Finite Group Databasebeta

← #10695869

Subgroups — #10695869

Order 768.

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

normal subgroups only

This group has 724 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×#451C211yesmaximal, characteristic, complemented
2C2×#5227C211yesmaximal, characteristic
3#82738C211yesmaximal, characteristic
4#72133C211yesmaximal, characteristic, complemented
5#71691C211yesmaximal, characteristic, complemented
6#69921C211yesmaximal, characteristic
7#67680C211yesmaximal, characteristic, complemented
8#9989—31nomaximal, Sylow
9C23×C3×Q8C2211yescharacteristic
10C22×C3×Q16—21no—
11C2×C3×C22⋊Q8C2211yescharacteristic
12C22×#215—21no—
13C2×#930C2211yescharacteristic
14C3×#451—22no—
15C3×#451—22no—
16C2×C3×#103—21no—
17C2×C3×C22⋊C8C2211yescharacteristic
18#5227—21no—
19#5227—22no—
20#5227—22no—
21C2×#836—22no—
22C2×#835—21no—
23#5207C2211yescharacteristic
24#5207C2211yescharacteristic
25C2×#830C2211yescharacteristic
26#5049—22no—
27#5046—21no—
28#5044—21no—
29#4967—21no—
30#4528—22no—
31#4468—22no—
32#3193—31no—
33C2×#451D311yescharacteristic, complemented
34C2×#448—31no—
35#1784—31no—
36#1472—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×Q8D411yescharacteristic
46C22×C3×Q8D412yes—
47C23×C3×C4—21no—
48C23×C3×C4C2311yescharacteristic
49C22×#67—21no—
50C22×D12D411yescharacteristic
51C2×C3×Q16—24no—
52C2×C3×Q16—44no—
53C22×C3×C8—21no—
54C3×C22⋊Q8—21no—
55C3×C22⋊Q8—21no—
56C3×C22⋊Q8—21no—
57C3×C22⋊Q8—21no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C22⋊C4—21no—
61C2×#215—24no—
62C2×#215—22no—
63C2×#215—44no—
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#835—22no—
79#830—21no—
80#830—21no—
81#830—21no—
82#830—21no—
83#830—21no—
84#830D412yes—
85#809—22no—
86#807—22no—
87C23×Q8C2×D311yescharacteristic
88C22×Q16—21no—
89C22×SD16—61no—
90C2×C22⋊Q8C2×D311yescharacteristic
91C2×#121—31no—
92#484—61no—
93#480—61no—
94#468—62no—
95#451—22no—
96#451—22no—
97#450—61no—
98#448—62no—
99#448—61no—
100#448—62no—
101C2×#103—21no—
102C2×#103—61no—
103C2×#101—62no—
104#407—31no—
105#407—31no—
106C2×C22⋊C8—31no—
107C2×C22⋊C8C2×D311yescharacteristic
108#324—62no—
109#323—62no—
110C24×C3#4111yescharacteristic
111C23×D3—21no—
112C2×C3×Q8—24no—
113C2×C3×Q8—24no—
114C2×C3×Q8—24no—
115C2×C3×Q8—22no—
116C2×C3×Q8—22no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—44no—
120C2×C3×Q8—42no—
121C2×C3×Q8—42no—
122C2×C3×Q8—42no—
123C2×C3×Q8—42no—
124C2×C3×Q8—44no—
125C2×C3×Q8—44no—
126C2×C3×Q8—41no—
127C2×C3×Q8—41no—
128C22×C3×C4—22no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—42no—
132C22×C3×C4—44no—
133C22×C3×C4—41no—
134C22×C3×C4—41no—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C2×#67—41no—
142C2×#67—41no—
143C2×#67—82no—
144C22×Dic3—21no—
145C2×D12—21no—
146C2×D12—21no—
147C2×D12—42no—
148C3×Q16—48no—
149C2×C3×C8—22no—
150C2×C3×C8—42no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—41no—
154C3×C4⋊C4—41no—
155C3×C22⋊C4—41no—
156C3×C22⋊C4—41no—
157#215—48no—
158#215—82no—
159(C2×D3)⋊C4—41no—
160(C2×D3)⋊C4—41no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163#208—42no—
164#208—42no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—42no—
168C2×C3⋊C8—41no—
169C22×Q8—21no—
170C22×Q8—21no—
171C22×Q8—21no—
172C22×Q8—21no—
173C22×Q8—21no—
174C22×Q8—44no—
175C22×Q8#6711yescharacteristic
176C22×Q8#6712yes—
177C22×D4—31no—
178C22×D4—61no—
179C23×C4—21no—
180C23×C4C22×D311yescharacteristic
181C2×Q16—24no—
182C2×Q16—44no—
183C2×SD16—64no—
184C2×SD16—62no—
185C2×SD16—124no—
186C2×M16—32no—
187C2×M16—61no—
188C22×C8—21no—
189C22×C8—61no—
190C22⋊Q8—21no—
191C22⋊Q8—21no—
192C22⋊Q8—21no—
193C22⋊Q8—21no—
194#121—61no—
195#121—61no—
196#121—61no—
197#121—61no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—31no—
200C2×C4⋊C4#6711yescharacteristic
201C2×C22⋊C4—21no—
202C2×C22⋊C4—61no—
203#107—62no—
204#103—24no—
205#103—64no—
206#103—62no—
207#102—62no—
208#101—124no—
209C22⋊C8—21no—
210C22⋊C8—32no—
211C22⋊C8—61no—
212C22⋊C8—61no—
213C22⋊C8—61no—
214C22⋊C8—61no—
