Finite Group Databasebeta

← #10697699

Subgroups — #10697699

Order 768.

678 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5019C211yesmaximal, characteristic, complemented
2C2×#5015C211yesmaximal, characteristic
3C2×#4694C211yesmaximal, characteristic, complemented
4C3×#1954C211yesmaximal, characteristic, complemented
5C2×#4486C211yesmaximal, characteristic, complemented
6C2×#4485C211yesmaximal, characteristic, complemented
7#67654C211yesmaximal, characteristic, complemented
8#10261—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C22×#214—21no—
11C22×#213—21no—
12C2×#874C2211yescharacteristic
13C2×#869C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×C3×C22⋊C8—22no—
16C2×C3×C4×C8C2211yescharacteristic
17C2×#821—22no—
18#5019—22no—
19#5019—22no—
20#5015—22no—
21#5015—22no—
22C2×#810—21no—
23C2×#809—21no—
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26#4694—22no—
27#4694—22no—
28#4486—22no—
29#4485—22no—
30C2×#486—31no—
31C2×#461—31no—
32C2×#459—31no—
33#1954D311yescharacteristic, complemented
34C2×#332—31no—
35C2×#331—31no—
36#1277—31no—
37C22×C3×D4D411yescharacteristic
38C23×C3×C4—21no—
39C22×D12—21no—
40C22×D12—21no—
41C22×D12—21no—
42C22×Dic6—21no—
43C22×C3×C8—21no—
44C22×C3×C8D412yes—
45C3×C4×D4—21no—
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C2×C3×C4⋊C4D411yescharacteristic
50C2×C3×C22⋊C4—21no—
51C2×C3×C42C2311yescharacteristic
52C2×#214—24no—
53C2×#214—42no—
54C2×#214—42no—
55C2×#213—24no—
56C2×#213—42no—
57C2×#213—42no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3—21no—
60C22×C3⋊C8—21no—
61C22×C3⋊C8—21no—
62#874—21no—
63#874—21no—
64#874—21no—
65#874D412yes—
66#869—21no—
67#869—21no—
68#869—21no—
69#869D412yes—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C22⋊C8—44no—
73C3×C4×C8—21no—
74C3×C4×C8—21no—
75#821—28no—
76#810—24no—
77#809—24no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)D412yes—
81C4⋊(C3⋊C8)D412yes—
82C22×SD16—61no—
83C22×D8—61no—
84C2×C4⋊Q8—31no—
85C2×#127—31no—
86C2×C4×D4C2×D311yescharacteristic
87#486—62no—
88#486—62no—
89#461—62no—
90#461—62no—
91#459—62no—
92#459—62no—
93C2×C4⋊C8—31no—
94C2×C4⋊C8—31no—
95C2×C4⋊C8C2×D311yescharacteristic
96C2×#103—61no—
97C2×#102—62no—
98C2×#102—61no—
99C2×C22⋊C8—22no—
100C2×C4×C8C2×D311yescharacteristic
101#332—62no—
102#331—62no—
103C24×C3—21no—
104C23×D3—41no—
105C2×C3×D4—21no—
106C2×C3×D4—21no—
107C2×C3×D4—21no—
108C2×C3×D4—21no—
109C2×C3×D4—21no—
110C2×C3×D4D812yes—
111C22×C3×C4—21no—
112C22×C3×C4—41no—
113C22×C3×C4—41no—
114C22×C3×C4—42no—
115C22×C3×C4—42no—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C22×Dic3—41no—
120C2×D12—22no—
121C2×D12—22no—
122C2×D12—22no—
123C2×D12—41no—
124C2×D12—41no—
125C2×D12—42no—
126C2×D12—41no—
127C2×D12—41no—
128C2×D12—42no—
129C2×D12—41no—
130C2×D12—41no—
131C2×D12—41no—
132C2×D12—82no—
133C2×Dic6—22no—
134C2×Dic6—41no—
135C2×Dic6—41no—
136C2×Dic6—41no—
137C2×Dic6—41no—
138C2×Dic6—41no—
139C2×C3×C8—22no—
140C2×C3×C8—22no—
141C2×C3×C8—22no—
142C2×C3×C8—41no—
143C2×C3×C8—42no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152#214—48no—
153#213—48no—
154C4⋊Dic3—22no—
