Finite Group Databasebeta

← #10695945

Subgroups — #10695945

Order 768.

632 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 101 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×#2535C211yesmaximal, characteristic
2#82314C211yesmaximal, characteristic, complemented
3#82296C211yesmaximal, characteristic, complemented
4#71598C211yesmaximal, characteristic, complemented
5#71597C211yesmaximal, characteristic, complemented
6#69546C211yesmaximal, characteristic, complemented
7#67631C211yesmaximal, characteristic, complemented
8#10000—31nomaximal, Sylow
9C2×C3×#121—21no—
10C2×C3×C4×D4C2211yescharacteristic
11C2×C3×C4×D4C2211yescharacteristic
12#5470C2211yescharacteristic
13C3×#397—21no—
14C3×#392—21no—
15C3×#386—22no—
16C3×#382C2211yescharacteristic
17C3×#380—21no—
18#5218—21no—
19#5215—21no—
20#5019—21no—
21C4×#214—22no—
22#5009—22no—
23#5008—21no—
24#5008—21no—
25#5007—21no—
26#4971—21no—
27#4964—21no—
28#4925C2211yescharacteristic
29#4925C2211yescharacteristic
30#4924C2211yescharacteristic
31D4⋊(C3⋊C8)—22no—
32D4⋊(C3⋊C8)—22no—
33#4452—22no—
34#3101—31no—
35#3077—31no—
36#2535D311yescharacteristic
37#1518—31no—
38#1486—31no—
39#1478—31no—
40#1286—31no—
41C22×C3×D4—22no—
42C22×C3×D4D411yescharacteristic
43C22×C3×D4D411yescharacteristic
44C23×C3×C4—21no—
45C23×C3×C4—21no—
46C3×#121—42no—
47C3×#121—42no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C22⋊C4—22no—
60C2×C3×C22⋊C4—21no—
61C2×C3×C22⋊C4—21no—
62C2×C3×C22⋊C4—22no—
63C2×C3×C42C2311yescharacteristic
64C2×#214—22no—
65#926—21no—
66#924—21no—
67#924—21no—
68C2×#208—21no—
69#870—21no—
70#869D411yescharacteristic, complemented
71#869D411yescharacteristic, complemented
72C4×Dic6—21no—
73C4×Dic6—21no—
74C3×#95—22no—
75C3×#95—21no—
76C3×#95D412yes—
77#832—22no—
78#832—22no—
79#832—22no—
80#830—21no—
81#830—21no—
82#810—22no—
83#808—22no—
84#808—22no—
85#807—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)D412yescomplemented
90C4×C3⋊C8—21no—
91C4×C3⋊C8D412yescomplemented
92#533—31no—
93C2×#121—21no—
94C2×C4×D4C2×D311yescharacteristic
95C2×C4×D4C2×D311yescharacteristic
96#483—61no—
97#478—62no—
98#476—61no—
99#476—61no—
100#474—61no—
101#461—61no—
102#450—61no—
103C4×SD16—62no—
104#431—31no—
105#431—31no—
106#428—61no—
107#423—31no—
108#418—61no—
109#397—21no—
110#392—21no—
111#386—22no—
112#382C2×D311yescharacteristic
113#380—21no—
114#328—62no—
115D4⋊C8—62no—
116D4⋊C8—62no—
117C24×C3—21no—
118C24×C3—21no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4—42no—
127C2×C3×D4—44no—
128C2×C3×D4—44no—
129C2×C3×D4—82no—
130C2×C3×D4SD1612yescomplemented
131C22×C3×C4—22no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—41no—
135C22×C3×C4—41no—
136C22×C3×C4—41no—
137C22×C3×C4—41no—
138C22×C3×C4—41no—
139C22×C3×C4—41no—
140C22×C3×C4—41no—
141C22×C3×C4—41no—
142C22×C3×C4—82no—
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C2×Dic6—21no—
147C2×Dic6—21no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—41no—
153C3×C4⋊C4—41no—
154C3×C22⋊C4—44no—
155C3×C22⋊C4—41no—
156C3×C22⋊C4—41no—
