2×D8—68no—
181C2×D8—124no—
182C2×D8—124no—
183C22×C8—21no—
184C22×C8—32no—
185C22×C8—61no—
186C22×C8—61no—
187#127—21no—
188#127—21no—
189#127—21no—
190#127—32no—
191#127—61no—
192#127—61no—
193#127—61no—
194#127#6712yes—
195C4×Q8—21no—
196C4×Q8—21no—
197C4×Q8—21no—
198C4×Q8—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4#6711yescharacteristic
201C2×C42—21no—
202C2×C42C22×D311yescharacteristic
203C4⋊C8—21no—
204C4⋊C8—32no—
205C4⋊C8—61no—
206C4⋊C8—61no—
207C4⋊C8—61no—
208C4⋊C8#6712yes—
209#102—24no—
210#102—64no—
211C4×C8—61no—
212C4×C8—61no—
213C23×C3—42no—
214C23×C3—82no—
215C23×C3—81no—
216C23×C3—81no—
217C23×C3—82no—
218C23×C3#12111yescharacteristic
219C22×D3—42no—
220C22×D3—82no—
221C22×D3—81no—
222C22×D3—81no—
223C22×D3—82no—
224C3×Q8—22no—
225C3×Q8—22no—
226C3×Q8—41no—
227C3×Q8—41no—
228C3×D4—42no—
229C3×D4—42no—
230C3×D4—44no—
231C3×D4—82no—
232C3×D4—82no—
233C3×D4—81no—
234C3×D4—81no—
235C3×D4—81no—
236C3×D4—81no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—42no—
243C2×C3×C4#12011yescharacteristic
244C2×C3×C4#12011yescharacteristic
245C2×C3×C4#12011yescharacteristic
246C2×C3×C4#12111yescharacteristic
247C2×C3×C4#12111yescharacteristic
248C2×C3×C4#12711yescharacteristic
249C2×C3×C4C2×D811yescharacteristic
250C2×C3×C4C2×SD1612yes—
251C2×C3×C4#13511yescharacteristic
252C2×C3×C4AGL(1,ℤ/8)12yes—
253C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
254C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
255D12—42no—
256D12—42no—
257D12—44no—
258D12—82no—
259D12—82no—
260D12—81no—
261D12—81no—
262D12—81no—
263D12—81no—
264C3×C8—42no—
265C3⋊C8—24no—
266C3⋊C8—42no—
267C3⋊C8—81no—
268C24—41no—
269C24—121no—
270C2×Q8—21no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8#21312yescomplemented
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—62no—
289C2×D4—62no—
290C2×D4—62no—
291C2×D4—82no—
292C2×D4—122no—
293C2×D4—122no—
294C2×D4—121no—
295C2×D4—121no—
296C2×D4—121no—
297C2×D4—121no—
298C2×D4—121no—
299C2×D4—121no—
300C2×D4—121no—
301C2×D4—242no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4D3×D411yescharacteristic
305C22×C4#23711yescharacteristic
306C22×C4C2×#6711yescharacteristic
307SD16—48no—
308SD16—128no—
309D8—1216no—
310C2×C8—22no—
311C2×C8—34no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—62no—
315C2×C8—62no—
316C2×C8—62no—
317C2×C8—121no—
318C2×C8—121no—
319C2×C8—122no—
320C2×C8—121no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C42—41no—
326C42—41no—
327C42D3×D411yescharacteristic
328C42D3×D411yescharacteristic
329C42C2×#6711yescharacteristic
330C42C2×#6711yescharacteristic
331C22×C3—82no—
332C22×C3—82no—
333C22×C3—82no—
334C22×C3—162no—
335C22×C3—162no—
336C22×C3#39211yescharacteristic
337C22×C3#46012yes—
338C22×C3#46611yescharacteristic
339C22×C3#46911yescharacteristic
340C22×C3#47212yes—
341C2×D3—82no—
342C2×D3—82no—
343C2×D3—82no—
344C2×D3—162no—
345C2×D3—162no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—42no—
349C3×C4—81no—
350C3×C4#44711yescharacteristic
351C3×C4#44711yescharacteristic
352C3×C4#44911yescharacteristic
353C3×C4#44911yescharacteristic
354C3×C4#45012yes—
355C3×C4#49211yescharacteristic
356C3×C4#49611yescharacteristic
357C23—42no—
358C23—82no—
359C23—82no—
360C23—81no—
361C23—81no—
362C23—122no—
363C23—242no—
364C23—242no—
365C23—241no—
366C23—241no—
367C23#93911yescharacteristic
368Q8—22no—
369Q8—22no—
370Q8—41no—
371Q8—41no—
372D4—44no—
373D4—42no—
374D4—42no—
375D4—82no—
376D4—82no—
377D4—81no—
378D4—81no—
379D4—81no—
380D4—81no—
381D4—124no—
382D4—122no—
383D4—122no—
384D4—242no—
385D4—242no—
386D4—241no—
387D4—241no—
388D4—241no—
389D4—241no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4D3×SD1612yes—
397C2×C4#91412yes—
398C2×C4C2×#21311yescharacteristic
399C2×C4#93211yescharacteristic
400C2×C4#93711yescharacteristic
401C2×C4#93711yescharacteristic
402C2×C4#93811yescharacteristic
403C2×C4#93911yescharacteristic
404C2×C4#94011yescharacteristic
405C2×C4#94911yescharacteristic
406C2×C4#95011yescharacteristic
407C2×C4#95311yescharacteristic
408C8—42no—
409C8—64no—
410C8—122no—
411C8—241no—
412C2×C3—162no—
413C2×C3#166812yes—
414C2×C3#168812yes—
415C2×C3#199911yescharacteristic
416C2×C3#201411yescharacteristic
417C2×C3#201811yescharacteristic
418D3—162no—
419C22—82no—
420C22—82no—
421C22—82no—
422C22—162no—
423C22—162no—
424C22—242no—
425C22—242no—
426C22—242no—
427C22—482no—
428C22—482no—
429C22#503811yescharacteristic
430C22#503911yescharacteristic
431C22#516312yes—
432C22#519912yes—
433C22#522311yescharacteristic
434C4—21no—
435C4—21no—
436C4—42no—
437C4—81no—
438C4#503611yescharacteristic
439C4#503711yescharacteristic
440C4#516912yes—
441C4#517511yescharacteristic
442C4#517611yescharacteristic
443C4#521711yescharacteristic
444C4#523811yescharacteristic
445C3#1026211yescharacteristic, complemented, Sylow
446C2—162no—
447C2—482no—
448C2#7102112yes—
449C2#7102512yes—
450C2#7168411yescharacteristic
451C2#7205611yescharacteristic
452C2#7209511yescharacteristic
subgroupG/H
centerC23#939
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#460C2
socleC23×C3#121
prime radicalC2×#5073C2
hypercenterC2×#460D3
hypocenterC3#10262
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
1. C2×C3×#460C2semidirect
3. C2×#5027C2semidirect
4. C2×#4544C2semidirect
5. #71634C2semidirect
6. C2×#4485C2semidirect
7. #67703C2semidirect
32. C2×#460D3semidirect
111. C2×C3×Q8D8semidirect
166. C2×C3⋊C8SD16semidirect
275. C2×Q8#213semidirect
445. C3#10262semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4602
C2×#50732
C2×#50272
C2×#45442
#716342
C2×#44852
#677032
#102623