Finite Group Databasebeta

← #10698977

Subgroups — #10698977

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×#5015C211yesmaximal, characteristic
2C3×#1802C211yesmaximal, characteristic, complemented
3#71614C211yesmaximal, characteristic, complemented
4C2×#4485C211yesmaximal, characteristic, complemented
5C2×#4479C211yesmaximal, characteristic, complemented
6#69580C211yesmaximal, characteristic, complemented
7#67634C211yesmaximal, characteristic, complemented
8#10472—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C22×#213—21no—
11C2×#874C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×#102—21no—
14C3×#384C2211yescharacteristic
15C2×#830—21no—
16C2×#830—21no—
17#5015—22no—
18#5015—22no—
19C2×#809—21no—
20C2×#809—21no—
21C2×#809—21no—
22C2×C4⋊(C3⋊C8)C2211yescharacteristic
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24C2×C4×C3⋊C8C2211yescharacteristic
25C3×#340—21no—
26#4485—22no—
27#4479—22no—
28C2×#459—31no—
29#1957—31no—
30#1954—31no—
31#1802D311yescharacteristic, complemented
32C2×#335—31no—
33C2×#331—31no—
34#1296—31no—
35C22×C3×D4D411yescharacteristic
36C23×C3×C4—21no—
37C22×D12—21no—
38C22×D12—21no—
39C22×D12—21no—
40C22×C3×C8—21no—
41C3×C4×D4—21no—
42C3×C4×D4—21no—
43C3×C4×D4—21no—
44C3×C4×D4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4D411yescharacteristic
49C2×C3×C22⋊C4—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#213—24no—
52C2×#213—42no—
53C2×#213—42no—
54C22×C3⋊C8—21no—
55C22×C3⋊C8—21no—
56C22×C3⋊C8D411yescharacteristic
57C22×C3⋊C8D411yescharacteristic
58#874—21no—
59#874—21no—
60#874—21no—
61#874D412yes—
62C3×C4⋊C8—21no—
63C3×C4⋊C8—21no—
64C3×#102—42no—
65#830—42no—
66#830—42no—
67#809—24no—
68#809—42no—
69#809—42no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)D412yes—
74C4×C3⋊C8—21no—
75C4×C3⋊C8—21no—
76C22×D8—61no—
77C2×#127—31no—
78C2×C4×D4C2×D311yescharacteristic
79#459—62no—
80#459—62no—
81C2×C4⋊C8—31no—
82C2×C4⋊C8—31no—
83C2×C4⋊C8C2×D311yescharacteristic
84C2×#102—21no—
85C2×#102—61no—
86C2×#102—61no—
87C2×#102—61no—
88C2×C22⋊C8—61no—
89C2×C22⋊C8—61no—
90C2×C4×C8—31no—
91#384C2×D311yescharacteristic
92#340—21no—
93#335—62no—
94#331—62no—
95C24×C3—21no—
96C23×D3—41no—
97C2×C3×D4—21no—
98C2×C3×D4—21no—
99C2×C3×D4—21no—
100C2×C3×D4—21no—
101C2×C3×D4—21no—
102C2×C3×D4D812yes—
103C22×C3×C4—21no—
104C22×C3×C4—41no—
105C22×C3×C4—41no—
106C22×C3×C4—41no—
107C22×C3×C4—41no—
108C22×C3×C4—41no—
109C22×C3×C4—41no—
110C22×C3×C4—41no—
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4#4111yescharacteristic
114C2×D12—22no—
115C2×D12—22no—
116C2×D12—22no—
117C2×D12—41no—
118C2×D12—41no—
119C2×D12—41no—
120C2×D12—41no—
121C2×D12—41no—
122C2×D12—41no—
123C2×D12—41no—
124C2×D12—41no—
125C2×D12—41no—
126C2×D12—41no—
127C2×D12—41no—
128C2×D12—81no—
129C2×D12—81no—
130C2×C3×C8—41no—
131C2×C3×C8—41no—
132C2×C3×C8—41no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—81no—
140C3×C22⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42#4111yescharacteristic
145C3×C42#4111yescharacteristic
146#213—48no—
147C2×C3⋊C8—21no—
148C2×C3⋊C8—21no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—21no—
151C2×C3⋊C8—21no—
152C2×C3⋊C8—21no—
153C2×C3⋊C8—41no—
154C2×C3⋊C8—41no—
155C2×C3⋊C8—41no—
