Finite Group Databasebeta

← #10692834

Subgroups — #10692834

Order 768.

770 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 770 conjugacy classes of subgroups — too many to draw. Only its 103 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
1#84092C211yesmaximal, characteristic
2#84074C211yesmaximal, characteristic, complemented
3C3×#1782C211yesmaximal, characteristic, complemented
4#69531C212yesmaximal, complemented
5#67628C212yesmaximal, complemented
6#9445—31nomaximal, Sylow
7C22×C3×#41C2211yescharacteristic
8C2×#947C411yescharacteristic, complemented
9C2×#940C411yescharacteristic, complemented
10C2×#930C2211yescharacteristic
11C22×#208—21no—
12C2×C3×#99—21no—
13C2×C3×#99—21no—
14C3×#381C2212yes—
15#5234—24no—
16#5229C411yescharacteristic, complemented
17#5223—21no—
18#5223—21no—
19#5221C411yescharacteristic, complemented
20#5213—21no—
21#5213—21no—
22C4×#217C2211yescharacteristic
23#5207—21no—
24#5207C2212yes—
25#4451—24no—
26#2879—31no—
27#2431—31no—
28#1782D311yescharacteristic, complemented
29#1510—32no—
30#1284—32no—
31C2×C3×#41—22no—
32C2×C3×#41—22no—
33C2×C3×#41—42no—
34C2×C3×#41—42no—
35C22×C3×Q8C2×C411yescharacteristic
36C22×C3×D4C2×C411yescharacteristic
37C22×C3×D4D411yescharacteristic
38C22×C3×D4D411yescharacteristic
39C23×C3×C4D411yescharacteristic
40C23×C3×C4D411yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C22×#67—21no—
43C22×#67—21no—
44C22×#67—21no—
45C22×D12C2×C411yescharacteristic
46C2×C3×C22⋊C4—22no—
47C2×#217—21no—
48C2×#217—21no—
49C2×#217—21no—
50#947—42no—
51#940—42no—
52#930—22no—
53#930—22no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4—21no—
57C2×C4⋊Dic3C2×C411yescharacteristic
58C2×C4×Dic3C2×C411yescharacteristic
59C2×C4×Dic3C2×C411yescharacteristic
60C2×#208—22no—
61C2×#208—22no—
62C2×#208—42no—
63C2×#208—42no—
64C22×C3⋊C8—22no—
65C3×#99—42no—
66C3×#99—42no—
67C3×#95—22no—
68C3×#95—22no—
69C3⋊#95—21no—
70C3⋊#95—21no—
71C3⋊#95—21no—
72#830—24no—
73#830—22no—
74#830D414yescomplemented
75C22×#41C2×D311yescharacteristic
76C22×M16—61no—
77C2×#127—31no—
78C2×#124—31no—
79C2×#121—31no—
80C2×#99—21no—
81C2×#99—21no—
82#408—64no—
83#407—32no—
84#407—61no—
85#392—61no—
86#392—61no—
87#391—31no—
88#390—31no—
89#386—61no—
90#386—61no—
91#381C2×D312yes—
92C4×C22⋊C4—31no—
93#327—64no—
94C24×C3C22⋊C411yescharacteristic
95C24×C3C22⋊C411yescharacteristic
96C24×C3C2×D411yescharacteristic
97C23×D3—21no—
98C3×#41—44no—
99C3×#41—44no—
100C3×#41—48no—
101C2×C3×Q8—22no—
102C2×C3×Q8—22no—
103C2×C3×Q8—41no—
104C2×C3×D4—22no—
105C2×C3×D4—22no—
106C2×C3×D4—22no—
107C2×C3×D4—22no—
108C2×C3×D4—22no—
109C2×C3×D4—22no—
110C2×C3×D4—41no—
111C2×C3×D4—41no—
112C2×C3×D4—41no—
113C22×C3×C4—22no—
114C22×C3×C4—22no—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4—41no—
120C22×C3×C4—41no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4—42no—
124C22×C3×C4C22⋊C411yescharacteristic
125C22×C3×C4C22⋊C411yescharacteristic
126C22×C3×C4C22×C411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D412yes—
129C22×C3×C4#4112yes—
130C2×#67—42no—
131C2×#67—42no—
132C2×#67—42no—
