Finite Group Databasebeta

← #10698975

Subgroups — #10698975

Order 768.

574 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5072C211yesmaximal, characteristic
2C2×C3×#331C211yesmaximal, characteristic, complemented
3#71764C211yesmaximal, characteristic, complemented
4#70479C211yesmaximal, characteristic, complemented
5#70470C211yesmaximal, characteristic, complemented
6C2×#4479C211yesmaximal, characteristic, complemented
7#67702C211yesmaximal, characteristic, complemented
8#10472—31nomaximal, Sylow
9C2×C3×#127C2211yescharacteristic
10C22×#213—21no—
11C2×C4×D12C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×#832—21no—
15C2×#832—21no—
16C2×#832—21no—
17C2×#821—21no—
18C2×(C2×D3)⋊C8—21no—
19C2×(C2×D3)⋊C8—21no—
20#5072—22no—
21#5072—22no—
22#4935C2211yescharacteristic
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24C2×C4×C3⋊C8C2211yescharacteristic
25C3×#331—22no—
26#4551—21no—
27#4479—22no—
28C2×#459—31no—
29#1957—31no—
30#1954—31no—
31#1802—31no—
32C2×#335—31no—
33C2×#331D311yescharacteristic, complemented
34#1296—31no—
35C22×C3×D4—21no—
36C22×C3×D4—21no—
37C22×C3×D4—21no—
38C22×D12D411yescharacteristic
39C22×C4×D3—21no—
40C22×C3×C8—21no—
41C22×C3×C8—21no—
42C3×#127—21no—
43C3×#127—21no—
44C3×#127—21no—
45C3×#127D412yes—
46C2×C3×C42C2311yescharacteristic
47C2×#213—24no—
48C2×#213—42no—
49C2×#213—42no—
50C2×(C2×D3)⋊C4—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3D411yescharacteristic
55C22×C3⋊C8—21no—
56C22×C3⋊C8D411yescharacteristic
57C22×C3⋊C8D411yescharacteristic
58C4×D12—21no—
59C4×D12—21no—
60C4×D12—21no—
61C4×D12—21no—
62C3×C4⋊C8—21no—
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66#832—24no—
67#832—42no—
68#832—42no—
69#821—42no—
70(C2×D3)⋊C8—42no—
71(C2×D3)⋊C8—42no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)D412yes—
74C4×C3⋊C8—21no—
75C4×C3⋊C8—21no—
76C22×D8—61no—
77C2×#127C2×D311yescharacteristic
78C2×C4×D4—31no—
79#459—62no—
80#459—62no—
81C2×C4⋊C8—31no—
82C2×C4⋊C8C2×D311yescharacteristic
83C2×C4⋊C8C2×D311yescharacteristic
84C2×#102—61no—
85C2×#102—61no—
86C2×#102—61no—
87C2×#102—61no—
88C2×C22⋊C8—61no—
89C2×C22⋊C8—61no—
90C2×C4×C8—31no—
91#384—31no—
92#340—61no—
93#335—62no—
94#331—22no—
95C24×C3—41no—
96C23×D3—21no—
97C2×C3×D4—22no—
98C2×C3×D4—22no—
99C2×C3×D4—22no—
100C2×C3×D4—41no—
101C2×C3×D4—41no—
102C2×C3×D4—41no—
103C2×C3×D4—41no—
104C2×C3×D4—41no—
105C2×C3×D4—41no—
106C2×C3×D4—41no—
107C2×C3×D4—41no—
108C2×C3×D4—41no—
109C2×C3×D4—41no—
110C2×C3×D4—41no—
111C2×C3×D4—81no—
112C2×C3×D4—81no—
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4#4111yescharacteristic
116C22×Dic3—21no—
117C22×Dic3—41no—
118C2×D12—21no—
119C2×D12—21no—
120C2×D12—21no—
121C2×D12—21no—
122C2×D12—21no—
123C2×D12D812yes—
124C2×C4×D3—41no—
125C2×C4×D3—41no—
126C2×C4×D3—41no—
127C2×C4×D3—41no—
128C2×C4×D3—41no—
129C2×C4×D3—41no—
130C2×C3×C8—41no—
131C2×C3×C8—41no—
132C2×C3×C8—41no—
133C2×C3×C8—41no—
134C2×C3×C8—41no—
135C2×C3×C8—41no—
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42#4111yescharacteristic
139C3×C42#4111yescharacteristic
140#213—48no—
141(C2×D3)⋊C4—41no—
142(C2×D3)⋊C4—41no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—41no—
146C4⋊Dic3—41no—
147C4⋊Dic3—41no—
148C4⋊Dic3—41no—
149C4⋊Dic3—81no—
150C2×C3⋊C8—21no—
151C2×C3⋊C8—21no—
152C2×C3⋊C8—22no—
153C2×C3⋊C8—21no—
154C2×C3⋊C8—21no—
155C2×C3⋊C8—21no—
