Finite Group Databasebeta

← #10831766

Subgroups — #10831766

Order 768.

988 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#557C211yesmaximal, characteristic
2C2×#5091C211yesmaximal, characteristic
3#82480C211yesmaximal, characteristic
4C2×#5030C211yesmaximal, characteristic
5#71788C211yesmaximal, characteristic
6#71744C212yesmaximal
7#71739C212yesmaximal
8#71635C211yesmaximal, characteristic
9#71632C212yesmaximal
10#71628C212yesmaximal
11#71627C211yesmaximal, characteristic
12#24344—31nomaximal, Sylow
13C3×#557—24no—
14C3×#557—24no—
15C2×C3×C4⋊Q8—21no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#125—22no—
18C2×C3×#125C2212yes—
19C2×C3×C4×Q8—21no—
20C2×C3×C4×Q8C2212yes—
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×Q8C2211yescharacteristic
23C22×#216C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25#5230—21no—
26C2×#835C2212yes—
27C2×#835C2211yescharacteristic
28#5091—24no—
29#5030—24no—
30#4985—21no—
31#4978—22no—
32C2×#810C2212yes—
33C2×#810C2211yescharacteristic
34C2×#808C2211yescharacteristic
35C2×#807C2212yes—
36#4936C2211yescharacteristic
37#4929C2212yes—
38#4928C2212yes—
39C2×C4⋊(C3⋊C8)C2211yescharacteristic
40C2×#803C2211yescharacteristic
41#4529C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#557D311yescharacteristic
47C2×#477—31no—
48C2×#441—31no—
49#2581—31no—
50#2059—32no—
51#2026—32no—
52#1993—31no—
53#1985—31no—
54#1878—32no—
55#1863—32no—
56#1805—31no—
57C22×C3×Q8—21no—
58C22×C3×Q8D411yescharacteristic
59C22×C3×Q8C2311yescharacteristic
60C22×Dic6C2311yescharacteristic
61C3×C4⋊Q8—28no—
62C3×C4⋊Q8—22no—
63C3×C4⋊Q8D414yes—
64C3×#125—216no—
65C3×#125—24no—
66C3×#125—24no—
67C3×C4×Q8—28no—
68C3×C4×Q8—24no—
69C3×C4×Q8—24no—
70C3×C4×Q8—22no—
71C3×C4×Q8—22no—
72C3×C4×Q8—22no—
73C3×C4×Q8D414yes—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4D412yes—
80C2×C3×C4⋊C4D411yescharacteristic
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C42—22no—
84C2×C3×C42—21no—
85C2×C3×C42C2312yes—
86C2×C3×C42C2311yescharacteristic
87C2×#216—22no—
88C2×#216—22no—
89C2×#216Q814yescomplemented
90C2×C4⋊Dic3C2311yescharacteristic
91C2×C4⋊Dic3C2312yes—
92C2×Dic3⋊C4—22no—
93C2×Dic3⋊C4—21no—
94C2×C4×Dic3—21no—
95C22×C3⋊C8C2311yescharacteristic
96C22×C3⋊C8C2312yes—
97C22×C3⋊C8C2311yescharacteristic
98C4×Dic6—22no—
99C4×Dic6—22no—
100#835—22no—
101#835—24no—
102C3⋊#95—22no—
103C3⋊#95—22no—
104C3⋊#95—22no—
105C3⋊#95—21no—
106C3⋊#95—21no—
107#810—22no—
108#810—24no—
109#808—22no—
110#807—24no—
111C4⋊(C3⋊C8)—22no—
112#803—22no—
113C22×Q16—31no—
114#557—24no—
115#557—24no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8C2×D311yescharacteristic
118C2×#125—22no—
119C2×#125C2×D312yes—
120C2×C4×Q8—21no—
121C2×C4×Q8—31no—
122C2×C4×Q8C2×D312yes—
123C2×C4×Q8C2×D311yescharacteristic
124C2×C4×Q8C2×D311yescharacteristic
125#477—64no—
126#441—64no—
127C2×#107—32no—
128C2×#106—31no—
129C2×C4⋊C8—31no—
130C2×#103—32no—
131C2×#103—31no—
132C2×#103—32no—
133C2×#103—31no—
134C2×#97—31no—
135#400—61no—
136#395—61no—
137#391—32no—
138#389—32no—
139#387—62no—