133C2×#67—42no—
134C2×#67—42no—
135C2×#67—81no—
136C2×#67—81no—
137C2×#67—81no—
138C2×#67—81no—
139C22×Dic3—21no—
140C22×Dic3—21no—
141C22×Dic3—21no—
142C2×D12—22no—
143C2×D12—41no—
144C2×D12—41no—
145C3×C22⋊C4—42no—
146C3×C22⋊C4—42no—
147#217—42no—
148#217—81no—
149#217—81no—
150(C2×D3)⋊C4—42no—
151(C2×D3)⋊C4—42no—
152(C2×D3)⋊C4—41no—
153(C2×D3)⋊C4—41no—
154C4⋊Dic3—22no—
155C4×Dic3—41no—
156C4×Dic3—41no—
157#208—44no—
158#208—42no—
159#208—42no—
160C2×C3⋊C8—24no—
161C2×C3⋊C8—44no—
162C2×#41—22no—
163C2×#41—22no—
164C2×#41—42no—
165C2×#41—42no—
166C22×Q8C4×D311yescharacteristic
167C22×D4—31no—
168C22×D4—61no—
169C22×D4—61no—
170C22×D4—61no—
171C22×D4C4×D311yescharacteristic
172C22×D4#6711yescharacteristic
173C22×D4#6711yescharacteristic
174C23×C4D1211yescharacteristic
175C23×C4D1211yescharacteristic
176C23×C4C22×D311yescharacteristic
177C2×M16—62no—
178C2×M16—62no—
179C2×M16—122no—
180C2×M16—122no—
181C22×C8—62no—
182#127—122no—
183#124—122no—
184#121—62no—
185#121—62no—
186C2×C4⋊C4—31no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—61no—
189C2×C22⋊C4—61no—
190C2×C22⋊C4—61no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4—61no—
193C2×C22⋊C4—61no—
194C2×C42—31no—
195C2×C42—31no—
196#99—42no—
197#99—42no—
198C22⋊C8—34no—
199C22⋊C8—64no—
200C22⋊C8—62no—
201#95—22no—
202#95—22no—
203#95—61no—
204#95—61no—
205#95—61no—
206C23×C3—21no—
207C23×C3—21no—
208C23×C3—21no—
209C23×C3—21no—
210C23×C3—21no—
211C23×C3—21no—
212C23×C3—21no—
213C23×C3—41no—
214C23×C3—41no—
215C23×C3—41no—
216C23×C3—41no—
217C23×C3—41no—
218C23×C3—41no—
219C23×C3#9912yes—
220C23×C3C2×C22⋊C411yescharacteristic
221C23×C3#12112yes—
222C22×D3—41no—
223C22×D3—41no—
224C22×D3—41no—
225C22×D3—41no—
226C22×D3—41no—
227C22×D3—41no—
228C22×D3—41no—
229C3×Q8—44no—
230C3×D4—44no—
231C3×D4—44no—
232C3×D4—44no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—24no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4—82no—
250C2×C3×C4—82no—
251C2×C3×C4—82no—
252C2×C3×C4#9912yes—
253C2×C3×C4C4×D412yes—
254C2×C3×C4#12011yescharacteristic
255C2×C3×C4#12311yescharacteristic
256#67—84no—
257#67—84no—
258#67—84no—
259C2×Dic3—22no—
260C2×Dic3—41no—
261C2×Dic3—41no—
262C2×Dic3—41no—
263C2×Dic3—41no—
264C2×Dic3—41no—
265C2×Dic3—41no—
266C2×Dic3—41no—
267C2×Dic3—41no—
268D12—82no—
269C3⋊C8—44no—
270C24—61no—
271C24(C2×D3)⋊C411yescharacteristic
272C24(C2×D3)⋊C411yescharacteristic
273C24C2×#6711yescharacteristic
274#41—44no—
275#41—44no—
276#41—48no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—41no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—62no—
290C2×D4—121no—
291C2×D4—121no—
292C2×D4—122no—
293C2×D4—122no—
294C2×D4—122no—
295C2×D4—122no—
296C2×D4—122no—
297C2×D4—241no—
298C2×D4—241no—
299C2×D4—241no—
300C2×D4—241no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—61no—
313C22×C4—61no—
314C22×C4—61no—
315C22×C4(C2×D3)⋊C411yescharacteristic
316C22×C4(C2×D3)⋊C411yescharacteristic
317C22×C4C2×C4×D311yescharacteristic
318C22×C4C2×D1211yescharacteristic
319C22×C4D3×D412yes—
320C22×C4#23712yes—
321M16—124no—
322M16—122no—