157C3×C22⋊C4—41no—
158C3×C22⋊C4—41no—
159C3×C22⋊C4—82no—
160C3×C42C2×D411yescharacteristic
161C3×C42C2×D411yescharacteristic
162C3×C42C2×D411yescharacteristic
163C3×C42#4111yescharacteristic
164#217—41no—
165#214—44no—
166C4⋊Dic3—21no—
167C4⋊Dic3—21no—
168C4⋊Dic3—21no—
169C4⋊Dic3—21no—
170Dic3⋊C4—41no—
171Dic3⋊C4—41no—
172C4×Dic3—41no—
173#208—41no—
174#208—41no—
175C2×C3⋊C8—22no—
176C2×C3⋊C8—22no—
177C2×C3⋊C8—41no—
178C22×D4—22no—
179C22×D4#6711yescharacteristic
180C22×D4#6711yescharacteristic
181C23×C4—21no—
182C23×C4—21no—
183C2×SD16—62no—
184C2×M16—61no—
185C4⋊Q8—31no—
186C4⋊Q8—31no—
187#125—61no—
188C22⋊Q8—61no—
189C22⋊Q8—61no—
190#121—42no—
191#121—42no—
192C4×Q8—61no—
193C4×Q8—61no—
194C4×D4—21no—
195C4×D4—21no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—21no—
200C4×D4—21no—
201C4×D4—21no—
202#117—61no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4#6711yescharacteristic
205C2×C4⋊C4#6711yescharacteristic
206C2×C22⋊C4—22no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—22no—
210C2×C42C22×D311yescharacteristic
211#107—62no—
212#106—62no—
213#106—62no—
214C4⋊C8—32no—
215C4⋊C8—61no—
216C4⋊C8—61no—
217C4⋊C8—61no—
218#103—62no—
219#102—62no—
220#102—62no—
221#102—62no—
222C22⋊C8—61no—
223C22⋊C8—61no—
224C4×C8—32no—
225C4×C8—61no—
226#95—22no—
227#95—21no—
228#95#6712yes—
229C23×C3—22no—
230C23×C3—22no—
231C23×C3—41no—
232C23×C3—42no—
233C23×C3—42no—
234C23×C3—41no—
235C23×C3—41no—
236C23×C3—41no—
237C23×C3—41no—
238C23×C3—41no—
239C23×C3—41no—
240C23×C3—41no—
241C23×C3#12011yescharacteristic
242C3×D4—22no—
243C3×D4—22no—
244C3×D4—22no—
245C3×D4—22no—
246C3×D4—41no—
247C3×D4—41no—
248C3×D4—41no—
249C3×D4—41no—
250C3×D4—88no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—42no—
260C2×C3×C4—41no—
261C2×C3×C4—82no—
262C2×C3×C4—82no—
263C2×C3×C4—81no—
264C2×C3×C4—81no—
265C2×C3×C4—81no—
266C2×C3×C4—81no—
267C2×C3×C4#12011yescharacteristic
268C2×C3×C4#12011yescharacteristic
269C2×C3×C4#12111yescharacteristic
270C2×C3×C4#12111yescharacteristic
271C2×C3×C4#12111yescharacteristic
272C2×C3×C4#12711yescharacteristic
273C2×C3×C4C2×SD1611yescharacteristic
274C2×C3×C4#13511yescharacteristic
275C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
276C2×C3×C4#13711yescharacteristic
277C2×Dic3—41no—
278C2×Dic3—41no—
279Dic6—42no—
280Dic6—81no—
281C3⋊C8—42no—
282C3⋊C8—42no—
283C3⋊C8—81no—
284C24—21no—
285C24—21no—
286C2×Q8—61no—
287C2×Q8—61no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—42no—
296C2×D4—44no—
297C2×D4—44no—
298C2×D4—82no—
299C2×D4#21412yescomplemented
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—82no—
312C22×C4C2×#6711yescharacteristic
313C22×C4C2×#6711yescharacteristic
314C22×C4C2×#6711yescharacteristic
315SD16—124no—
316M16—121no—
317M16—121no—
318C2×C8—62no—
319C2×C8—62no—
320C2×C8—121no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—61no—
328C4⋊C4—61no—
329C4⋊C4—61no—
330C4⋊C4—61no—
331C4⋊C4—121no—
332C4⋊C4—121no—
333C22⋊C4—44no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—82no—
