Finite Group Databasebeta

← #10695864

Subgroups — #10695864

Order 768.

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

normal subgroups only

This group has 750 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×#448C211yesmaximal, characteristic, complemented
2C2×#5227C211yesmaximal, characteristic, complemented
3#82627C211yesmaximal, characteristic
4#72133C211yesmaximal, characteristic, complemented
5#71676C211yesmaximal, characteristic, complemented
6#69921C211yesmaximal, characteristic, complemented
7#67680C211yesmaximal, characteristic, complemented
8#9988—31nomaximal, Sylow
9C23×C3×Q8C2211yescharacteristic
10C22×C3×SD16—21no—
11C2×C3×#121C2211yescharacteristic
12C22×#215—21no—
13C2×#930C2211yescharacteristic
14C3×#448—22no—
15C3×#448—22no—
16C2×C3×#103—21no—
17C2×C3×C22⋊C8C2211yescharacteristic
18#5227—22no—
19#5227—22no—
20C2×#836—22no—
21C2×#835—21no—
22#5217—21no—
23#5207C2211yescharacteristic
24#5207C2211yescharacteristic
25C2×#830C2211yescharacteristic
26#5039—22no—
27#5036—21no—
28#5034—21no—
29#4968—21no—
30#4525—22no—
31#4468—22no—
32#3192—31no—
33C2×#448—31no—
34C2×#448D311yescharacteristic, complemented
35#1784—31no—
36#1471—31no—
37#1471—31no—
38#1298—31no—
39C22×C3×Q8—21no—
40C22×C3×Q8—21no—
41C22×C3×Q8—21no—
42C22×C3×Q8—21no—
43C22×C3×Q8—21no—
44C22×C3×Q8—44no—
45C22×C3×Q8D412yes—
46C22×C3×D4—21no—
47C22×C3×D4D411yescharacteristic
48C23×C3×C4—21no—
49C23×C3×C4C2311yescharacteristic
50C22×#67—21no—
51C22×D12D411yescharacteristic
52C2×C3×SD16—24no—
53C2×C3×SD16—44no—
54C22×C3×C8—21no—
55C3×#121—21no—
56C3×#121—21no—
57C3×#121—21no—
58C3×#121—21no—
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C22⋊C4—21no—
61C2×#215—24no—
62C2×#215—44no—
63C2×#213—22no—
64#930—21no—
65#930—21no—
66#930—21no—
67#930—21no—
68C2×(C2×D3)⋊C4—21no—
69C2×C4⋊Dic3D411yescharacteristic
70C2×#208—21no—
71C2×#208D412yes—
72C22×C3⋊C8—21no—
73C3×#103—24no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8D412yes—
76#836—44no—
77#835—24no—
78#832—22no—
79#830—21no—
80#830—21no—
81#830—21no—
82#830—21no—
83#830—21no—
84#830D412yes—
85#809—22no—
86#808—22no—
87C23×Q8C2×D311yescharacteristic
88C22×SD16—21no—
89C22×SD16—61no—
90C2×#121—31no—
91C2×#121C2×D311yescharacteristic
92#481—61no—
93#481—61no—
94#469—62no—
95#448—22no—
96#448—22no—
97#448—62no—
98#448—62no—
99#447—61no—
100#447—61no—
101C2×#103—21no—
102C2×#103—61no—
103C2×#101—62no—
104#407—31no—
105#407—31no—
106C2×C22⋊C8—31no—
107C2×C22⋊C8C2×D311yescharacteristic
108#323—62no—
109#323—62no—
110C24×C3—21no—
111C24×C3#4111yescharacteristic
112C23×D3—21no—
113C2×C3×Q8—24no—
114C2×C3×Q8—24no—
115C2×C3×Q8—24no—
116C2×C3×Q8—22no—
117C2×C3×Q8—22no—
118C2×C3×Q8—44no—
119C2×C3×Q8—42no—
120C2×C3×Q8—42no—
121C2×C3×Q8—42no—
122C2×C3×Q8—44no—
123C2×C3×Q8—44no—
124C2×C3×Q8—41no—
125C2×C3×Q8—41no—
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4—42no—
129C2×C3×D4—41no—
130C2×C3×D4—41no—
131C2×C3×D4—82no—
132C22×C3×C4—22no—
133C22×C3×C4—21no—
134C22×C3×C4—42no—
135C22×C3×C4—44no—
136C22×C3×C4—41no—
137C22×C3×C4—41no—
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C2×#67—41no—
145C2×#67—41no—
146C2×#67—82no—
147C22×Dic3—21no—
148C2×D12—21no—
149C2×D12—21no—
150C2×D12—42no—
151C3×SD16—48no—
152C2×C3×C8—22no—
153C2×C3×C8—42no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—21no—
156C3×C22⋊C4—41no—
157C3×C22⋊C4—41no—
158#215—48no—
159#213—82no—
160(C2×D3)⋊C4—41no—
161(C2×D3)⋊C4—41no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164#208—42no—
165#208—42no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—42no—
169C2×C3⋊C8—41no—
170C22×Q8—21no—
171C22×Q8—21no—
172C22×Q8—21no—
173C22×Q8—21no—
174C22×Q8—21no—
175C22×Q8—44no—
176C22×Q8#6712yes—
177C22×D4—21no—
178C22×D4—31no—
179C22×D4—61no—
180C22×D4#6711yescharacteristic
181C23×C4—21no—
182C23×C4C22×D311yescharacteristic
183C2×SD16—24no—
184C2×SD16—44no—
185C2×SD16—64no—
186C2×SD16—124no—
187C2×D8—62no—
188C2×M16—32no—
189C2×M16—61no—
190C22×C8—21no—
191C22×C8—61no—
192#121—21no—
193#121—21no—
194#121—21no—
195#121—21no—
196#121—61no—
197#121—61no—
198#121—61no—
199#121—61no—
200C2×C4⋊C4—31no—
201C2×C4⋊C4#6711yescharacteristic
202C2×C22⋊C4—21no—
203C2×C22⋊C4—61no—
204#106—62no—
205#103—24no—
206#103—64no—
207#102—62no—
208#102—62no—
209#101—124no—
210C22⋊C8—21no—
211C22⋊C8—32no—
212C22⋊C8—61no—
213C22⋊C8—61no—
214C22⋊C8—61no—