156C2×C3⋊C8—41no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8D812yescomplemented
159C22×D4—61no—
160C22×D4—61no—
161C22×D4—61no—
162C22×D4#6711yescharacteristic
163C23×C4—21no—
164C2×D8—64no—
165C2×D8—122no—
166C2×D8—122no—
167C22×C8—21no—
168C22×C8—31no—
169C22×C8—31no—
170C22×C8—61no—
171C22×C8—61no—
172#127—32no—
173#127—61no—
174#127—61no—
175#127—61no—
176C4×D4—21no—
177C4×D4—21no—
178C4×D4—21no—
179C4×D4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4#6711yescharacteristic
184C2×C22⋊C4—21no—
185C2×C42C22×D311yescharacteristic
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—32no—
189C4⋊C8—61no—
190C4⋊C8—61no—
191C4⋊C8—61no—
192#102—42no—
193#102—64no—
194#102—122no—
195#102—122no—
196C22⋊C8—122no—
197C22⋊C8—122no—
198C4×C8—61no—
199C4×C8—61no—
200C23×C3—22no—
201C23×C3—41no—
202C23×C3—41no—
203C23×C3—41no—
204C23×C3—41no—
205C23×C3—41no—
206C23×C3—41no—
207C23×C3#12111yescharacteristic
208C22×D3—42no—
209C22×D3—81no—
210C22×D3—81no—
211C22×D3—81no—
212C22×D3—81no—
213C22×D3—81no—
214C22×D3—81no—
215C3×D4—22no—
216C3×D4—22no—
217C3×D4—41no—
218C3×D4—41no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—81no—
225C2×C3×C4—81no—
226C2×C3×C4—81no—
227C2×C3×C4—81no—
228C2×C3×C4—81no—
229C2×C3×C4—81no—
230C2×C3×C4—81no—
231C2×C3×C4#12011yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12111yescharacteristic
234C2×C3×C4#12311yescharacteristic
235C2×C3×C4#12311yescharacteristic
236C2×C3×C4#12411yescharacteristic
237C2×C3×C4C2×D811yescharacteristic
238C2×C3×C4C2×D811yescharacteristic
239C2×C3×C4#13511yescharacteristic
240C2×C3×C4#13511yescharacteristic
241C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
242C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
243C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
244C2×C3×C4#13711yescharacteristic
245D12—42no—
246D12—42no—
247D12—42no—
248D12—42no—
249D12—82no—
250D12—82no—
251D12—81no—
252D12—81no—
253D12—81no—
254D12—81no—
255C3×C8—81no—
256C3⋊C8—22no—
257C3⋊C8—42no—
258C3⋊C8—41no—
259C3⋊C8—81no—
260C24—21no—
261C24—121no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—62no—
268C2×D4—62no—
269C2×D4—62no—
270C2×D4—121no—
271C2×D4—121no—
272C2×D4—121no—
273C2×D4—121no—
274C2×D4—121no—
275C2×D4—121no—
276C2×D4—121no—
277C2×D4—121no—
278C2×D4—121no—
279C2×D4—121no—
280C2×D4—121no—
281C2×D4—241no—
282C2×D4—241no—
283C2×D4#21312yes—
284C22×C4—21no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4D3×D411yescharacteristic
293C22×C4#23711yescharacteristic
294C22×C4C2×#6711yescharacteristic
295D8—128no—
296C2×C8—32no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—61no—
301C2×C8—61no—
302C2×C8—61no—
303C2×C8—61no—
304C2×C8—61no—
305C2×C8—62no—
306C2×C8—121no—
307C2×C8—121no—
308C2×C8—121no—
309C2×C8—121no—
310C2×C8—121no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—81no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C42#23511yescharacteristic
321C42#23511yescharacteristic
322C42D3×D411yescharacteristic
323C42D3×D411yescharacteristic
324C22×C3—42no—
325C22×C3—42no—
326C22×C3—42no—
327C22×C3—81no—
328C22×C3—81no—
329C22×C3—81no—
330C22×C3—81no—
331C22×C3#39411yescharacteristic
332C22×C3#45911yescharacteristic
333C22×C3#46311yescharacteristic
