Finite Group Databasebeta

← #10697572

Subgroups — #10697572

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×#3037C211yesmaximal, characteristic, complemented
2#83302C211yesmaximal, characteristic
3#82476C211yesmaximal, characteristic, complemented
4#71811C211yesmaximal, characteristic, complemented
5#71621C211yesmaximal, characteristic, complemented
6#70502C211yesmaximal, characteristic, complemented
7#67715C211yesmaximal, characteristic, complemented
8#10242—31nomaximal, Sylow
9C2×C3×#137—21no—
10C3×#532C2211yescharacteristic
11C2×C3×C4×Q8C2211yescharacteristic
12C2×#947—21no—
13C2×#874C2211yescharacteristic
14C3×#464—22no—
15C3×#462—21no—
16C3×#460—21no—
17C3×#423C2211yescharacteristic
18C3×#417—21no—
19#5088—21no—
20#5073—21no—
21#5036—22no—
22#5015—22no—
23#4991—22no—
24#4988—21no—
25#4939—22no—
26C4×C4⋊Dic3C2211yescharacteristic
27#4925C2211yescharacteristic
28#4925C2211yescharacteristic
29#4537—22no—
30#4535—22no—
31#4485—22no—
32#3205—31no—
33#3037D311yescharacteristic, complemented
34#2548—31no—
35#1545—31no—
36#1502—31no—
37#1479—31no—
38#1303—31no—
39C2×C3×#41—21no—
40C22×C3×Q8D411yescharacteristic
41C22×D12—21no—
42C22×D12—21no—
43C22×D12D412yes—
44C3×#137—42no—
45C3×#137—42no—
46C2×C3×Q16—22no—
47C2×C3×SD16—22no—
48C2×C3×M16—21no—
49C3×C4⋊Q8D411yescharacteristic
50C3×#127D411yescharacteristic
51C3×#124—21no—
52C3×#121—21no—
53C3×#121—21no—
54C3×C4×Q8—21no—
55C3×C4×Q8—21no—
56C3×C4×Q8—21no—
57C3×C4×Q8—21no—
58C3×C4×D4—21no—
59C3×C4×D4—21no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4D411yescharacteristic
62C2×C3×C42—21no—
63C2×C3×C42C2311yescharacteristic
64#947—42no—
65#947—42no—
66C2×#213—24no—
67C2×(C2×D3)⋊C4—24no—
68C2×(C2×D3)⋊C4—24no—
69C2×C4⋊Dic3—21no—
70C2×C4×Dic3—21no—
71#874—21no—
72#874—21no—
73#874—21no—
74#874—21no—
75C3×C4⋊C8—21no—
76C3×C4⋊C8D412yes—
77C3×#103—22no—
78C3×#102—22no—
79C3⋊#95D412yes—
80#830—21no—
81#830—21no—
82#809—24no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)D412yes—
85C4×C3⋊C8—21no—
86C4×C3⋊C8—21no—
87C2×#137—21no—
88#532C2×D311yescharacteristic
89C2×#127—31no—
90C2×#124—61no—
91C2×C4×Q8C2×D311yescharacteristic
92#493—61no—
93#486—61no—
94#464—22no—
95#462—21no—
96#460—21no—
97#459—62no—
98#447—62no—
99#431—31no—
100#431—31no—
101#423C2×D311yescharacteristic
102#417—21no—
103#394—62no—
104#390—62no—
105#390—61no—
106C4×C4⋊C4—31no—
107#331—62no—
108#327—62no—
109Q8⋊C8—62no—
110C23×D3—22no—
111C3×#41—41no—
112C3×#41—41no—
113C3×#41—41no—
114C3×#41—41no—
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—21no—
121C2×C3×Q8D812yescomplemented
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—41no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—41no—
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×Dic3—22no—
133C2×D12—22no—
134C2×D12—22no—
135C2×D12—44no—
136C2×D12—41no—
137C2×D12—41no—
138C2×D12—41no—
139C2×D12—41no—
140C2×D12—41no—
141C2×D12—41no—
142C2×D12—41no—
143C2×D12—41no—
144C2×D12—82no—
145C3×Q16—44no—
146C3×SD16—44no—
147C3×M16—41no—
148C3×M16—41no—
149C2×C3×C8—22no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—41no—
155C3×C4⋊C4—41no—
156C3×C22⋊C4—41no—
157C3×C22⋊C4—41no—
158C3×C42—41no—
159C3×C42—41no—
160C3×C42C2×D411yescharacteristic
161C3×C42C2×D411yescharacteristic
162C3×C42C2×D411yescharacteristic
163C3×C42#4111yescharacteristic
164#213—84no—
165(C2×D3)⋊C4—48no—
166(C2×D3)⋊C4—84no—
167C4⋊Dic3—42no—
168C4×Dic3—41no—
169C4×Dic3—41no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—41no—
172C2×#41—21no—
173C22×Q8#6711yescharacteristic
174C22×D4—32no—
175C22×D4—61no—
176C22×D4—61no—
177#137—42no—
178#137—42no—
179C2×Q16—22no—
180C2×SD16—22no—
181C2×D8—64no—
182C2×M16—21no—
183C4⋊Q8#6711yescharacteristic
184#127—61no—
185#127—61no—
186#127—61no—
187#127—61no—
188#127#6711yescharacteristic
189#124—21no—
190#124—122no—
191#124—122no—
192#121—21no—
193#121—21no—
194C4×Q8—21no—
195C4×Q8—21no—
196C4×Q8—21no—
197C4×Q8—21no—
198C4×D4—21no—
199C4×D4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—61no—
202C2×C4⋊C4#6711yescharacteristic
203C2×C22⋊C4—64no—
204C2×C22⋊C4—64no—
205C2×C42—21no—
206C2×C42—61no—
207C2×C42C22×D311yescharacteristic
208C4⋊C8—21no—
209C4⋊C8—32no—
210C4⋊C8—61no—
211C4⋊C8#6712yes—
212#103—22no—
213#102—22no—
214#102—64no—
215C22⋊C8—61no—
216C22⋊C8—61no—
217C4×C8—61no—
218C4×C8—61no—
219#95—32no—
220C23×C3—41no—
