Finite Group Databasebeta

← #10695860

Subgroups — #10695860

Order 768.

852 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#447C211yesmaximal, characteristic
2C2×#5218C211yesmaximal, characteristic, complemented
3#82631C211yesmaximal, characteristic, complemented
4#72129C211yesmaximal, characteristic, complemented
5#71676C211yesmaximal, characteristic, complemented
6#69923C211yesmaximal, characteristic, complemented
7#67680C211yesmaximal, characteristic, complemented
8#9987—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C22×C3×D8—21no—
11C2×C3×#121C2211yescharacteristic
12C22×#214—21no—
13C2×#924C2211yescharacteristic
14C3×#447—22no—
15C3×#447—22no—
16C2×C3×#102—21no—
17C2×C3×C22⋊C8C2211yescharacteristic
18#5218—22no—
19#5218—21no—
20#5218—22no—
21C2×#833—22no—
22C2×#832—21no—
23#5207C2211yescharacteristic
24#5207C2211yescharacteristic
25C2×#830C2211yescharacteristic
26#5042—22no—
27#5040—21no—
28#5033—21no—
29#4970—21no—
30#4525—22no—
31#4469—22no—
32#3193—31no—
33C2×#450—31no—
34C2×#447D311yescharacteristic
35#1783—31no—
36#1472—31no—
37#1471—31no—
38#1298—31no—
39C25×C3—21no—
40C22×C3×D4—21no—
41C22×C3×D4—21no—
42C22×C3×D4—21no—
43C22×C3×D4—21no—
44C22×C3×D4—21no—
45C22×C3×D4—21no—
46C22×C3×D4—44no—
47C22×C3×D4D411yescharacteristic
48C22×C3×D4D412yes—
49C23×C3×C4C2311yescharacteristic
50C22×Dic6D411yescharacteristic
51C2×C3×D8—24no—
52C2×C3×D8—44no—
53C22×C3×C8—21no—
54C3×#121—21no—
55C3×#121—21no—
56C3×#121—21no—
57C3×#121—21no—
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C22⋊C4—21no—
60C2×#217—21no—
61C2×#214—24no—
62C2×#214—22no—
63C2×#214—44no—
64C2×C4⋊Dic3D411yescharacteristic
65#924—21no—
66#924—21no—
67#924—21no—
68#924—21no—
69C2×Dic3⋊C4—21no—
70C2×#208—21no—
71C2×#208D412yes—
72C22×C3⋊C8—21no—
73C3×#102—24no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8D412yes—
76#833—44no—
77#832—24no—
78#832—22no—
79#830—21no—
80#830—21no—
81#830—21no—
82#830—21no—
83#830—21no—
84#830D412yes—
85#810—22no—
86#807—22no—
87C23×D4C2×D311yescharacteristic
88C22×SD16—61no—
89C22×D8—21no—
90C2×C22⋊Q8—31no—
91C2×#121C2×D311yescharacteristic
92#484—61no—
93#480—61no—
94#468—62no—
95#450—62no—
96#450—61no—
97#450—62no—
98#448—61no—
99#447—22no—
100#447—22no—
101C2×#102—21no—
102C2×#102—61no—
103C2×#100—62no—
104#407—31no—
105#407—31no—
106C2×C22⋊C8—31no—
107C2×C22⋊C8C2×D311yescharacteristic
108#324—62no—
109#323—62no—
110C24×C3—22no—
111C24×C3—21no—
112C24×C3—44no—
113C24×C3—41no—
114C24×C3—41no—
115C24×C3—82no—
116C24×C3—82no—
117C24×C3#4111yescharacteristic
118C2×C3×D4—24no—
119C2×C3×D4—24no—
120C2×C3×D4—24no—
121C2×C3×D4—22no—
122C2×C3×D4—22no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—42no—
126C2×C3×D4—42no—
127C2×C3×D4—42no—
128C2×C3×D4—42no—
129C2×C3×D4—44no—
130C2×C3×D4—44no—
131C2×C3×D4—44no—
132C2×C3×D4—41no—
133C2×C3×D4—41no—
134C2×C3×D4—41no—
135C2×C3×D4—41no—
136C2×C3×D4—82no—
137C22×C3×C4—21no—