334C22×C3#46611yescharacteristic
335C22×C3#46811yescharacteristic
336C22×C3#47111yescharacteristic
337C22×C3#47211yescharacteristic
338C2×D3—82no—
339C2×D3—82no—
340C2×D3—82no—
341C2×D3—161no—
342C2×D3—161no—
343C2×D3—161no—
344C2×D3—161no—
345C3×C4—21no—
346C3×C4—21no—
347C3×C4—81no—
348C3×C4—161no—
349C3×C4#44711yescharacteristic
350C3×C4#45211yescharacteristic
351C3×C4#45911yescharacteristic
352C3×C4#46311yescharacteristic
353C3×C4#46611yescharacteristic
354C3×C4#46911yescharacteristic
355C3×C4#48011yescharacteristic
356C3×C4#48211yescharacteristic
357C23—22no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—122no—
365C23—241no—
366C23—241no—
367C23—241no—
368C23—241no—
369C23—241no—
370C23—241no—
371C23#93911yescharacteristic
372D4—22no—
373D4—22no—
374D4—41no—
375D4—41no—
376D4—122no—
377D4—122no—
378D4—122no—
379D4—122no—
380D4—242no—
381D4—242no—
382D4—241no—
383D4—241no—
384D4—241no—
385D4—241no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4#88311yescharacteristic
399C2×C4#88411yescharacteristic
400C2×C4#88411yescharacteristic
401C2×C4#88511yescharacteristic
402C2×C4D3×D811yescharacteristic
403C2×C4#91111yescharacteristic
404C2×C4#91411yescharacteristic
405C2×C4#91611yescharacteristic
406C2×C4#92911yescharacteristic
407C2×C4C2×#21311yescharacteristic
408C2×C4#93711yescharacteristic
409C2×C4#94211yescharacteristic
410C2×C4#94911yescharacteristic
411C2×C4#95011yescharacteristic
412C8—62no—
413C8—81no—
414C8—122no—
415C8—121no—
416C8—241no—
417C2×C3—82no—
418C2×C3#166711yescharacteristic
419C2×C3#168111yescharacteristic
420C2×C3#168511yescharacteristic
421C2×C3#169711yescharacteristic
422C2×C3#203311yescharacteristic
423C2×C3#205111yescharacteristic
424C2×C3#205411yescharacteristic
425D3—162no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—242no—
434C22—242no—
435C22—242no—
436C22—481no—
437C22—481no—
438C22—481no—
439C22—481no—
440C22#495911yescharacteristic
441C22#503811yescharacteristic
442C22#504911yescharacteristic
443C22#515011yescharacteristic
444C22#516511yescharacteristic
445C22#519911yescharacteristic
446C22#520311yescharacteristic
447C4—21no—
448C4—21no—
449C4—81no—
450C4—161no—
451C4#477511yescharacteristic
452C4#478211yescharacteristic
453C4#478611yescharacteristic
454C4#478711yescharacteristic
455C4#496711yescharacteristic
456C4#500311yescharacteristic
457C4#503611yescharacteristic
458C4#504711yescharacteristic
459C3#1047211yescharacteristic, complemented, Sylow
460C2—82no—
461C2—482no—
462C2#6952311yescharacteristic
463C2#7021211yescharacteristic
464C2#7030711yescharacteristic
465C2#7100511yescharacteristic
466C2#7103911yescharacteristic
467C2#7110411yescharacteristic
468C2#7113811yescharacteristic
subgroupG/H
centerC23#939
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1802C2
socleC23×C3#121
prime radicalC2×#5015C2
hypercenter#1802D3
hypocenterC3#10472
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. C3×#1802C2semidirect
3. #71614C2semidirect
4. C2×#4485C2semidirect
5. C2×#4479C2semidirect
6. #69580C2semidirect
7. #67634C2semidirect
31. #1802D3semidirect
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×#50152
C3×#18022
#716142
C2×#44852
C2×#44792
#695802
#676342
#104723