Finite Group Databasebeta

← #10697575

Subgroups — #10697575

Order 768.

658 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2548C211yesmaximal, characteristic, complemented
2#83166C211yesmaximal, characteristic
3#82441C211yesmaximal, characteristic, complemented
4#71759C211yesmaximal, characteristic, complemented
5#71621C211yesmaximal, characteristic, complemented
6#69469C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#10242—31nomaximal, Sylow
9C2×C3×#127C2211yescharacteristic
10C2×C3×#124—21no—
11C2×C3×C4×Q8C2211yescharacteristic
12C2×#942—21no—
13#5477C2211yescharacteristic
14C3×#394—22no—
15C3×#390—22no—
16C3×#390—21no—
17C3×C4×C4⋊C4C2211yescharacteristic
18#5217—22no—
19#5073—21no—
20#5072—22no—
21#5070—21no—
22#5031—21no—
23#5028—22no—
24#5027—21no—
25#4966—21no—
26#4925C2211yescharacteristic
27#4925C2211yescharacteristic
28#4924C2211yescharacteristic
29#4544—22no—
30#4535—22no—
31#4451—22no—
32#3205—31no—
33#3037—31no—
34#2548D311yescharacteristic, complemented
35#1545—31no—
36#1502—31no—
37#1479—31no—
38#1303—31no—
39C22×C3×Q8D411yescharacteristic
40C22×C3×D4—21no—
41C22×C3×D4—21no—
42C22×C3×D4D412yes—
43C2×#235—21no—
44C3×#127—21no—
45C3×#127—21no—
46C3×#127—21no—
47C3×#127—21no—
48C3×#124—42no—
49C3×#124—42no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C22⋊C4—24no—
58C2×C3×C22⋊C4—24no—
59C2×C3×C42—21no—
60C2×C3×C42—21no—
61C2×C3×C42C2311yescharacteristic
62C2×#216—22no—
63#942—42no—
64#942—42no—
65C2×#215—22no—
66C2×#213—24no—
67#930—21no—
68#930—21no—
69C2×#208—21no—
70#875—21no—
71#874D411yescharacteristic
72C4×D12—21no—
73C4×D12—21no—
74#869D411yescharacteristic
75C3×#95D412yes—
76#832—24no—
77#830—21no—
78#830—21no—
79#810—22no—
80#809—22no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)D412yes—
84C4⋊(C3⋊C8)D412yes—
85C4×C3⋊C8—21no—
86C4×C3⋊C8—21no—
87C2×#137—61no—
88#532—31no—
89C2×#127C2×D311yescharacteristic
90C2×#124—21no—
91C2×C4×Q8C2×D311yescharacteristic
92#493—61no—
93#486—61no—
94#464—62no—
95#462—61no—
96#460—61no—
97#459—62no—
98#447—62no—
99#431—31no—
100#431—31no—
101#423—31no—
102#417—61no—
103#394—22no—
104#390—22no—
105#390—21no—
106C4×C4⋊C4C2×D311yescharacteristic
107#331—62no—
108#327—62no—
109Q8⋊C8—62no—
110C24×C3—22no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×Q8—21no—
115C2×C3×Q8—21no—
116C2×C3×Q8D812yescomplemented
117C2×C3×D4—22no—
118C2×C3×D4—22no—
119C2×C3×D4—44no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—41no—
123C2×C3×D4—41no—
124C2×C3×D4—41no—
125C2×C3×D4—41no—
126C2×C3×D4—41no—
127C2×C3×D4—41no—
128C2×C3×D4—82no—
129C22×C3×C4—21no—
130C22×C3×C4—22no—
131C22×C3×C4—21no—
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C2×#67—41no—
136#235—41no—
137#235—41no—
138#235—41no—
139#235—41no—
140C2×D12—21no—
141C2×D12—21no—
142C2×D12—21no—
143C2×C4×D3—41no—
144C2×Dic6—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—42no—
148C3×C4⋊C4—41no—
149C3×C4⋊C4—41no—
150C3×C22⋊C4—48no—
151C3×C22⋊C4—84no—
152C3×C42—41no—
153C3×C42—41no—
154C3×C42—41no—
155C3×C42—41no—
156C3×C42C2×D411yescharacteristic
157C3×C42C2×D411yescharacteristic
158C3×C42C2×D411yescharacteristic
159C3×C42#4111yescharacteristic
160#216—44no—
161#215—44no—
162#213—84no—
163(C2×D3)⋊C4—41no—
164(C2×D3)⋊C4—41no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167#208—41no—
168#208—41no—
169C2×C3⋊C8—22no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—41no—
172C2×#41—61no—
173C22×Q8#6711yescharacteristic
174C22×D4—21no—
175C22×D4—21no—
176C22×D4#6712yes—
177#137—122no—
178#137—122no—
179C2×Q16—62no—
180C2×SD16—62no—
181C2×D8—64no—
182C2×M16—61no—
183C4⋊Q8—31no—
184#127—21no—
185#127—21no—
186#127—21no—
187#127—21no—
188#127—31no—
189#124—42no—
190#124—42no—
191#124—61no—
192#121—61no—
193#121—61no—
194C4×Q8—21no—
195C4×Q8—21no—
196C4×Q8—21no—
197C4×Q8—21no—
198C4×D4—61no—
199C4×D4—61no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4#6711yescharacteristic
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42C22×D311yescharacteristic
208C4⋊C8—32no—
209C4⋊C8—32no—
210C4⋊C8—61no—
211C4⋊C8—61no—
212#103—62no—
213#102—62no—
214#102—64no—
215C22⋊C8—61no—
216C22⋊C8—61no—
217C4×C8—61no—
218C4×C8—61no—
219#95#6712yes—
220C23×C3—44no—
221C23×C3—44no—
222C23×C3—42no—
223C23×C3—42no—
224C23×C3—42no—
225C23×C3#12011yescharacteristic
226C22×D3—41no—
227C3×Q8—22no—
228C3×Q8—22no—
229C3×Q8—41no—
230C3×Q8—41no—
231C3×D4—44no—
232C3×D4—82no—
233C3×D4—82no—
234C3×D4—82no—
235C3×D4—81no—
236C3×D4—81no—
237C3×D4—81no—