221C23×C3#12011yescharacteristic
222C22×D3—44no—
223C22×D3—44no—
224C22×D3—42no—
225C22×D3—42no—
226C22×D3—42no—
227C3×Q8—22no—
228C3×Q8—22no—
229C3×Q8—41no—
230C3×Q8—41no—
231C3×Q8—41no—
232C3×Q8—41no—
233C3×D4—41no—
234C3×D4—41no—
235C3×D4—42no—
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—41no—
246C2×C3×C4—41no—
247C2×C3×C4—81no—
248C2×C3×C4—81no—
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
259C2×Dic3—42no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262D12—44no—
263D12—82no—
264D12—82no—
265D12—82no—
266D12—81no—
267D12—81no—
268D12—81no—
269D12—81no—
270C3×C8—42no—
271C3⋊C8—42no—
272C3⋊C8—81no—
273C24—62no—
274#41—41no—
275#41—41no—
276#41—41no—
277#41—41no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8#21312yescomplemented
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—62no—
290C2×D4—62no—
291C2×D4—124no—
292C2×D4—121no—
293C2×D4—121no—
294C2×D4—121no—
295C2×D4—121no—
296C2×D4—121no—
297C2×D4—121no—
298C2×D4—121no—
299C2×D4—121no—
300C2×D4—242no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—41no—
304C22×C4—62no—
305C22×C4D3×D411yescharacteristic
306C22×C4D3×D411yescharacteristic
307C22×C4C2×#6711yescharacteristic
308Q16—44no—
309SD16—44no—
310D8—244no—
311M16—41no—
312M16—41no—
313C2×C8—22no—
314C2×C8—62no—
315C2×C8—121no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—122no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—128no—
326C22⋊C4—244no—
327C42—41no—
328C42—41no—
329C42—121no—
330C42—121no—
331C42D3×D411yescharacteristic
332C42#23711yescharacteristic
333C42C2×#6711yescharacteristic
334C42C2×#6711yescharacteristic
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—81no—
338C22×C3—81no—
339C22×C3—81no—
340C22×C3#39211yescharacteristic
341C22×C3#44711yescharacteristic
342C22×C3#44911yescharacteristic
343C2×D3—84no—
344C2×D3—84no—
345C2×D3—82no—
346C2×D3—82no—
347C2×D3—82no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—42no—
352C3×C4—81no—
353C3×C4—81no—
354C3×C4#44711yescharacteristic
355C3×C4#44911yescharacteristic
356C3×C4#45512yes—
357C3×C4#46611yescharacteristic
358C3×C4#46911yescharacteristic
359Dic3—82no—
360C23—41no—
361C23—124no—
362C23—124no—
363C23—122no—
364C23—122no—
365C23—122no—
366C23#93711yescharacteristic
367Q8—22no—
368Q8—22no—
369Q8—41no—
370Q8—41no—
371Q8—41no—
372Q8—41no—
373D4—42no—
374D4—41no—
375D4—41no—
376D4—81no—
377D4—81no—
378D4—81no—
379D4—124no—
380D4—242no—
381D4—242no—
382D4—242no—
383D4—241no—
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—41no—
394C2×C4—41no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—122no—
398C2×C4—122no—
399C2×C4—122no—
400C2×C4C2×#21311yescharacteristic
401C2×C4#93211yescharacteristic
402C2×C4#93711yescharacteristic
403C2×C4#93811yescharacteristic
404C2×C4#93911yescharacteristic
405C2×C4#93911yescharacteristic
406C2×C4#94011yescharacteristic
407C2×C4#94911yescharacteristic
408C2×C4#95011yescharacteristic
409C2×C4#95311yescharacteristic
410C8—42no—
411C8—122no—
412C8—241no—
413C2×C3—21no—
414C2×C3—21no—
415C2×C3—161no—
416C2×C3#199911yescharacteristic
417C2×C3#200411yescharacteristic
418C2×C3#200811yescharacteristic
419D3—162no—
420C22—21no—
421C22—21no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—244no—
426C22—244no—
427C22—242no—
428C22—242no—
429C22—242no—
430C22#503611yescharacteristic
431C22#503711yescharacteristic
432C22#522311yescharacteristic
433C4—21no—
434C4—21no—
435C4—21no—
436C4—42no—
437C4—81no—
438C4—81no—
439C4—242no—
440C4#503811yescharacteristic
441C4#503911yescharacteristic
442C4#520212yes—
443C4#521711yescharacteristic
444C4#523811yescharacteristic
445C3#1024211yescharacteristic, complemented, Sylow
446C2—21no—
447C2—21no—
448C2—161no—
449C2—482no—
450C2#7168411yescharacteristic
451C2#7210611yescharacteristic
452C2#7211911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3037C2
socleC22×C3#392
prime radical#83302C2
hypercenter#3037D3
hypocenterC3#10242
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3037

Complements

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

NH ≅ G/Ntype
1. C3×#3037C2semidirect
3. #82476C2semidirect
4. #71811C2semidirect
5. #71621C2semidirect
6. #70502C2semidirect
7. #67715C2semidirect
33. #3037D3semidirect
121. 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×#30372
#833022
#824762
#718112
#716212
#705022
#677152
#102423