Finite Group Databasebeta

← #10707832

Subgroups — #10707832

Order 768.

544 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 55 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#77705C211yesmaximal, characteristic, complemented
2C3×#1607C211yesmaximal, characteristic, complemented
3#71851C211yesmaximal, characteristic
4#71844C211yesmaximal, characteristic, complemented
5#67649C211yesmaximal, characteristic, complemented
6#67451C211yesmaximal, characteristic, complemented
7#67444C211yesmaximal, characteristic, complemented
8#12192—31nomaximal, Sylow
9C3×#534C2211yescharacteristic
10C2×#937—21no—
11C3×#480—21no—
12C3×#397C2211yescharacteristic
13C2×#834—21no—
14#5034—21no—
15#4994—21no—
16#4955—21no—
17#4955C2211yescharacteristic
18#4951—21no—
19#4949C2211yescharacteristic
20#4674—21no—
21#4670—21no—
22#4669—21no—
23C3×#327—21no—
24C3×#323C2211yescharacteristic
25#4537—21no—
26#4526C2211yescharacteristic
27#4525C2211yescharacteristic
28#2735—31no—
29#2021—31no—
30#1613—31no—
31#1607D311yescharacteristic, complemented
32#1352—31no—
33#1344—31no—
34#1316—31no—
35C24×D3D411yescharacteristic
36C22×C3×D4—21no—
37C22×#67—41no—
38C3×#123—41no—
39C3×#121—21no—
40C3×#121—21no—
41C3×#120—41no—
42C2×C3×C4⋊C4D411yescharacteristic, complemented
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4C2311yescharacteristic
45C2×#217—21no—
46C2×#217—21no—
47C2×#217D411yescharacteristic
48#937—42no—
49#937—42no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4—21no—
53C2×(C2×D3)⋊C4—21no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4—41no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3D411yescharacteristic, complemented
59C2×Dic3⋊C4—21no—
60C2×C4×Dic3—21no—
61C3×#107—41no—
62C3×#102—41no—
63C3×C22⋊C8—21no—
64C3×#95—21no—
65C3×#95D411yescharacteristic, complemented
66#834—42no—
67#832—41no—
68C3⋊#95—21no—
69C3⋊#95—21no—
70C3⋊#95D411yescharacteristic, complemented
71#830—21no—
72#808—41no—
73#534C2×D311yescharacteristic
74C2×#120—61no—
75#481—61no—
76#480—21no—
77C2×#99—61no—
78#399—31no—
79#397—61no—
80#397C2×D311yescharacteristic
81#396—61no—
82#388—61no—
83#388—61no—
84#388—61no—
85#379—31no—
86#379—61no—
87#342—31no—
88#327—21no—
89#327—61no—
90#323—31no—
91#323C2×D311yescharacteristic
92C24×C3—21no—
93C24×C3C2×D411yescharacteristic
94C23×D3—21no—
95C23×D3—21no—
96C23×D3—21no—
97C23×D3—42no—
98C23×D3—41no—
99C23×D3—41no—
100C23×D3—81no—
101C2×C3×D4—21no—
102C2×C3×D4—21no—
103C2×C3×D4—41no—
104C2×C3×D4—41no—
105C2×C3×D4—41no—
106C2×C3×D4—41no—
107C2×C3×D4—41no—
108C2×C3×D4—41no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C2×#67—44no—
115C2×#67—82no—
116C2×#67—82no—
117C22×Dic3—21no—
118C22×Dic3—21no—
119C22×Dic3—21no—
120C22×Dic3—41no—
121C2×C3×C8—41no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—41no—
125C3×C22⋊C4—41no—
126C3×C22⋊C4—41no—
127C3×C22⋊C4—41no—
128C3×C22⋊C4—41no—
129C3×C22⋊C4—41no—
130#217—22no—
131#217—41no—
132#217—41no—
133#217—81no—
134(C2×D3)⋊C4—42no—
135(C2×D3)⋊C4—42no—
136(C2×D3)⋊C4—44no—
137(C2×D3)⋊C4—81no—
138(C2×D3)⋊C4—81no—
139(C2×D3)⋊C4—81no—
140(C2×D3)⋊C4—81no—
141C4⋊Dic3—41no—
142C4⋊Dic3—81no—
143Dic3⋊C4—81no—
144C4×Dic3—81no—
145C2×C3⋊C8—41no—
146C25—31no—
147C22×D4—21no—
148C22×D4—121no—
149#123—41no—
150#121—21no—
151#121—21no—
152#120—41no—
153#120—122no—
154#120—122no—
155C2×C4⋊C4—31no—
156C2×C4⋊C4—61no—
157C2×C4⋊C4—61no—
158C2×C4⋊C4#6711yescharacteristic, complemented
159C2×C22⋊C4—21no—
160C2×C22⋊C4—31no—
161C2×C22⋊C4—61no—
162C2×C22⋊C4—61no—
163C2×C22⋊C4—61no—
164C2×C22⋊C4—61no—
165C2×C22⋊C4—61no—
166C2×C22⋊C4—61no—
167C2×C22⋊C4—61no—
168C2×C22⋊C4—61no—
169C2×C22⋊C4—121no—
170C2×C22⋊C4C22×D311yescharacteristic
171C2×C42—61no—
172#107—41no—
173#106—121no—
174#102—41no—
175#102—121no—
176#99—122no—
177C22⋊C8—21no—
178C22⋊C8—61no—
179#95—21no—
180#95—31no—
181#95—61no—
182#95—61no—
183#95#6711yescharacteristic, complemented
184C23×C3—21no—
185C23×C3—21no—
186C23×C3—41no—
187C23×C3—41no—
188C23×C3—41no—
189C23×C3—41no—
190C23×C3—41no—
191C23×C3—81no—
192C23×C3—81no—
193C23×C3#12011yescharacteristic
194C23×C3#13511yescharacteristic
195C23×C3AGL(1,ℤ/8)11yescharacteristic
196C22×D3—22no—
197C22×D3—42no—
198C22×D3—42no—
199C22×D3—41no—
200C22×D3—41no—
201C22×D3—41no—
202C22×D3—42no—
203C22×D3—41no—
204C22×D3—41no—
