Finite Group Databasebeta

← #10697547

Subgroups — #10697547

Order 768.

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

normal subgroups only

This group has 794 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×#2541C211yesmaximal, characteristic, complemented
2#83166C211yesmaximal, characteristic, complemented
3#82307C211yesmaximal, characteristic, complemented
4#71759C211yesmaximal, characteristic, complemented
5#71597C211yesmaximal, characteristic, complemented
6#69469C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#10238—31nomaximal, Sylow
9C2×C3×#127—21no—
10C2×C3×#127C2211yescharacteristic
11C2×C3×#121—22no—
12C2×C3×C4×D4C2211yescharacteristic
13C2×#932—21no—
14#5477C2211yescharacteristic
15C3×#392—22no—
16C3×#390—21no—
17C3×C4×C4⋊C4C2211yescharacteristic
18#5217—22no—
19#5073—21no—
20#5072—22no—
21#5070—21no—
22#5019—21no—
23#5016—22no—
24#5015—21no—
25#4966—21no—
26#4925C2211yescharacteristic
27#4925C2211yescharacteristic
28#4924C2211yescharacteristic
29#4544—22no—
30D4⋊(C3⋊C8)—22no—
31#4451—22no—
32#3205—31no—
33#3036—31no—
34#2541D311yescharacteristic, complemented
35#1545—31no—
36#1502—31no—
37#1478—31no—
38#1303—31no—
39C22×C3×D4—24no—
40C22×C3×D4—21no—
41C22×C3×D4—21no—
42C22×C3×D4D411yescharacteristic
43C22×C3×D4D412yes—
44C23×C3×C4—21no—
45C2×#235—21no—
46C3×#127—21no—
47C3×#127—21no—
48C3×#127—21no—
49C3×#127—21no—
50C3×#127—42no—
51C3×#127—42no—
52C3×#121—48no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C3×C4×D4—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C22⋊C4—24no—
60C2×C3×C22⋊C4—21no—
61C2×C3×C42—21no—
62C2×C3×C42C2311yescharacteristic
63C2×#214—22no—
64#932—42no—
65#932—42no—
66C2×#213—22no—
67C2×#213—24no—
68#930—21no—
69#930—21no—
70C2×#208—21no—
71#875—21no—
72#874D411yescharacteristic, complemented
73C4×D12—21no—
74C4×D12—21no—
75#869D411yescharacteristic, complemented
76C3×#95D412yes—
77#832—24no—
78#830—21no—
79#830—21no—
80#810—22no—
81#809—22no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)D412yescomplemented
85C4⋊(C3⋊C8)D412yescomplemented
86C4×C3⋊C8—21no—
87C4×C3⋊C8—21no—
88C2×AGL(1,ℤ/8)—61no—
89#532—31no—
90C2×#127—21no—
91C2×#127C2×D311yescharacteristic
92C2×#121—22no—
93C2×C4×D4C2×D311yescharacteristic
94#493—61no—
95#486—61no—
96#463—62no—
97#461—61no—
98#459—62no—
99#459—61no—
100#447—62no—
101#431—31no—
102#431—31no—
103#423—31no—
104#417—61no—
105#392—22no—
106#390—21no—
107C4×C4⋊C4C2×D311yescharacteristic
108#331—62no—
109#327—62no—
110D4⋊C8—62no—
111C24×C3—21no—
112C24×C3—22no—
113C2×C3×D4—21no—
114C2×C3×D4—22no—
115C2×C3×D4—21no—
116C2×C3×D4—22no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—44no—
121C2×C3×D4—48no—
122C2×C3×D4—48no—
123C2×C3×D4—41no—
124C2×C3×D4—41no—
125C2×C3×D4—41no—
126C2×C3×D4—41no—
127C2×C3×D4—41no—
128C2×C3×D4—41no—
129C2×C3×D4—41no—
130C2×C3×D4—41no—
131C2×C3×D4—82no—
132C2×C3×D4—84no—
133C2×C3×D4D812yescomplemented
134C22×C3×C4—22no—
135C22×C3×C4—21no—
136C22×C3×C4—44no—
137C22×C3×C4—41no—
138C22×C3×C4—41no—
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C2×#67—41no—
143#235—41no—
144#235—41no—
145#235—41no—
146#235—41no—
147C2×D12—21no—
148C2×D12—21no—
149C2×D12—21no—
150C2×C4×D3—41no—
151C2×Dic6—21no—
152C3×C4⋊C4—24no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—21no—