156C2×C3⋊C8—41no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8D812yescomplemented
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—31no—
163C23×C4—61no—
164C2×D8—64no—
165C2×D8—122no—
166C2×D8—122no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—31no—
170C22×C8—31no—
171C22×C8—61no—
172#127—21no—
173#127—21no—
174#127—21no—
175#127#6712yes—
176C4×D4—61no—
177C4×D4—61no—
178C4×D4—61no—
179C4×D4—61no—
180C2×C4⋊C4—31no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—61no—
184C2×C22⋊C4—61no—
185C2×C42C22×D311yescharacteristic
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—21no—
190C4⋊C8—32no—
191C4⋊C8—61no—
192#102—64no—
193#102—122no—
194#102—122no—
195#102—122no—
196C22⋊C8—122no—
197C22⋊C8—122no—
198C4×C8—61no—
199C4×C8—61no—
200C23×C3—42no—
201C23×C3—81no—
202C23×C3—81no—
203C23×C3—81no—
204C23×C3—81no—
205C23×C3—81no—
206C23×C3—81no—
207C23×C3#12111yescharacteristic
208C22×D3—22no—
209C22×D3—41no—
210C22×D3—41no—
211C22×D3—41no—
212C22×D3—41no—
213C22×D3—41no—
214C22×D3—41no—
215C3×D4—42no—
216C3×D4—42no—
217C3×D4—42no—
218C3×D4—42no—
219C3×D4—82no—
220C3×D4—82no—
221C3×D4—81no—
222C3×D4—81no—
223C3×D4—81no—
224C3×D4—81no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12311yescharacteristic
231C2×C3×C4#12311yescharacteristic
232C2×C3×C4#12411yescharacteristic
233C2×C3×C4C2×D811yescharacteristic
234C2×C3×C4C2×D811yescharacteristic
235C2×C3×C4#13511yescharacteristic
236C2×C3×C4#13511yescharacteristic
237C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
238C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
239C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
240C2×C3×C4#13711yescharacteristic
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—81no—
245C2×Dic3—81no—
246C2×Dic3—81no—
247D12—22no—
248D12—22no—
249D12—41no—
250D12—41no—
251C4×D3—81no—
252C4×D3—81no—
253C4×D3—81no—
254C4×D3—81no—
255C3×C8—81no—
256C3×C8—81no—
257C3⋊C8—22no—
258C3⋊C8—42no—
259C3⋊C8—41no—
260C24—41no—
261C24—61no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—32no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—61no—
278C2×D4—61no—
279C2×D4—61no—
280C2×D4—61no—
281C2×D4—61no—
282C2×D4—81no—
283C2×D4—81no—
284C22×C4—61no—
285C22×C4—121no—
286C22×C4—121no—
287C22×C4—121no—
288C22×C4—121no—
289C22×C4—121no—
290C22×C4—121no—
291C22×C4—121no—
292C22×C4#23511yescharacteristic
293C22×C4D3×D411yescharacteristic
294C22×C4D3×D411yescharacteristic
295D8—128no—
296C2×C8—32no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—61no—
304C2×C8—61no—
305C2×C8—61no—
306C2×C8—61no—
307C2×C8—61no—
308C2×C8—62no—
309C2×C8—121no—
310C2×C8—121no—
311C4⋊C4—61no—
312C4⋊C4—61no—
313C4⋊C4—121no—
314C4⋊C4—121no—
315C4⋊C4—121no—
316C4⋊C4—121no—
317C4⋊C4—241no—
318C22⋊C4—121no—
319C22⋊C4—121no—
320C42#23511yescharacteristic
321C42D3×D411yescharacteristic
322C42#23711yescharacteristic
323C42C2×#6711yescharacteristic
324C22×C3—82no—
325C22×C3—82no—
326C22×C3—82no—
327C22×C3—161no—
328C22×C3—161no—
329C22×C3—161no—
330C22×C3—161no—
331C22×C3#39411yescharacteristic
332C22×C3#45911yescharacteristic
333C22×C3#46311yescharacteristic
334C22×C3#46611yescharacteristic
335C22×C3#46811yescharacteristic
336C22×C3#47111yescharacteristic
337C22×C3#47211yescharacteristic
338C2×D3—42no—
339C2×D3—42no—
340C2×D3—42no—
341C2×D3—81no—
342C2×D3—81no—
343C2×D3—81no—
