Finite Group Databasebeta

← #10697724

Subgroups — #10697724

Order 768.

710 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#474C211yesmaximal, characteristic, complemented
2C2×#5085C211yesmaximal, characteristic, complemented
3C2×#5015C211yesmaximal, characteristic
4C2×#4547C211yesmaximal, characteristic, complemented
5#71614C211yesmaximal, characteristic, complemented
6C2×#4485C211yesmaximal, characteristic, complemented
7#67703C211yesmaximal, characteristic, complemented
8#10266—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C22×#215—22no—
12C22×#213—21no—
13C2×#874C2211yescharacteristic
14C3×#474—22no—
15C3×#474—22no—
16C2×C3×#107—21no—
17C2×C3×C4⋊C8C2211yescharacteristic
18C2×C3×#102—21no—
19C2×#830—22no—
20#5085—22no—
21#5085—22no—
22#5015—22no—
23#5015—22no—
24C2×#809—21no—
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26C2×C4⋊(C3⋊C8)C2211yescharacteristic
27C2×C4×C3⋊C8C2211yescharacteristic
28#4547—22no—
29#4485—22no—
30C2×#492—31no—
31C2×#474D311yescharacteristic, complemented
32C2×#459—31no—
33#1954—31no—
34C2×#332—31no—
35C2×#331—31no—
36#1277—31no—
37C22×C3×Q8—21no—
38C22×C3×D4D411yescharacteristic
39C23×C3×C4—21no—
40C22×D12—21no—
41C22×D12—21no—
42C22×D12—21no—
43C22×C3×C8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8D412yes—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C22⋊C4—21no—
56C2×C3×C42C2311yescharacteristic
57C2×#215—28no—
58C2×#215—44no—
59C2×#215—44no—
60C2×#213—24no—
61C2×#213—42no—
62C2×#213—42no—
63C22×C3⋊C8—21no—
64C22×C3⋊C8—21no—
65C22×C3⋊C8D412yes—
66#874—21no—
67#874—21no—
68#874—21no—
69#874D412yes—
70C3×#107—24no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8D412yes—
73C3×#102—24no—
74#830—44no—
75#809—24no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)D412yes—
80C4×C3⋊C8—21no—
81C4×C3⋊C8—21no—
82C22×SD16—62no—
83C22×D8—61no—
84C2×C4⋊Q8C2×D311yescharacteristic
85C2×#127—31no—
86C2×C4×D4C2×D311yescharacteristic
87#492—62no—
88#492—62no—
89#474—22no—
90#474—22no—
91#459—62no—
92#459—62no—
93C2×#107—21no—
94C2×C4⋊C8—31no—
95C2×C4⋊C8—31no—
96C2×C4⋊C8C2×D311yescharacteristic
97C2×#102—21no—
98C2×#102—61no—
99C2×C22⋊C8—62no—
100C2×C4×C8—31no—
101#332—62no—
102#331—62no—
103C24×C3—21no—
104C23×D3—41no—
105C2×C3×Q8—22no—
106C2×C3×Q8—41no—
107C2×C3×Q8—42no—
108C2×C3×Q8—42no—
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4SD1612yescomplemented
115C22×C3×C4—21no—
116C22×C3×C4—41no—
117C22×C3×C4—41no—
118C22×C3×C4—41no—
119C22×C3×C4—42no—
120C22×C3×C4—42no—
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4#4111yescharacteristic
124C2×D12—22no—
125C2×D12—22no—
126C2×D12—22no—
127C2×D12—42no—
128C2×D12—41no—
129C2×D12—41no—
130C2×D12—41no—
131C2×D12—42no—
132C2×D12—41no—
133C2×D12—41no—
134C2×D12—41no—
135C2×D12—41no—
136C2×D12—82no—
137C2×C3×C8—22no—
138C2×C3×C8—41no—
139C2×C3×C8—41no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—22no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—41no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152#215—416no—
153#213—48no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8—41no—
160C2×C3⋊C8—42no—
161C2×C3⋊C8D814yescomplemented
162C22×Q8—21no—
163C22×D4—61no—
164C22×D4—61no—
165C22×D4—61no—
166C22×D4#6711yescharacteristic
167C23×C4—21no—
168C2×SD16—68no—
169C2×SD16—124no—
170C2×SD16—124no—
171C2×D8—64no—
172C2×D8—122no—
173C2×D8—122no—
174C22×C8—21no—
175C22×C8—32no—
176C22×C8—61no—
177C22×C8—61no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8#6712yes—
182#127—32no—
183#127—61no—
184#127—61no—
185#127—61no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4#6711yescharacteristic
193C2×C22⋊C4—21no—
194C2×C42C22×D311yescharacteristic
195#107—24no—
196C4⋊C8—21no—
197C4⋊C8—32no—
198C4⋊C8—61no—
199C4⋊C8—61no—
200C4⋊C8—61no—
201C4⋊C8#6712yes—
202#102—24no—
203#102—64no—
204C22⋊C8—124no—
205C4×C8—61no—
206C4×C8—61no—
207C23×C3—22no—
208C23×C3—41no—
209C23×C3—42no—
210C23×C3—41no—
211C23×C3—42no—
212C23×C3#12111yescharacteristic
213C22×D3—42no—
214C22×D3—82no—
215C22×D3—81no—
216C22×D3—81no—
217C22×D3—82no—
218C3×Q8—44no—
219C3×Q8—81no—
220C3×Q8—81no—
221C3×D4—22no—
222C3×D4—22no—
223C3×D4—41no—
224C3×D4—41no—
225C2×C3×C4—22no—
226C2×C3×C4—41no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4—82no—
230C2×C3×C4—82no—
231C2×C3×C4—81no—
232C2×C3×C4—81no—
233C2×C3×C4#12011yescharacteristic
234C2×C3×C4#12011yescharacteristic
