Finite Group Databasebeta

← #10705641

Subgroups — #10705641

Order 768.

778 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 778 conjugacy classes of subgroups — too many to draw. Only its 87 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#82125C211yesmaximal, characteristic
2C2×C3×#342C211yesmaximal, characteristic, complemented
3#71539C211yesmaximal, characteristic, complemented
4#67490C212yesmaximal, complemented
5#67489C212yesmaximal, complemented
6#11753—31nomaximal, Sylow
7C22×#235C411yescharacteristic, complemented
8C22×C3×C22⋊C4—21no—
9C22×C3×C22⋊C4C2211yescharacteristic
10C2×#953—21no—
11#5213—21no—
12#5211C411yescharacteristic, complemented
13#5207—21no—
14#4960C411yescharacteristic, complemented
15#4960C2211yescharacteristic
16#4956—21no—
17#4956—21no—
18#4947C411yescharacteristic, complemented
19#4947C2211yescharacteristic
20C3×#342—21no—
21C3×#342C2212yes—
22#4525—21no—
23#4525C2212yes—
24#2404—31no—
25C2×#342D311yescharacteristic, complemented
26#1462—31no—
27#1351—32no—
28#1343—32no—
29C25×C3D411yescharacteristic
30C23×C3×C4—41no—
31C23×C3×C4C2×C411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C22×#67—21no—
34C22×#67—21no—
35C22×#67—21no—
36C2×#235—41no—
37C2×#235—41no—
38C2×#235—44no—
39C22×D12C2×C411yescharacteristic, complemented
40C22×C4×D3—21no—
41C22×Dic6C2×C411yescharacteristic, complemented
42C2×C3×C22⋊C4—24no—
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4—41no—
45C2×C3×C22⋊C4—41no—
46C2×C3×C22⋊C4—41no—
47C2×C3×C22⋊C4—42no—
48C2×C3×C22⋊C4C2×C411yescharacteristic
49C2×C3×C22⋊C4D411yescharacteristic
50C2×C3×C22⋊C4D411yescharacteristic
51C2×C3×C22⋊C4C2311yescharacteristic
52#953—44no—
53C2×#217—21no—
54C2×#217—21no—
55C2×#217—21no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4—21no—
58C2×#208—41no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95C2×C411yescharacteristic, complemented
63C3⋊#95C2×C411yescharacteristic, complemented
64#830—22no—
65#830—41no—
66C22×#41—31no—
67C2×#120—61no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4C2×D311yescharacteristic
70#407—61no—
71#393—31no—
72#393—31no—
73#392—31no—
74#392—31no—
75#386—61no—
76#386—61no—
77#386—61no—
78#381—31no—
79#342—21no—
80#342C2×D312yes—
81#323—32no—
82#323—61no—
83C24×C3—21no—
84C24×C3—21no—
85C24×C3—22no—
86C24×C3—21no—
87C24×C3—21no—
88C24×C3—42no—
89C24×C3—81no—
90C24×C3C22⋊C411yescharacteristic
91C24×C3C22⋊C411yescharacteristic
92C24×C3C2×D411yescharacteristic
93C23×D3—21no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4—42no—
99C22×C3×C4—44no—
100C22×C3×C4—42no—
101C22×C3×C4—81no—
102C22×C3×C4—82no—
103C22×C3×C4—82no—
104C22×C3×C4C22⋊C411yescharacteristic
105C22×C3×C4C22⋊C411yescharacteristic
106C22×C3×C4C22×C411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C2×#67—24no—
109C2×#67—42no—
110C2×#67—41no—
111C2×#67—41no—
112C2×#67—42no—
113C2×#67—42no—
114C2×#67—42no—
115C2×#67—41no—
116C2×#67—41no—
117C2×#67—81no—
118C2×#67—81no—
119C22×Dic3—21no—
120C22×Dic3—21no—
121C22×Dic3—21no—
122#235—44no—
123#235—44no—
124#235—44no—
125#235—44no—
126C2×D12—41no—
127C2×D12—42no—
128C2×C4×D3—42no—
129C2×C4×D3—42no—
130C2×C4×D3—42no—
131C2×Dic6—41no—
132C2×Dic6—42no—
133C3×C22⋊C4—24no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C22⋊C4—44no—