155C3×C22⋊C4—48no—
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#214—44no—
165#213—44no—
166#213—84no—
167(C2×D3)⋊C4—41no—
168(C2×D3)⋊C4—41no—
169C4⋊Dic3—21no—
170C4⋊Dic3—21no—
171#208—41no—
172#208—41no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—22no—
175C2×C3⋊C8—41no—
176C2×#41—61no—
177C22×D4—24no—
178C22×D4—21no—
179C22×D4—21no—
180C22×D4#6711yescharacteristic
181C22×D4#6712yes—
182C23×C4—21no—
183AGL(1,ℤ/8)—122no—
184AGL(1,ℤ/8)—122no—
185C2×SD16—62no—
186C2×D8—62no—
187C2×D8—64no—
188C2×M16—61no—
189C4⋊Q8—31no—
190#127—21no—
191#127—21no—
192#127—21no—
193#127—21no—
194#127—31no—
195#127—42no—
196#127—42no—
197#124—61no—
198#121—48no—
199#121—61no—
200#121—61no—
201C4×D4—21no—
202C4×D4—21no—
203C4×D4—21no—
204C4×D4—21no—
205C4×D4—61no—
206C4×D4—61no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4#6711yescharacteristic
209C2×C22⋊C4—24no—
210C2×C22⋊C4—21no—
211C2×C42—21no—
212C2×C42C22×D311yescharacteristic
213C4⋊C8—32no—
214C4⋊C8—32no—
215C4⋊C8—61no—
216C4⋊C8—61no—
217#103—62no—
218#102—62no—
219#102—64no—
220C22⋊C8—61no—
221C22⋊C8—61no—
222C4×C8—61no—
223C4×C8—61no—
224#95#6712yes—
225C23×C3—22no—
226C23×C3—42no—
227C23×C3—42no—
228C23×C3—44no—
229C23×C3—44no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3—42no—
233C23×C3—42no—
234C23×C3—42no—
235C23×C3#12011yescharacteristic
236C22×D3—41no—
237C3×D4—22no—
238C3×D4—22no—
239C3×D4—41no—
240C3×D4—44no—
241C3×D4—41no—
242C3×D4—82no—
243C3×D4—82no—
244C3×D4—82no—
245C3×D4—816no—
246C3×D4—81no—
247C3×D4—81no—
248C3×D4—81no—
249C3×D4—81no—
250C2×C3×C4—22no—
251C2×C3×C4—24no—
252C2×C3×C4—41no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—41no—
257C2×C3×C4—82no—
258C2×C3×C4—82no—
259C2×C3×C4#12011yescharacteristic
260C2×C3×C4#12011yescharacteristic
261C2×C3×C4#12111yescharacteristic
262C2×C3×C4#12111yescharacteristic
263C2×C3×C4#12111yescharacteristic
264C2×C3×C4#12711yescharacteristic
265C2×C3×C4C2×D811yescharacteristic
266C2×C3×C4#13511yescharacteristic
267C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
268C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
269#67—81no—
270#67—81no—
271C2×Dic3—41no—
272D12—41no—
273D12—41no—
274D12—42no—
275D12—81no—
276C4×D3—81no—
277C4×D3—81no—
278Dic6—41no—
279Dic6—41no—
280C3⋊C8—42no—
281C3⋊C8—42no—
282C3⋊C8—81no—
283C24—21no—
284C24—22no—
285#41—121no—
286#41—121no—
287#41—121no—
288#41—121no—
289C2×Q8—61no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—22no—
293C2×D4—21no—
294C2×D4—22no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—44no—
298C2×D4—48no—
299C2×D4—48no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—61no—
309C2×D4—61no—
310C2×D4—61no—
311C2×D4—82no—
312C2×D4—84no—
313C2×D4—121no—
314C2×D4#21312yescomplemented
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—44no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—121no—
321C22×C4C2×#6711yescharacteristic
322C22×C4C2×#6711yescharacteristic
323C22×C4C2×#6711yescharacteristic
324SD16—124no—
325D8—124no—
326D8—244no—
327M16—121no—
328M16—121no—
329C2×C8—62no—
330C2×C8—62no—
331C2×C8—121no—
332C4⋊C4—24no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—61no—
336C4⋊C4—61no—
337C22⋊C4—48no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—121no—
341C22⋊C4—121no—