155C4⋊Dic3—22no—
156C4⋊Dic3—41no—
157C4⋊Dic3—41no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—41no—
161C2×C3⋊C8—41no—
162C2×C3⋊C8—41no—
163C2×C3⋊C8—41no—
164C22×Q8—61no—
165C22×D4—61no—
166C22×D4—61no—
167C22×D4—61no—
168C22×D4D1211yescharacteristic
169C23×C4—21no—
170C2×SD16—64no—
171C2×SD16—122no—
172C2×SD16—122no—
173C2×D8—64no—
174C2×D8—122no—
175C2×D8—122no—
176C22×C8—21no—
177C22×C8—61no—
178C22×C8—61no—
179C22×C8#6712yes—
180C4⋊Q8—32no—
181C4⋊Q8—61no—
182C4⋊Q8—61no—
183C4⋊Q8—61no—
184#127—32no—
185#127—61no—
186#127—61no—
187#127—61no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—21no—
192C2×C4⋊C4—61no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4D1211yescharacteristic
195C2×C22⋊C4—21no—
196C2×C42C22×D311yescharacteristic
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—32no—
200C4⋊C8—32no—
201C4⋊C8—61no—
202C4⋊C8—61no—
203#103—64no—
204#102—64no—
205#102—68no—
206C22⋊C8—44no—
207C4×C8—21no—
208C4×C8—21no—
209C23×C3—22no—
210C23×C3—41no—
211C23×C3—42no—
212C23×C3—41no—
213C23×C3—42no—
214C23×C3#12111yescharacteristic
215C22×D3—42no—
216C22×D3—82no—
217C22×D3—81no—
218C22×D3—81no—
219C22×D3—82no—
220C3×D4—22no—
221C3×D4—22no—
222C3×D4—41no—
223C3×D4—41no—
224C2×C3×C4—22no—
225C2×C3×C4—41no—
226C2×C3×C4—42no—
227C2×C3×C4—82no—
228C2×C3×C4—81no—
229C2×C3×C4—81no—
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4#12011yescharacteristic
232C2×C3×C4#12011yescharacteristic
233C2×C3×C4#12111yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4#12711yescharacteristic
236C2×C3×C4C2×D811yescharacteristic
237C2×C3×C4#13511yescharacteristic
238C2×C3×C4#13512yes—
239C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
240C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
241C2×C3×C4AGL(1,ℤ/8)12yes—
242C2×Dic3—42no—
243C2×Dic3—81no—
244C2×Dic3—81no—
245D12—42no—
246D12—42no—
247D12—44no—
248D12—82no—
249D12—82no—
250D12—81no—
251D12—81no—
252D12—81no—
253D12—81no—
254Dic6—42no—
255Dic6—42no—
256Dic6—81no—
257Dic6—81no—
258C3×C8—42no—
259C3×C8—81no—
260C3⋊C8—42no—
261C3⋊C8—42no—
262C24—21no—
263C24—121no—
264C2×Q8—62no—
265C2×Q8—121no—
266C2×Q8—121no—
267C2×Q8—121no—
268C2×Q8—121no—
269C2×Q8—121no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—62no—
276C2×D4—62no—
277C2×D4—62no—
278C2×D4—121no—
279C2×D4—121no—
280C2×D4—121no—
281C2×D4—121no—
282C2×D4—121no—
283C2×D4—121no—
284C2×D4—122no—
285C2×D4—121no—
286C2×D4—122no—
287C2×D4—242no—
288C2×D4D2412yes—
289C22×C4—21no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—121no—
295C22×C4C2×D1211yescharacteristic
296C22×C4#23511yescharacteristic
297C22×C4C2×#6711yescharacteristic
298SD16—128no—
299D8—128no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—42no—
304C2×C8—41no—
305C2×C8—62no—
306C2×C8—62no—
307C2×C8—121no—
308C2×C8—121no—
309C2×C8—121no—
310C2×C8—121no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—62no—
314C4⋊C4—62no—
315C4⋊C4—121no—
316C4⋊C4—121no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C42D3×D411yescharacteristic
320C42D3×D411yescharacteristic
321C42D3×D411yescharacteristic
322C42D3×D411yescharacteristic