339C22⋊C4—121no—
340C42—121no—
341C42D3×D411yescharacteristic
342C42D3×D411yescharacteristic
343C42D3×D411yescharacteristic
344C42#23711yescharacteristic
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—42no—
348C22×C3—44no—
349C22×C3—42no—
350C22×C3—42no—
351C22×C3—42no—
352C22×C3—81no—
353C22×C3—82no—
354C22×C3—81no—
355C22×C3—81no—
356C22×C3—81no—
357C22×C3—81no—
358C22×C3—81no—
359C22×C3#39211yescharacteristic
360C22×C3#45011yescharacteristic
361C22×C3#45211yescharacteristic
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—82no—
367C3×C4—81no—
368C3×C4—81no—
369C3×C4#44811yescharacteristic
370C3×C4#44911yescharacteristic
371C3×C4#46511yescharacteristic
372C3×C4#46811yescharacteristic
373Dic3—81no—
374Dic3—81no—
375C23—22no—
376C23—22no—
377C23—41no—
378C23—42no—
379C23—42no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23#95311yescharacteristic
388Q8—122no—
389Q8—241no—
390D4—22no—
391D4—22no—
392D4—22no—
393D4—22no—
394D4—41no—
395D4—41no—
396D4—41no—
397D4—41no—
398D4—88no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—121no—
416C2×C4—121no—
417C2×C4#93211yescharacteristic
418C2×C4C2×#21411yescharacteristic
419C2×C4#93711yescharacteristic
420C2×C4#93711yescharacteristic
421C2×C4#93811yescharacteristic
422C2×C4#93911yescharacteristic
423C2×C4#93911yescharacteristic
424C2×C4#94011yescharacteristic
425C2×C4#95011yescharacteristic
426C2×C4#95111yescharacteristic
427C8—122no—
428C8—122no—
429C8—241no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—82no—
433C2×C3—82no—
434C2×C3#200011yescharacteristic
435C2×C3#200811yescharacteristic
436C2×C3#201011yescharacteristic
437C22—21no—
438C22—21no—
439C22—42no—
440C22—44no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—81no—
445C22—82no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22#521811yescharacteristic
452C22#522311yescharacteristic
453C22#523911yescharacteristic
454C4—21no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—82no—
459C4—81no—
460C4—81no—
461C4—241no—
462C4—241no—
463C4#503711yescharacteristic
464C4#504011yescharacteristic
465C4#504111yescharacteristic
466C4#504211yescharacteristic
467C3#1000011yescharacteristic, complemented, Sylow
468C2—21no—
469C2—21no—
470C2—82no—
471C2—82no—
472C2#7168511yescharacteristic
473C2#7171511yescharacteristic
474C2#7185811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#2535C2
socleC22×C3#392
hypercenter#2535D3
hypocenterC3#10000
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2535

Complements

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

NH ≅ G/Ntype
2. #82314C2semidirect
3. #82296C2semidirect
4. #71598C2semidirect
5. #71597C2semidirect
6. #69546C2semidirect
7. #67631C2semidirect
70. #869D4semidirect
71. #869D4semidirect
89. C4⋊(C3⋊C8)D4semidirect
91. C4×C3⋊C8D4semidirect
130. C2×C3×D4SD16semidirect
299. C2×D4#214semidirect
467. C3#10000semidirect

Maximal subgroups

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

subgroupindex
C3×#25352
#823142
#822962
#715982
#715972
#695462
#676312
#100003