238C3×D4—81no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—41no—
248C2×C3×C4—42no—
249C2×C3×C4#12011yescharacteristic
250C2×C3×C4#12011yescharacteristic
251C2×C3×C4#12111yescharacteristic
252C2×C3×C4#12111yescharacteristic
253C2×C3×C4#12111yescharacteristic
254C2×C3×C4#12711yescharacteristic
255C2×C3×C4C2×D811yescharacteristic
256C2×C3×C4#13511yescharacteristic
257C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
258C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
259#67—81no—
260#67—81no—
261C2×Dic3—41no—
262D12—41no—
263D12—41no—
264D12—42no—
265D12—81no—
266C4×D3—81no—
267C4×D3—81no—
268Dic6—41no—
269Dic6—41no—
270C3⋊C8—42no—
271C3⋊C8—42no—
272C3⋊C8—81no—
273C24—22no—
274#41—121no—
275#41—121no—
276#41—121no—
277#41—121no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—61no—
284C2×Q8#21312yescomplemented
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—44no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—61no—
297C2×D4—61no—
298C2×D4—61no—
299C2×D4—82no—
300C2×D4—121no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—121no—
305C22×C4C2×#6711yescharacteristic
306C22×C4C2×#6711yescharacteristic
307C22×C4C2×#6711yescharacteristic
308Q16—124no—
309SD16—124no—
310D8—244no—
311M16—121no—
312M16—121no—
313C2×C8—62no—
314C2×C8—62no—
315C2×C8—121no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—42no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—61no—
322C4⋊C4—61no—
323C22⋊C4—48no—
324C22⋊C4—84no—
325C22⋊C4—121no—
326C22⋊C4—121no—
327C42—41no—
328C42—41no—
329C42—41no—
330C42—41no—
331C42D3×D411yescharacteristic
332C42D3×D411yescharacteristic
333C42D3×D411yescharacteristic
334C42#23711yescharacteristic
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—84no—
338C22×C3—84no—
339C22×C3—82no—
340C22×C3—82no—
341C22×C3—82no—
342C22×C3#39211yescharacteristic
343C22×C3#44711yescharacteristic
344C22×C3#44911yescharacteristic
345C2×D3—81no—
346C2×D3—81no—
347C2×D3—81no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—42no—
352C3×C4—82no—
353C3×C4—81no—
354C3×C4#44711yescharacteristic
355C3×C4#44911yescharacteristic
356C3×C4#45512yes—
357C3×C4#46611yescharacteristic
358C3×C4#46911yescharacteristic
359Dic3—81no—
360C23—44no—
361C23—44no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—121no—
366C23#95311yescharacteristic
367Q8—22no—
368Q8—22no—
369Q8—41no—
370Q8—41no—
371Q8—121no—
372Q8—121no—
373D4—44no—
374D4—82no—
375D4—82no—
376D4—82no—
377D4—81no—
378D4—81no—
379D4—81no—
380D4—81no—
381D4—121no—
382D4—121no—
383D4—122no—
384D4—241no—
385D4—241no—
386D4—241no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—121no—
398C2×C4—241no—
399C2×C4—241no—
400C2×C4C2×#21311yescharacteristic
401C2×C4#93211yescharacteristic
402C2×C4#93711yescharacteristic
403C2×C4#93711yescharacteristic
404C2×C4#93811yescharacteristic
405C2×C4#93911yescharacteristic
406C2×C4#93911yescharacteristic
407C2×C4#94011yescharacteristic
408C2×C4#94911yescharacteristic
409C2×C4#95011yescharacteristic
410C8—122no—
411C8—122no—
412C8—241no—
413C2×C3—21no—
414C2×C3—21no—
415C2×C3—162no—
416C2×C3#199911yescharacteristic
417C2×C3#200411yescharacteristic
418C2×C3#200811yescharacteristic
419D3—161no—
420C22—21no—
421C22—21no—
422C22—84no—
423C22—84no—
424C22—82no—
425C22—82no—
426C22—82no—
427C22—241no—
428C22—241no—
429C22—241no—
430C22#521711yescharacteristic
431C22#522311yescharacteristic
432C22#523811yescharacteristic
433C4—21no—
434C4—21no—
435C4—21no—
436C4—42no—
437C4—82no—
438C4—81no—
439C4—241no—
440C4#503611yescharacteristic
441C4#503711yescharacteristic
442C4#503811yescharacteristic
443C4#503911yescharacteristic
444C4#506112yes—
445C3#1024211yescharacteristic, complemented, Sylow
446C2—21no—
447C2—21no—
448C2—162no—
449C2—481no—
450C2#7168411yescharacteristic
451C2#7173111yescharacteristic
452C2#7185811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2548C2
socleC22×C3#392
prime radical#83166C2
hypercenter#2548D3
hypocenterC3#10242
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2548

Complements

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

NH ≅ G/Ntype
1. C3×#2548C2semidirect
3. #82441C2semidirect
4. #71759C2semidirect
5. #71621C2semidirect
6. #69469C2semidirect
7. #67620C2semidirect
34. #2548D3semidirect
116. C2×C3×Q8D8semidirect
284. C2×Q8#213semidirect
445. C3#10242semidirect

Maximal subgroups

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

subgroupindex
C3×#25482
#831662
#824412
#717592
#716212
#694692
#676202
#102423