139C3×C22⋊C4—42no—
140#217—42no—
141#217—42no—
142#217—42no—
143(C2×D3)⋊C4—81no—
144(C2×D3)⋊C4—81no—
145#208—82no—
146C2×C3⋊C8—42no—
147C25#6711yescharacteristic
148C2×#41—121no—
149C2×#41—121no—
150C2×#41—124no—
151C22×Q8—31no—
152C22×D4—31no—
153C22×D4—61no—
154C22×D4—61no—
155C22×D4—61no—
156C23×C4—41no—
157C23×C4—61no—
158C23×C4C4×D311yescharacteristic
159C23×C4D1211yescharacteristic
160C2×M16—121no—
161#120—124no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—42no—
168C2×C22⋊C4—61no—
169C2×C22⋊C4—61no—
170C2×C22⋊C4—61no—
171C2×C22⋊C4—61no—
172C2×C22⋊C4—61no—
173C2×C22⋊C4C4×D311yescharacteristic
174C2×C22⋊C4D1211yescharacteristic
175C2×C22⋊C4#6711yescharacteristic
176C2×C22⋊C4C22×D311yescharacteristic
177C22⋊C8—62no—
178C22⋊C8—121no—
179#95—31no—
180#95—31no—
181#95—61no—
182#95—61no—
183#95—61no—
184C23×C3—22no—
185C23×C3—22no—
186C23×C3—22no—
187C23×C3—22no—
188C23×C3—22no—
189C23×C3—21no—
190C23×C3—21no—
191C23×C3—21no—
192C23×C3—21no—
193C23×C3—21no—
194C23×C3—21no—
195C23×C3—41no—
196C23×C3—42no—
197C23×C3—41no—
198C23×C3—41no—
199C23×C3—41no—
200C23×C3—41no—
201C23×C3—42no—
202C23×C3—81no—
203C23×C3—81no—
204C23×C3—84no—
205C23×C3—82no—
206C23×C3—82no—
207C23×C3#9911yescharacteristic
208C23×C3#9911yescharacteristic
209C23×C3#10412yes—
210C23×C3C2×C22⋊C411yescharacteristic
211C23×C3AGL(1,ℤ/8)11yescharacteristic
212C23×C3#13711yescharacteristic
213C22×D3—22no—
214C22×D3—41no—
215C22×D3—41no—
216C22×D3—41no—
217C22×D3—41no—
218C22×D3—41no—
219C22×D3—41no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—42no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—44no—
232C2×C3×C4—81no—
233C2×C3×C4—88no—
234C2×C3×C4—82no—
235C2×C3×C4—82no—
236#67—44no—
237#67—44no—
238#67—84no—
239#67—82no—
240#67—82no—
241C2×Dic3—22no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250D12—42no—
251D12—42no—
252C4×D3—48no—
253Dic6—42no—
254Dic6—42no—
255C3⋊C8—82no—
256C24—21no—
257C24—22no—
258C24—21no—
259C24—21no—
260C24—21no—
261C24—42no—
262C24—61no—
263C24—81no—
264C24(C2×D3)⋊C411yescharacteristic
265C24(C2×D3)⋊C411yescharacteristic
266C24C2×#6711yescharacteristic
267#41—124no—
268#41—124no—
269#41—124no—
270#41—124no—
271C2×Q8—121no—
272C2×Q8—122no—
273C2×D4—64no—
274C2×D4—121no—
275C2×D4—122no—
276C2×D4—122no—
277C2×D4—121no—
278C2×D4—121no—
279C2×D4—122no—
280C2×D4—122no—
281C2×D4—122no—
282C2×D4—121no—
283C2×D4—121no—
284C2×D4—241no—
285C2×D4—241no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—42no—
291C22×C4—44no—
292C22×C4—42no—
293C22×C4—61no—
294C22×C4—61no—
295C22×C4—61no—
296C22×C4—81no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4—122no—
300C22×C4—122no—
301C22×C4—122no—
302C22×C4(C2×D3)⋊C411yescharacteristic
303C22×C4(C2×D3)⋊C411yescharacteristic
304C22×C4C2×C4×D311yescharacteristic
305C22×C4C2×D1211yescharacteristic
306M16—242no—
307C2×C8—122no—
308C22⋊C4—24no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—44no—
314C22⋊C4—42no—
315C22⋊C4—122no—
316C22⋊C4—122no—
317C22⋊C4—122no—
318C22⋊C4—241no—
319C22⋊C4—241no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—22no—
329C22×C3—21no—
330C22×C3—42no—
331C22×C3—42no—
332C22×C3—42no—
333C22×C3—41no—
334C22×C3—41no—
335C22×C3—44no—
336C22×C3—44no—