205C22×D3—41no—
206C22×D3—82no—
207C22×D3—81no—
208C22×D3—84no—
209C22×D3—82no—
210C22×D3—81no—
211C22×D3—81no—
212C3×D4—81no—
213C3×D4—81no—
214C3×D4—81no—
215C3×D4—81no—
216C3×D4—81no—
217C3×D4—81no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231#67—88no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—42no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—81no—
243C2×Dic3—81no—
244C3×C8—81no—
245C3⋊C8—81no—
246C24—21no—
247C24—61no—
248C24—61no—
249C24—61no—
250C24—121no—
251C24—121no—
252C24—122no—
253C24—241no—
254C24D3×D411yescharacteristic
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—124no—
264C2×D4—242no—
265C2×D4—242no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—61no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4—121no—
273C22×C4D3×D411yescharacteristic
274C22×C4C2×#6711yescharacteristic
275C2×C8—41no—
276C2×C8—121no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—41no—
280C4⋊C4—121no—
281C4⋊C4—241no—
282C4⋊C4—241no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—62no—
289C22⋊C4—122no—
290C22⋊C4—122no—
291C22⋊C4—124no—
292C22⋊C4—121no—
293C22⋊C4—121no—
294C22⋊C4—241no—
295C22⋊C4—241no—
296C22⋊C4—241no—
297C22⋊C4—241no—
298C22⋊C4—241no—
299C42—241no—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3—41no—
308C22×C3—81no—
309C22×C3—82no—
310C22×C3—81no—
311C22×C3—81no—
312C22×C3—81no—
313C22×C3—161no—
314C22×C3#44911yescharacteristic
315C22×C3#45311yescharacteristic
316C22×C3B(4,𝔽2)11yescharacteristic
317C2×D3—42no—
318C2×D3—42no—
319C2×D3—44no—
320C2×D3—42no—
321C2×D3—81no—
322C2×D3—81no—
323C2×D3—82no—
324C2×D3—82no—
325C2×D3—81no—
326C2×D3—81no—
327C2×D3—82no—
328C3×C4—41no—
329C3×C4—41no—
330C3×C4—81no—
331C3×C4—81no—
332C3×C4—81no—
333Dic3—81no—
334Dic3—81no—
335Dic3—82no—
336Dic3—81no—
337C23—21no—
338C23—21no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—62no—
345C23—81no—
346C23—81no—
347C23—122no—
348C23—121no—
349C23—121no—
350C23—121no—
351C23—122no—
352C23—122no—
353C23—121no—
354C23—121no—
355C23—121no—
356C23—241no—
357C23—242no—
358C23—241no—
359C23—242no—
360C23—241no—
361C23—244no—
362C23#91111yescharacteristic
363C23#91211yescharacteristic
364C23#93711yescharacteristic
365D4—81no—
366D4—81no—
367D4—81no—
368D4—81no—
369D4—81no—
370D4—81no—
371D4—248no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4—122no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4—121no—
392C2×C4—121no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—241no—
396C2×C4—241no—
397C8—81no—
398C8—241no—
399C2×C3—21no—
400C2×C3—21no—
401C2×C3—41no—
402C2×C3—41no—
403C2×C3—161no—
404C2×C3#160011yescharacteristic
405C2×C3#219711yescharacteristic
406C2×C3#220011yescharacteristic
407D3—82no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—81no—
417C22—82no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—122no—
422C22—122no—
423C22—124no—
424C22—122no—
425C22—161no—
426C22—241no—
427C22—241no—
428C22—242no—
429C22—242no—
430C22—242no—
431C22—241no—
432C22—241no—
433C22#503211yescharacteristic
434C22#515811yescharacteristic
435C22#519811yescharacteristic
436C4—41no—
437C4—41no—
438C4—81no—
439C4—81no—
440C4—81no—
441C4—241no—
442C4—241no—
443C4—242no—
444C4—241no—
445C3#1219211yescharacteristic, complemented, Sylow
446C2—21no—
447C2—21no—
448C2—41no—
449C2—41no—
450C2—161no—
451C2—242no—
452C2#7096911yescharacteristic
453C2#7099011yescharacteristic
454C2#7100111yescharacteristic
subgroupG/H
centerC22#5032
commutator, cosocleC2×C3×C22⋊C4C23
FrattiniC2×C22⋊C4C22×D3
FittingC3×#1607C2
socleC22×C3B(4,𝔽2)
prime radical#71851C2
hypercenter#1607D3
hypocenterC3#12192
LCSG ▷ C2×C3×C22⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1607

Complements

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

NH ≅ G/Ntype
1. #77705C2semidirect
2. C3×#1607C2semidirect
4. #71844C2semidirect
5. #67649C2semidirect
6. #67451C2semidirect
7. #67444C2semidirect
31. #1607D3semidirect
42. C2×C3×C4⋊C4D4semidirect
58. C2×C4⋊Dic3D4semidirect
65. C3×#95D4semidirect
70. C3⋊#95D4semidirect
158. C2×C4⋊C4#67semidirect
183. #95#67semidirect
445. C3#12192semidirect

Maximal subgroups

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

subgroupindex
#777052
C3×#16072
#718512
#718442
#676492
#674512
#674442
#121923