138C22×C3×C4—42no—
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×Dic3—21no—
146C22×Dic3—21no—
147C2×Dic6—21no—
148C2×Dic6—21no—
149C2×Dic6—42no—
150C3×D8—48no—
151C2×C3×C8—22no—
152C2×C3×C8—42no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—21no—
155C3×C22⋊C4—41no—
156C3×C22⋊C4—41no—
157#217—41no—
158#217—41no—
159#214—48no—
160#214—82no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163Dic3⋊C4—41no—
164Dic3⋊C4—41no—
165#208—42no—
166#208—42no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—22no—
169C2×C3⋊C8—42no—
170C2×C3⋊C8—41no—
171C25—21no—
172C22×Q8—31no—
173C22×D4—21no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—44no—
180C22×D4#6711yescharacteristic
181C22×D4#6712yes—
182C23×C4C22×D311yescharacteristic
183C2×SD16—64no—
184C2×SD16—62no—
185C2×SD16—124no—
186C2×D8—24no—
187C2×D8—44no—
188C2×M16—32no—
189C2×M16—61no—
190C22×C8—21no—
191C22×C8—61no—
192C22⋊Q8—61no—
193C22⋊Q8—61no—
194C22⋊Q8—61no—
195C22⋊Q8—61no—
196#121—21no—
197#121—21no—
198#121—21no—
199#121—21no—
200C2×C4⋊C4—31no—
201C2×C4⋊C4—61no—
202C2×C4⋊C4#6711yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—61no—
205#107—62no—
206#103—62no—
207#102—24no—
208#102—64no—
209#102—62no—
210#100—124no—
211C22⋊C8—21no—
212C22⋊C8—32no—
213C22⋊C8—61no—
214C22⋊C8—61no—
215C22⋊C8—61no—
216C22⋊C8—61no—
217C22⋊C8—61no—
218C22⋊C8#6712yes—
219C23×C3—22no—
220C23×C3—22no—
221C23×C3—24no—
222C23×C3—41no—
223C23×C3—44no—
224C23×C3—44no—
225C23×C3—42no—
226C23×C3—42no—
227C23×C3—41no—
228C23×C3—42no—
229C23×C3—82no—
230C23×C3—82no—
231C23×C3—82no—
232C23×C3—82no—
233C23×C3—81no—
234C23×C3—81no—
235C23×C3—816no—
236C23×C3—82no—
237C23×C3—82no—
238C23×C3#12111yescharacteristic
239C23×C3#12111yescharacteristic
240C23×C3#12111yescharacteristic
241C3×D4—24no—
242C3×D4—24no—
243C3×D4—42no—
244C3×D4—44no—
245C3×D4—42no—
246C3×D4—42no—
247C3×D4—44no—
248C3×D4—42no—
249C3×D4—82no—
250C3×D4—82no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—81no—
259C2×C3×C4#12011yescharacteristic
260C2×C3×C4#12011yescharacteristic
261C2×C3×C4#12011yescharacteristic
262C2×C3×C4#12711yescharacteristic
263C2×Dic3—41no—
264C2×Dic3—41no—
265C2×Dic3—42no—
266C2×Dic3—81no—
267Dic6—42no—
268Dic6—42no—
269C3×C8—42no—
270C3⋊C8—42no—
271C3⋊C8—42no—
272C3⋊C8—81no—
273C24—22no—
274C24—21no—
275C24—44no—
276C24—41no—
277C24—41no—
278C24—82no—
279C24—82no—
280C24#23711yescharacteristic
281C2×Q8—61no—
282C2×Q8—61no—
283C2×Q8—122no—
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—22no—
288C2×D4—21no—
289C2×D4—22no—
290C2×D4—21no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—44no—
296C2×D4—44no—
297C2×D4—44no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—82no—
303C22×C4—21no—
304C22×C4—42no—
305C22×C4—61no—
306C22×C4—61no—
307C22×C4D3×D411yescharacteristic
308C22×C4D3×D411yescharacteristic
309C22×C4D3×D411yescharacteristic
310C22×C4C2×#6711yescharacteristic