140#382—31no—
141#340—32no—
142#340—32no—
143#340—32no—
144#336—32no—
145#336—32no—
146C2×C3×Q8—22no—
147C2×C3×Q8—24no—
148C2×C3×Q8—21no—
149C2×C3×Q8—21no—
150C2×C3×Q8—21no—
151C2×C3×Q8—42no—
152C2×C3×Q8—44no—
153C2×C3×Q8C2×D412yes—
154C2×C3×Q8C2×Q812yes—
155C2×C3×Q8#4112yes—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—22no—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2411yescharacteristic
164C22×Dic3—21no—
165C22×Dic3—22no—
166C22×Dic3—21no—
167C2×Dic6—21no—
168C2×Dic6—21no—
169C2×Dic6—21no—
170C2×Dic6—21no—
171C2×Dic6—21no—
172C2×Dic6C2×Q812yes—
173C3×C4⋊C4—28no—
174C3×C4⋊C4—28no—
175C3×C4⋊C4—28no—
176C3×C4⋊C4—24no—
177C3×C4⋊C4—24no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—24no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4C2×D414yes—
183C3×C4⋊C4C2×D412yes—
184C3×C42—28no—
185C3×C42—24no—
186C3×C42—22no—
187C3×C42—22no—
188C3×C42C2×D412yes—
189C3×C42C2×D412yes—
190#216—28no—
191C4⋊Dic3—21no—
192C4⋊Dic3—21no—
193C4⋊Dic3—22no—
194C4⋊Dic3—22no—
195Dic3⋊C4—44no—
196Dic3⋊C4—42no—
197C4×Dic3—42no—
198C2×C3⋊C8—21no—
199C2×C3⋊C8—21no—
200C2×C3⋊C8—21no—
201C2×C3⋊C8—22no—
202C2×C3⋊C8—22no—
203C2×C3⋊C8—21no—
204C2×C3⋊C8—21no—
205C2×C3⋊C8—22no—
206C2×C3⋊C8C2×Q812yes—
207C22×Q8—21no—
208C22×Q8—31no—
209C22×Q8#6711yescharacteristic
210C22×Q8C22×D311yescharacteristic
211C2×Q16—34no—
212C2×Q16—62no—
213C2×Q16—62no—
214C22×C8—31no—
215C22×C8—31no—
216C22×C8—32no—
217C4⋊Q8—28no—
218C4⋊Q8—22no—
219C4⋊Q8#6714yes—
220#125—216no—
221#125—24no—
222#125—24no—
223C4×Q8—28no—
224C4×Q8—24no—
225C4×Q8—24no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×Q8—62no—
230C4×Q8—62no—
231C4×Q8#6714yes—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—31no—
238C2×C4⋊C4—32no—
239C2×C4⋊C4—61no—
240C2×C4⋊C4—62no—
241C2×C4⋊C4#6711yescharacteristic
242C2×C4⋊C4#6712yes—
243C2×C4⋊C4C22×D311yescharacteristic
244C2×C4⋊C4C22×D312yes—
245C2×C42—22no—
246C2×C42—21no—
247C2×C42—61no—
248C2×C42C22×D312yes—
249C2×C42C22×D311yescharacteristic
250#107—64no—
251#106—62no—
252C4⋊C8—62no—
253#103—64no—
254#103—62no—
255#103—62no—
256#103—64no—
257#97—62no—
258#95—61no—
259#95—61no—
260#95—62no—
261#95—62no—
262#95—62no—
263C23×C3C22×D411yescharacteristic
264C3×Q8—22no—
265C3×Q8—22no—
266C3×Q8—22no—
267C3×Q8—44no—
268C3×Q8—48no—
269C3×Q8C22⋊Q814yes—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—24no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—24no—
282C2×C3×C4—44no—
283C2×C3×C4—42no—
284C2×C3×C4#12012yes—
285C2×C3×C4#12012yes—
286C2×C3×C4C22×D411yescharacteristic
287C2×C3×C4C22×D411yescharacteristic
288C2×C3×C4C22×Q811yescharacteristic
289C2×C3×C4C2×#4111yescharacteristic
290C2×C3×C42−1+411yescharacteristic
291C2×C3×C42−1+411yescharacteristic
292C2×Dic3—22no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296C2×Dic3—42no—
297C2×Dic3—42no—
298C2×Dic3—42no—
299C2×Dic3—42no—
300Dic6—22no—
301Dic6—22no—
302Dic6—41no—
303Dic6—41no—
304C3⋊C8—22no—
305C3⋊C8—41no—
306C3⋊C8—42no—
307C2×Q8—22no—
308C2×Q8—24no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—32no—