342C42—41no—
343C42—41no—
344C42D3×D411yescharacteristic
345C42D3×D411yescharacteristic
346C42D3×D411yescharacteristic
347C42#23711yescharacteristic
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—44no—
351C22×C3—42no—
352C22×C3—82no—
353C22×C3—84no—
354C22×C3—84no—
355C22×C3—81no—
356C22×C3—81no—
357C22×C3—82no—
358C22×C3—82no—
359C22×C3—82no—
360C22×C3#39211yescharacteristic
361C22×C3#44711yescharacteristic
362C22×C3#44911yescharacteristic
363C2×D3—81no—
364C2×D3—81no—
365C2×D3—81no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—44no—
370C3×C4—81no—
371C3×C4#44711yescharacteristic
372C3×C4#44911yescharacteristic
373C3×C4#45312yes—
374C3×C4#46611yescharacteristic
375C3×C4#46911yescharacteristic
376Dic3—81no—
377C23—22no—
378C23—42no—
379C23—42no—
380C23—44no—
381C23—44no—
382C23—41no—
383C23—41no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—121no—
388C23#95311yescharacteristic
389Q8—121no—
390Q8—121no—
391D4—22no—
392D4—22no—
393D4—41no—
394D4—41no—
395D4—44no—
396D4—82no—
397D4—82no—
398D4—82no—
399D4—816no—
400D4—81no—
401D4—81no—
402D4—81no—
403D4—81no—
404D4—121no—
405D4—121no—
406D4—122no—
407D4—241no—
408D4—241no—
409D4—241no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—121no—
420C2×C4—241no—
421C2×C4—241no—
422C2×C4C2×#21311yescharacteristic
423C2×C4#93211yescharacteristic
424C2×C4#93711yescharacteristic
425C2×C4#93711yescharacteristic
426C2×C4#93811yescharacteristic
427C2×C4#93911yescharacteristic
428C2×C4#93911yescharacteristic
429C2×C4#94011yescharacteristic
430C2×C4#94911yescharacteristic
431C2×C4#95011yescharacteristic
432C8—122no—
433C8—122no—
434C8—241no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—82no—
438C2×C3—162no—
439C2×C3#199911yescharacteristic
440C2×C3#200211yescharacteristic
441C2×C3#200811yescharacteristic
442D3—161no—
443C22—21no—
444C22—21no—
445C22—44no—
446C22—42no—
447C22—82no—
448C22—84no—
449C22—84no—
450C22—81no—
451C22—81no—
452C22—82no—
453C22—82no—
454C22—82no—
455C22—241no—
456C22—241no—
457C22—241no—
458C22#521711yescharacteristic
459C22#522311yescharacteristic
460C22#523811yescharacteristic
461C4—21no—
462C4—21no—
463C4—21no—
464C4—44no—
465C4—81no—
466C4—241no—
467C4#503611yescharacteristic
468C4#503711yescharacteristic
469C4#503811yescharacteristic
470C4#503911yescharacteristic
471C4#507712yes—
472C3#1023811yescharacteristic, complemented, Sylow
473C2—21no—
474C2—21no—
475C2—82no—
476C2—162no—
477C2—481no—
478C2#7168411yescharacteristic
479C2#7176911yescharacteristic
480C2#7185811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2541C2
socleC22×C3#392
hypercenter#2541D3
hypocenterC3#10238
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2541

Complements

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

NH ≅ G/Ntype
1. C3×#2541C2semidirect
2. #83166C2semidirect
3. #82307C2semidirect
4. #71759C2semidirect
5. #71597C2semidirect
6. #69469C2semidirect
7. #67620C2semidirect
34. #2541D3semidirect
72. #874D4semidirect
75. #869D4semidirect
84. C4⋊(C3⋊C8)D4semidirect
85. C4⋊(C3⋊C8)D4semidirect
133. C2×C3×D4D8semidirect
314. C2×D4#213semidirect
472. C3#10238semidirect

Maximal subgroups

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

subgroupindex
C3×#25412
#831662
#823072
#717592
#715972
#694692
#676202
#102383