215C22⋊C8—61no—
216C22⋊C8#6712yes—
217C23×C3—22no—
218C23×C3—22no—
219C23×C3—41no—
220C23×C3#12111yescharacteristic
221C23×C3#12111yescharacteristic
222C23×C3#12111yescharacteristic
223C22×D3—41no—
224C22×D3—42no—
225C22×D3—81no—
226C22×D3—81no—
227C3×Q8—24no—
228C3×Q8—24no—
229C3×Q8—42no—
230C3×Q8—44no—
231C3×Q8—42no—
232C3×Q8—42no—
233C3×Q8—44no—
234C3×Q8—42no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—24no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—42no—
243C2×C3×C4—44no—
244C2×C3×C4—44no—
245C2×C3×C4—42no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—42no—
249C2×C3×C4—81no—
250C2×C3×C4#12011yescharacteristic
251C2×C3×C4#12011yescharacteristic
252C2×C3×C4#12011yescharacteristic
253C2×C3×C4#12711yescharacteristic
254#67—82no—
255#67—82no—
256C2×Dic3—41no—
257C2×Dic3—81no—
258D12—42no—
259D12—42no—
260C3×C8—42no—
261C3⋊C8—42no—
262C3⋊C8—42no—
263C3⋊C8—81no—
264C24—61no—
265C24#23711yescharacteristic
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—22no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—21no—
273C2×Q8—44no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—44no—
279C2×Q8—44no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×D4—61no—
283C2×D4—61no—
284C2×D4—122no—
285C2×D4—121no—
286C2×D4—121no—
287C2×D4—242no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—42no—
292C22×C4—44no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—61no—
296C22×C4D3×D411yescharacteristic
297C22×C4D3×D411yescharacteristic
298C22×C4D3×D411yescharacteristic
299C22×C4C2×#6711yescharacteristic
300C22×C4C2×#6711yescharacteristic
301C22×C4C2×#6711yescharacteristic
302Q16—48no—
303SD16—128no—
304SD16—242no—
305M16—122no—
306M16—122no—
307C2×C8—22no—
308C2×C8—42no—
309C2×C8—62no—
310C2×C8—62no—
311C2×C8—122no—
312C2×C8—121no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—61no—
318C4⋊C4—61no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—121no—
322C22⋊C4—121no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—44no—
331C22×C3#39211yescharacteristic
332C22×C3#47112yes—
333C22×C3#47312yes—
334C2×D3—81no—
335C2×D3—81no—
336C2×D3—81no—
337C2×D3—162no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—44no—
342C3×C4—81no—
343C3×C4—81no—
344C3×C4B(4,𝔽2)12yes—
345Dic3—81no—
346C23—22no—
347C23—22no—
348C23—41no—
349C23—121no—
350C23—122no—
351C23—241no—
352C23—241no—
353C23#93811yescharacteristic
354C23#93911yescharacteristic
355C23#93911yescharacteristic
356Q8—24no—
357Q8—24no—
358Q8—42no—
359Q8—42no—
360Q8—44no—
361Q8—42no—
362Q8—44no—
363Q8—42no—
364D4—122no—
365D4—122no—
366D4—242no—
367D4—242no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—81no—
383C2×C4—121no—
384C2×C4—241no—
385C2×C4#93711yescharacteristic
386C2×C4#93711yescharacteristic
387C2×C4#94011yescharacteristic
388C2×C4#95311yescharacteristic
389C8—42no—
390C8—122no—
391C8—122no—
392C8—241no—
393C2×C3—22no—
394C2×C3—22no—
395C2×C3—41no—
396C2×C3#201911yescharacteristic
397C2×C3#202011yescharacteristic
398C2×C3#202111yescharacteristic
399D3—161no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—44no—
408C22—241no—
409C22—241no—
410C22—241no—
411C22—482no—
412C22#519212yes—
413C22#520312yes—
414C22#522311yescharacteristic
415C4—21no—
416C4—21no—
417C4—21no—
418C4—44no—
419C4—81no—
420C4—81no—
421C4—241no—
422C4#503212yes—
423C3#998911yescharacteristic, complemented, Sylow
424C2—22no—
425C2—22no—
426C2—41no—
427C2—481no—
428C2#7167111yescharacteristic
429C2#7207411yescharacteristic
430C2#7210711yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#451C2
socleC22×C3#392
prime radicalC2×#930C22
hypercenterC2×#451D3
hypocenterC3#9989
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#451

Complements

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

NH ≅ G/Ntype
1. C2×C3×#451C2semidirect
4. #72133C2semidirect
5. #71691C2semidirect
7. #67680C2semidirect
33. C2×#451D3semidirect
423. C3#9989semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4512
C2×#52272
#827382
#721332
#716912
#699212
#676802
#99893