323C22×C3—44no—
324C22×C3—42no—
325C22×C3—82no—
326C22×C3—81no—
327C22×C3—81no—
328C22×C3#39211yescharacteristic
329C22×C3#46312yes—
330C22×C3#46611yescharacteristic
331C22×C3#46911yescharacteristic
332C22×C3#47112yes—
333C2×D3—82no—
334C2×D3—82no—
335C2×D3—82no—
336C2×D3—162no—
337C2×D3—162no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—81no—
341C3×C4#44711yescharacteristic
342C3×C4#44711yescharacteristic
343C3×C4#44911yescharacteristic
344C3×C4#44911yescharacteristic
345C3×C4#44912yes—
346C3×C4#49511yescharacteristic
347C3×C4#49611yescharacteristic
348Dic3—82no—
349C23—22no—
350C23—41no—
351C23—42no—
352C23—41no—
353C23—42no—
354C23—122no—
355C23—241no—
356C23—242no—
357C23—242no—
358C23—241no—
359C23#93011yescharacteristic
360Q8—122no—
361Q8—122no—
362Q8—241no—
363Q8—241no—
364D4—22no—
365D4—22no—
366D4—41no—
367D4—41no—
368D4—122no—
369D4—122no—
370D4—124no—
371D4—242no—
372D4—242no—
373D4—241no—
374D4—241no—
375D4—241no—
376D4—241no—
377C2×C4—22no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—82no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—122no—
384C2×C4—241no—
385C2×C4—241no—
386C2×C4#88211yescharacteristic
387C2×C4#88211yescharacteristic
388C2×C4#88411yescharacteristic
389C2×C4#88411yescharacteristic
390C2×C4C2×D2411yescharacteristic
391C2×C4#90411yescharacteristic
392C2×C4#90811yescharacteristic
393C2×C4#90811yescharacteristic
394C2×C4#93212yes—
395C2×C4#93711yescharacteristic
396C2×C4#94011yescharacteristic
397C2×C4#95012yes—
398C8—42no—
399C8—81no—
400C8—122no—
401C8—122no—
402C2×C3—82no—
403C2×C3#168112yes—
404C2×C3#170112yes—
405C2×C3#199911yescharacteristic
406C2×C3#201211yescharacteristic
407C2×C3#201911yescharacteristic
408D3—162no—
409C22—42no—
410C22—44no—
411C22—81no—
412C22—81no—
413C22—82no—
414C22—242no—
415C22—242no—
416C22—242no—
417C22—482no—
418C22—482no—
419C22#496011yescharacteristic
420C22#501612yes—
421C22#511511yescharacteristic
422C22#513511yescharacteristic
423C22#514112yes—
424C4—21no—
425C4—21no—
426C4—81no—
427C4—242no—
428C4#473211yescharacteristic
429C4#473211yescharacteristic
430C4#473411yescharacteristic
431C4#473411yescharacteristic
432C4#503712yes—
433C4#505911yescharacteristic
434C4#507411yescharacteristic
435C3#1026111yescharacteristic, complemented, Sylow
436C2—82no—
437C2—482no—
438C2#6965911yescharacteristic
439C2#6996111yescharacteristic
440C2#7000711yescharacteristic
441C2#7084912yes—
442C2#7087012yes—
subgroupG/H
centerC23#930
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1954C2
socleC23×C3#121
prime radicalC2×#5015C2
hypercenter#1954D3
hypocenterC3#10261
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1954

Complements

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

NH ≅ G/Ntype
1. C2×#5019C2semidirect
3. C2×#4694C2semidirect
4. C3×#1954C2semidirect
5. C2×#4486C2semidirect
6. C2×#4485C2semidirect
7. #67654C2semidirect
33. #1954D3semidirect
435. C3#10261semidirect

Maximal subgroups

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

subgroupindex
C2×#50192
C2×#50152
C2×#46942
C3×#19542
C2×#44862
C2×#44852
#676542
#102613