215C22⋊C8—61no—
216C22⋊C8—61no—
217C22⋊C8#6712yes—
218C23×C3—22no—
219C23×C3—22no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3—42no—
223C23×C3—81no—
224C23×C3—81no—
225C23×C3#12111yescharacteristic
226C23×C3#12111yescharacteristic
227C23×C3#12111yescharacteristic
228C22×D3—41no—
229C22×D3—42no—
230C22×D3—81no—
231C22×D3—81no—
232C3×Q8—24no—
233C3×Q8—24no—
234C3×Q8—42no—
235C3×Q8—44no—
236C3×Q8—42no—
237C3×Q8—44no—
238C3×D4—42no—
239C3×D4—42no—
240C3×D4—82no—
241C3×D4—82no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—24no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—42no—
250C2×C3×C4—44no—
251C2×C3×C4—44no—
252C2×C3×C4—42no—
253C2×C3×C4—41no—
254C2×C3×C4—81no—
255C2×C3×C4#12011yescharacteristic
256C2×C3×C4#12011yescharacteristic
257C2×C3×C4#12011yescharacteristic
258C2×C3×C4#12711yescharacteristic
259#67—82no—
260#67—82no—
261C2×Dic3—41no—
262C2×Dic3—81no—
263D12—42no—
264D12—42no—
265C3×C8—42no—
266C3⋊C8—42no—
267C3⋊C8—42no—
268C3⋊C8—81no—
269C24—21no—
270C24—61no—
271C24#23711yescharacteristic
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—44no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—44no—
282C2×Q8—44no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—42no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—61no—
291C2×D4—61no—
292C2×D4—82no—
293C2×D4—122no—
294C2×D4—121no—
295C2×D4—121no—
296C2×D4—242no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—42no—
300C22×C4—44no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—61no—
304C22×C4D3×D411yescharacteristic
305C22×C4D3×D411yescharacteristic
306C22×C4D3×D411yescharacteristic
307C22×C4C2×#6711yescharacteristic
308C22×C4C2×#6711yescharacteristic
309C22×C4C2×#6711yescharacteristic
310SD16—48no—
311SD16—128no—
312D8—242no—
313M16—122no—
314M16—122no—
315C2×C8—22no—
316C2×C8—42no—
317C2×C8—62no—
318C2×C8—62no—
319C2×C8—122no—
320C2×C8—121no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—61no—
324C4⋊C4—61no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—121no—
328C22⋊C4—121no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—44no—
337C22×C3—81no—
338C22×C3—81no—
339C22×C3—81no—
340C22×C3—162no—
341C22×C3#39211yescharacteristic
342C22×C3#47112yes—
343C22×C3#47112yes—
344C2×D3—81no—
345C2×D3—81no—
346C2×D3—81no—
347C2×D3—162no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—44no—
352C3×C4—81no—
353C3×C4B(4,𝔽2)12yes—
354Dic3—81no—
355C23—22no—
356C23—22no—
357C23—41no—
358C23—41no—
359C23—42no—
360C23—81no—
361C23—81no—
362C23—121no—
363C23—122no—
364C23—241no—
365C23—241no—
366C23#93811yescharacteristic
367C23#93911yescharacteristic
368C23#93911yescharacteristic
369Q8—24no—
370Q8—24no—
371Q8—42no—
372Q8—42no—
373Q8—44no—
374Q8—44no—
375D4—42no—
376D4—42no—
377D4—82no—
378D4—82no—
379D4—122no—
380D4—122no—
381D4—242no—
382D4—242no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—81no—
396C2×C4—121no—
397C2×C4—241no—
398C2×C4#93711yescharacteristic
399C2×C4#93711yescharacteristic
400C2×C4#94011yescharacteristic
401C2×C4#95311yescharacteristic
402C8—42no—
403C8—122no—
404C8—122no—
405C8—241no—
406C2×C3—22no—
407C2×C3—22no—
408C2×C3—41no—
409C2×C3—161no—
410C2×C3#201911yescharacteristic
411C2×C3#201911yescharacteristic
412C2×C3#202111yescharacteristic
413D3—161no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—44no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—162no—
426C22—241no—
427C22—241no—
428C22—241no—
429C22—482no—
430C22#517712yes—
431C22#520312yes—
432C22#522311yescharacteristic
433C4—21no—
434C4—21no—
435C4—21no—
436C4—44no—
437C4—81no—
438C4—241no—
439C4#503212yes—
440C3#998811yescharacteristic, complemented, Sylow
441C2—22no—
442C2—22no—
443C2—41no—
444C2—161no—
445C2—481no—
446C2#7167111yescharacteristic
447C2#7205911yescharacteristic
448C2#7210711yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#448C2
socleC22×C3#392
prime radical#82627C2
hypercenterC2×#448D3
hypocenterC3#9988
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#448

Complements

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

NH ≅ G/Ntype
1. C2×C3×#448C2semidirect
2. C2×#5227C2semidirect
4. #72133C2semidirect
5. #71676C2semidirect
6. #69921C2semidirect
7. #67680C2semidirect
34. C2×#448D3semidirect
440. C3#9988semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4482
C2×#52272
#826272
#721332
#716762
#699212
#676802
#99883