311C22×C4C2×#6711yescharacteristic
312C22×C4C2×#6711yescharacteristic
313SD16—128no—
314SD16—242no—
315D8—48no—
316M16—122no—
317M16—122no—
318C2×C8—22no—
319C2×C8—42no—
320C2×C8—62no—
321C2×C8—62no—
322C2×C8—122no—
323C2×C8—121no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—61no—
327C4⋊C4—61no—
328C4⋊C4—121no—
329C4⋊C4—121no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—121no—
333C22⋊C4—121no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—44no—
342C22×C3—44no—
343C22×C3—88no—
344C22×C3—88no—
345C22×C3—84no—
346C22×C3—84no—
347C22×C3—82no—
348C22×C3—82no—
349C22×C3—81no—
350C22×C3—81no—
351C22×C3—81no—
352C22×C3—162no—
353C22×C3#39211yescharacteristic
354C22×C3#47112yes—
355C22×C3#47212yes—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—81no—
360C3×C4B(4,𝔽2)12yes—
361Dic3—81no—
362Dic3—81no—
363C23—22no—
364C23—22no—
365C23—24no—
366C23—41no—
367C23—44no—
368C23—44no—
369C23—42no—
370C23—42no—
371C23—41no—
372C23—42no—
373C23—82no—
374C23—82no—
375C23—82no—
376C23—82no—
377C23—81no—
378C23—81no—
379C23—816no—
380C23—82no—
381C23—82no—
382C23#93811yescharacteristic
383C23#93911yescharacteristic
384C23#93911yescharacteristic
385Q8—122no—
386Q8—122no—
387D4—24no—
388D4—24no—
389D4—42no—
390D4—42no—
391D4—44no—
392D4—42no—
393D4—44no—
394D4—42no—
395D4—82no—
396D4—82no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—81no—
405C2×C4—122no—
406C2×C4—121no—
407C2×C4—121no—
408C2×C4—241no—
409C2×C4#93711yescharacteristic
410C2×C4#93711yescharacteristic
411C2×C4#94011yescharacteristic
412C2×C4#95311yescharacteristic
413C8—42no—
414C8—122no—
415C8—122no—
416C8—241no—
417C2×C3—22no—
418C2×C3—22no—
419C2×C3—41no—
420C2×C3—84no—
421C2×C3—161no—
422C2×C3#201811yescharacteristic
423C2×C3#201911yescharacteristic
424C2×C3#202111yescharacteristic
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—44no—
433C22—44no—
434C22—88no—
435C22—88no—
436C22—84no—
437C22—84no—
438C22—82no—
439C22—82no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—162no—
444C22#516012yes—
445C22#520012yes—
446C22#522311yescharacteristic
447C4—21no—
448C4—21no—
449C4—21no—
450C4—81no—
451C4—241no—
452C4—241no—
453C4#503212yes—
454C3#998711yescharacteristic, complemented, Sylow
455C2—22no—
456C2—22no—
457C2—41no—
458C2—84no—
459C2—161no—
460C2#7167111yescharacteristic
461C2#7204611yescharacteristic
462C2#7209711yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
Fitting, prime radicalC2×C3×#447C2
socleC22×C3#392
hypercenterC2×#447D3
hypocenterC3#9987
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#447

Complements

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

NH ≅ G/Ntype
2. C2×#5218C2semidirect
3. #82631C2semidirect
4. #72129C2semidirect
5. #71676C2semidirect
6. #69923C2semidirect
7. #67680C2semidirect
454. C3#9987semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4472
C2×#52182
#826312
#721292
#716762
#699232
#676802
#99873