235C2×C3×C4#12011yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12711yescharacteristic
239C2×C3×C4C2×D812yes—
240C2×C3×C4C2×SD1611yescharacteristic
241C2×C3×C4#13511yescharacteristic
242C2×C3×C4AGL(1,ℤ/8)12yes—
243C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
244C2×C3×C4#13711yescharacteristic
245D12—44no—
246D12—42no—
247D12—42no—
248D12—82no—
249D12—82no—
250D12—81no—
251D12—81no—
252D12—81no—
253D12—81no—
254C3×C8—42no—
255C3⋊C8—24no—
256C3⋊C8—42no—
257C3⋊C8—81no—
258C24—21no—
259C24—121no—
260C2×Q8—22no—
261C2×Q8—41no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—62no—
270C2×D4—62no—
271C2×D4—62no—
272C2×D4—122no—
273C2×D4—122no—
274C2×D4—121no—
275C2×D4—121no—
276C2×D4—121no—
277C2×D4—121no—
278C2×D4—121no—
279C2×D4—121no—
280C2×D4—121no—
281C2×D4—242no—
282C2×D4#21512yescomplemented
283C22×C4—21no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4D3×D411yescharacteristic
290C22×C4#23711yescharacteristic
291C22×C4C2×#6711yescharacteristic
292SD16—1216no—
293D8—128no—
294C2×C8—22no—
295C2×C8—34no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—62no—
299C2×C8—62no—
300C2×C8—62no—
301C2×C8—121no—
302C2×C8—121no—
303C2×C8—122no—
304C2×C8—121no—
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—21no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C42D3×D411yescharacteristic
314C42D3×D411yescharacteristic
315C42C2×#6711yescharacteristic
316C42C2×#6711yescharacteristic
317C22×C3—44no—
318C22×C3—42no—
319C22×C3—82no—
320C22×C3—81no—
321C22×C3—81no—
322C22×C3#39211yescharacteristic
323C22×C3#45912yes—
324C22×C3#46511yescharacteristic
325C22×C3#46811yescharacteristic
326C22×C3#47112yes—
327C2×D3—82no—
328C2×D3—82no—
329C2×D3—82no—
330C2×D3—162no—
331C2×D3—162no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—81no—
335C3×C4—82no—
336C3×C4#44712yes—
337C3×C4#44811yescharacteristic
338C3×C4#44911yescharacteristic
339C3×C4#45011yescharacteristic
340C3×C4#45211yescharacteristic
341C3×C4#49311yescharacteristic
342C3×C4#49611yescharacteristic
343C23—22no—
344C23—42no—
345C23—42no—
346C23—41no—
347C23—41no—
348C23—122no—
349C23—242no—
350C23—242no—
351C23—241no—
352C23—241no—
353C23#93911yescharacteristic
354Q8—44no—
355Q8—81no—
356Q8—81no—
357D4—22no—
358D4—22no—
359D4—41no—
360D4—41no—
361D4—124no—
362D4—122no—
363D4—122no—
364D4—242no—
365D4—242no—
366D4—241no—
367D4—241no—
368D4—241no—
369D4—241no—
370C2×C4—22no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—82no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—82no—
378C2×C4D3×D812yes—
379C2×C4#91112yes—
380C2×C4#93711yescharacteristic
381C2×C4#93711yescharacteristic
382C2×C4#93811yescharacteristic
383C2×C4#93911yescharacteristic
384C2×C4#94011yescharacteristic
385C2×C4C2×#21511yescharacteristic
386C2×C4#94211yescharacteristic
387C2×C4#94911yescharacteristic
388C2×C4#95011yescharacteristic
389C2×C4#95311yescharacteristic
390C8—42no—
391C8—64no—
392C8—122no—
393C8—241no—
394C2×C3—82no—
395C2×C3#166512yes—
396C2×C3#168512yes—
397C2×C3#200011yescharacteristic
398C2×C3#201111yescharacteristic
399C2×C3#201911yescharacteristic
400D3—162no—
401C22—42no—
402C22—44no—
403C22—81no—
404C22—81no—
405C22—82no—
406C22—242no—
407C22—242no—
408C22—242no—
409C22—482no—
410C22—482no—
411C22#504811yescharacteristic
412C22#504911yescharacteristic
413C22#515012yes—
414C22#520312yes—
415C22#522311yescharacteristic
416C4—21no—
417C4—21no—
418C4—81no—
419C4—82no—
420C4#504611yescharacteristic
421C4#504711yescharacteristic
422C4#515111yescharacteristic
423C4#515411yescharacteristic
424C4#515612yes—
425C4#522711yescharacteristic
426C4#523811yescharacteristic
427C3#1026611yescharacteristic, complemented, Sylow
428C2—82no—
429C2—482no—
430C2#7110012yes—
431C2#7110412yes—
432C2#7169911yescharacteristic
433C2#7204411yescharacteristic
434C2#7210711yescharacteristic
subgroupG/H
centerC23#939
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#474C2
socleC23×C3#121
prime radicalC2×#5015C2
hypercenterC2×#474D3
hypocenterC3#10266
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. C2×C3×#474C2semidirect
2. C2×#5085C2semidirect
4. C2×#4547C2semidirect
5. #71614C2semidirect
6. C2×#4485C2semidirect
7. #67703C2semidirect
31. C2×#474D3semidirect
114. C2×C3×D4SD16semidirect
161. C2×C3⋊C8D8semidirect
282. C2×D4#215semidirect
427. C3#10266semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4742
C2×#50852
C2×#50152
C2×#45472
#716142
C2×#44852
#677032
#102663