313C2×Q8—42no—
314C2×Q8—44no—
315C2×Q8—61no—
316C2×Q8—61no—
317C2×Q8—61no—
318C2×Q8—61no—
319C2×Q8—61no—
320C2×Q8D3×Q812yes—
321C2×Q8#23912yes—
322C2×Q8C2×#6712yes—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—61no—
327C22×C4—62no—
328C22×C4—61no—
329C22×C4C2×#6712yes—
330C22×C4C2×#6712yes—
331C22×C4C2×#6711yescharacteristic
332C22×C4C2×#6711yescharacteristic
333C22×C4C23×D311yescharacteristic
334Q16—68no—
335C2×C8—32no—
336C2×C8—61no—
337C2×C8—61no—
338C2×C8—62no—
339C2×C8—61no—
340C2×C8—61no—
341C2×C8—62no—
342C2×C8—61no—
343C2×C8—62no—
344C4⋊C4—28no—
345C4⋊C4—28no—
346C4⋊C4—28no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—22no—
350C4⋊C4—24no—
351C4⋊C4—21no—
352C4⋊C4—22no—
353C4⋊C4—62no—
354C4⋊C4—62no—
355C4⋊C4—61no—
356C4⋊C4—61no—
357C4⋊C4—124no—
358C4⋊C4—122no—
359C4⋊C4C2×#6712yes—
360C4⋊C4C2×#6714yes—
361C42—28no—
362C42—24no—
363C42—22no—
364C42—22no—
365C42—122no—
366C42C2×#6712yes—
367C42C2×#6712yes—
368C22×C3C2×#12011yescharacteristic
369C22×C3#54811yescharacteristic
370C22×C3D4×Q811yescharacteristic
371C22×C3#57512yes—
372C22×C3#57812yes—
373C3×C4—24no—
374C3×C4—22no—
375C3×C4—22no—
376C3×C4—42no—
377C3×C4—44no—
378C3×C4—42no—
379C3×C4—42no—
380C3×C4C2×C22⋊Q811yescharacteristic
381C3×C4#53611yescharacteristic
382C3×C4#54811yescharacteristic
383C3×C4D4×Q811yescharacteristic
384Dic3—42no—
385Dic3—82no—
386Dic3—81no—
387C23C22×#6711yescharacteristic
388Q8—22no—
389Q8—22no—
390Q8—22no—
391Q8—44no—
392Q8—48no—
393Q8—62no—
394Q8—62no—
395Q8—121no—
396Q8—121no—
397Q8#94614yes—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—62no—
413C2×C4—121no—
414C2×C4—121no—
415C2×C4—122no—
416C2×C4—122no—
417C2×C4—122no—
418C2×C4—122no—
419C2×C4—121no—
420C2×C4#95312yes—
421C2×C4#95312yes—
422C2×C4C2×D3×Q811yescharacteristic
423C2×C4C2×#23911yescharacteristic
424C2×C4#100711yescharacteristic
425C2×C4#100711yescharacteristic
426C2×C4C22×#6711yescharacteristic
427C2×C4C22×#6711yescharacteristic
428C8—62no—
429C8—121no—
430C8—122no—
431C2×C3#240911yescharacteristic
432C2×C3#300511yescharacteristic
433C2×C3#301111yescharacteristic
434C2×C3#337712yes—
435C2×C3#339012yes—
436C22Q8×#6711yescharacteristic
437C22#581711yescharacteristic
438C22#582212yes—
439C22#583812yes—
440C22C2×#95311yescharacteristic
441C4—24no—
442C4—22no—
443C4—22no—
444C4—42no—
445C4—44no—
446C4—42no—
447C4—42no—
448C4—122no—
449C4—241no—
450C4—242no—
451C4C2×#94611yescharacteristic
452C4Q8×#6711yescharacteristic
453C4#581611yescharacteristic
454C4#581711yescharacteristic
455C3#2434411yescharacteristic, complemented, Sylow
456C2#8368412yes—
457C2#8370012yes—
458C2#8406811yescharacteristic
459C2#8408111yescharacteristic
460C2#8411711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#557C2
socle, prime radicalC23×C3C22×D4
hypercenterC2×#557D3
hypocenterC3#24344
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
89. C2×#216Q8semidirect
455. C3#24344semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5572
C2×#50912
#824802
C2×#50302
#717882
#717442
#717392
#716352
#716322
#716282
#716272
#243443