344C2×D3—81no—
345C3×C4—21no—
346C3×C4—21no—
347C3×C4#44711yescharacteristic
348C3×C4#45211yescharacteristic
349C3×C4#45911yescharacteristic
350C3×C4#46311yescharacteristic
351C3×C4#46611yescharacteristic
352C3×C4#46911yescharacteristic
353C3×C4#48011yescharacteristic
354C3×C4#48211yescharacteristic
355Dic3—81no—
356Dic3—161no—
357C23—42no—
358C23—62no—
359C23—81no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23—81no—
364C23—81no—
365C23—121no—
366C23—121no—
367C23—121no—
368C23—121no—
369C23—121no—
370C23—121no—
371C23#88411yescharacteristic
372D4—42no—
373D4—42no—
374D4—42no—
375D4—42no—
376D4—62no—
377D4—62no—
378D4—82no—
379D4—82no—
380D4—81no—
381D4—81no—
382D4—81no—
383D4—81no—
384D4—121no—
385D4—121no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4—241no—
392C2×C4—241no—
393C2×C4—241no—
394C2×C4—241no—
395C2×C4—241no—
396C2×C4—241no—
397C2×C4—241no—
398C2×C4#88311yescharacteristic
399C2×C4#88411yescharacteristic
400C2×C4#88511yescharacteristic
401C2×C4D3×D811yescharacteristic
402C2×C4D3×D811yescharacteristic
403C2×C4#91111yescharacteristic
404C2×C4#91111yescharacteristic
405C2×C4#91411yescharacteristic
406C2×C4#91511yescharacteristic
407C2×C4#91611yescharacteristic
408C2×C4#91611yescharacteristic
409C2×C4#92911yescharacteristic
410C2×C4#93711yescharacteristic
411C2×C4#93911yescharacteristic
412C8—62no—
413C8—81no—
414C8—81no—
415C8—122no—
416C8—121no—
417C2×C3—162no—
418C2×C3#166711yescharacteristic
419C2×C3#168111yescharacteristic
420C2×C3#168511yescharacteristic
421C2×C3#169711yescharacteristic
422C2×C3#203311yescharacteristic
423C2×C3#205111yescharacteristic
424C2×C3#205411yescharacteristic
425D3—82no—
426C22—82no—
427C22—82no—
428C22—82no—
429C22—122no—
430C22—122no—
431C22—122no—
432C22—161no—
433C22—161no—
434C22—161no—
435C22—161no—
436C22—241no—
437C22—241no—
438C22—241no—
439C22—241no—
440C22#474811yescharacteristic
441C22#474911yescharacteristic
442C22#477511yescharacteristic
443C22#478011yescharacteristic
444C22#478611yescharacteristic
445C22#478711yescharacteristic
446C22#495911yescharacteristic
447C4—21no—
448C4—21no—
449C4—241no—
450C4—481no—
451C4#477411yescharacteristic
452C4#477511yescharacteristic
453C4#477911yescharacteristic
454C4#478211yescharacteristic
455C4#515011yescharacteristic
456C4#515611yescharacteristic
457C4#516511yescharacteristic
458C4#517311yescharacteristic
459C3#1047211yescharacteristic, complemented, Sylow
460C2—162no—
461C2—242no—
462C2#6839511yescharacteristic
463C2#6839711yescharacteristic
464C2#6839811yescharacteristic
465C2#6840011yescharacteristic
466C2#7009211yescharacteristic
467C2#7020611yescharacteristic
468C2#7021211yescharacteristic
subgroupG/H
centerC23#884
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#331C2
socleC23×C3#121
prime radicalC2×#5072C2
hypercenterC2×#331D3
hypocenterC3#10472
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#331

Complements

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

NH ≅ G/Ntype
2. C2×C3×#331C2semidirect
3. #71764C2semidirect
4. #70479C2semidirect
5. #70470C2semidirect
6. C2×#4479C2semidirect
7. #67702C2semidirect
33. C2×#331D3semidirect
158. C2×C3⋊C8D8semidirect
459. C3#10472semidirect

Maximal subgroups

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

subgroupindex
C2×#50722
C2×C3×#3312
#717642
#704792
#704702
C2×#44792
#677022
#104723