323M16—122no—
324C2×C8—64no—
325C2×C8—124no—
326C4⋊C4—62no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—122no—
330C22⋊C4—122no—
331C22⋊C4—122no—
332C22⋊C4—121no—
333C22⋊C4—121no—
334C22⋊C4—241no—
335C22⋊C4—241no—
336C42—121no—
337C42—121no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—22no—
341C22×C3—22no—
342C22×C3—22no—
343C22×C3—22no—
344C22×C3—42no—
345C22×C3—42no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—81no—
352C22×C3—81no—
353C22×C3—81no—
354C22×C3—81no—
355C22×C3#38611yescharacteristic
356C22×C3C2×#9911yescharacteristic
357C22×C3C2×#9911yescharacteristic
358C2×D3—81no—
359C2×D3—81no—
360C2×D3—81no—
361C2×D3—81no—
362C2×D3—81no—
363C2×D3—81no—
364C2×D3—81no—
365C3×C4—21no—
366C3×C4—22no—
367C3×C4—21no—
368C3×C4—22no—
369C3×C4—41no—
370C3×C4—162no—
371Dic3—42no—
372Dic3—81no—
373Dic3—81no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—121no—
388C23—121no—
389C23—121no—
390C23—121no—
391C23—121no—
392C23—121no—
393C23—121no—
394C23#80612yes—
395C23C2×(C2×D3)⋊C411yescharacteristic
396C23#93912yes—
397Q8—44no—
398D4—44no—
399D4—44no—
400D4—44no—
401D4—242no—
402D4—244no—
403D4—244no—
404D4—244no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—62no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—121no—
426C2×C4—121no—
427C2×C4—121no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4#80612yes—
434C2×C4#67⋊C412yes—
435C2×C4#88211yescharacteristic
436C2×C4#88611yescharacteristic
437C8—124no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—42no—
441C2×C3—42no—
442C2×C3—81no—
443C2×C3#188411yescharacteristic
444C2×C3#189811yescharacteristic
445C2×C3#191011yescharacteristic
446D3—161no—
447C22—21no—
448C22—21no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—42no—
454C22—42no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—81no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22—241no—
465C22—241no—
466C22—241no—
467C22—241no—
468C22—241no—
469C22—241no—
470C22—241no—
471C22C2×#80611yescharacteristic
472C22C2×#80611yescharacteristic
473C22#495811yescharacteristic
474C4—21no—
475C4—21no—
476C4—22no—
477C4—22no—
478C4—41no—
479C4—122no—
480C4—162no—
481C4—241no—
482C4—241no—
483C3#944511yescharacteristic, complemented, Sylow
484C2—21no—
485C2—21no—
486C2—42no—
487C2—42no—
488C2—81no—
489C2—481no—
490C2#6949611yescharacteristic
491C2#7004511yescharacteristic
492C2#7004711yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1782C2
socleC22×C3#386
prime radical#84092C2
cosocleC23×C3×C4C23
hypercenter#1782D3
hypocenterC3#9445
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1782

Complements

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

NH ≅ G/Ntype
2. #84074C2semidirect
3. C3×#1782C2semidirect
4. #69531C2semidirect
5. #67628C2semidirect
8. C2×#947C4semidirect
9. C2×#940C4semidirect
16. #5229C4semidirect
19. #5221C4semidirect
28. #1782D3semidirect
74. #830D4semidirect
483. C3#9445semidirect

Maximal subgroups

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

subgroupindex
#840922
#840742
C3×#17822
#695312
#676282
#94453