337C22×C3—41no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—41no—
341C22×C3—81no—
342C22×C3—81no—
343C22×C3—82no—
344C22×C3—82no—
345C22×C3AΣL(1,𝔽16)12yes—
346C22×C3C2×#9911yescharacteristic
347C22×C3#41711yescharacteristic
348C22×C3C2×#10411yescharacteristic
349C2×D3—42no—
350C2×D3—42no—
351C2×D3—42no—
352C2×D3—81no—
353C2×D3—81no—
354C2×D3—81no—
355C2×D3—81no—
356C3×C4—41no—
357C3×C4—41no—
358C3×C4—41no—
359C3×C4—41no—
360C3×C4—84no—
361Dic3—42no—
362Dic3—81no—
363Dic3—81no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—22no—
373C23—21no—
374C23—21no—
375C23—41no—
376C23—42no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—42no—
382C23—62no—
383C23—81no—
384C23—81no—
385C23—84no—
386C23—82no—
387C23—82no—
388C23—121no—
389C23—121no—
390C23—121no—
391C23—121no—
392C23—121no—
393C23—121no—
394C23#80512yes—
395C23#80611yescharacteristic
396C23#80611yescharacteristic
397C23C2×(C2×D3)⋊C411yescharacteristic
398C23#93211yescharacteristic
399C23#94211yescharacteristic
400Q8—122no—
401Q8—122no—
402D4—122no—
403D4—122no—
404D4—124no—
405D4—124no—
406D4—244no—
407D4—242no—
408D4—242no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—44no—
421C2×C4—62no—
422C2×C4—81no—
423C2×C4—88no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—128no—
427C2×C4—121no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—121no—
434C2×C4—121no—
435C8—242no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—22no—
439C2×C3—41no—
440C2×C3—41no—
441C2×C3—42no—
442C2×C3—41no—
443C2×C3#150411yescharacteristic
444C2×C3C2×AΣL(1,𝔽16)11yescharacteristic
445C2×C3#212711yescharacteristic
446D3—82no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—22no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—41no—
461C22—41no—
462C22—44no—
463C22—44no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—81no—
469C22—81no—
470C22—82no—
471C22—82no—
472C22—122no—
473C22—122no—
474C22—122no—
475C22—241no—
476C22—241no—
477C22—241no—
478C22—241no—
479C22#447112yes—
480C22C2×#80511yescharacteristic
481C22C2×#80611yescharacteristic
482C22#496611yescharacteristic
483C4—41no—
484C4—41no—
485C4—41no—
486C4—41no—
487C4—84no—
488C4—122no—
489C4—241no—
490C4—241no—
491C3#1175311yescharacteristic, complemented, Sylow
492C2—21no—
493C2—21no—
494C2—22no—
495C2—41no—
496C2—41no—
497C2—42no—
498C2—41no—
499C2—242no—
500C2#6939811yescharacteristic
501C2C2×#447111yescharacteristic
502C2#7001911yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C22⋊C4C22×D3
FittingC2×C3×#342C2
socleC22×C3C2×#99
prime radical#82125C2
cosocleC2×C3×C22⋊C4C23
hypercenterC2×#342D3
hypocenterC3#11753
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#342

Complements

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

NH ≅ G/Ntype
2. C2×C3×#342C2semidirect
3. #71539C2semidirect
4. #67490C2semidirect
5. #67489C2semidirect
7. C22×#235C4semidirect
12. #5211C4semidirect
14. #4960C4semidirect
18. #4947C4semidirect
25. C2×#342D3semidirect
39. C22×D12C2×C4semidirect
41. C22×Dic6C2×C4semidirect
62. C3⋊#95C2×C4semidirect
63. C3⋊#95C2×C4semidirect
491. C3#11753semidirect

Maximal subgroups

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

subgroupindex
#821252
C2×C3×#3422
#715392
#674902
#674892
#117533