Finite Group Databasebeta

← #10695870

Subgroups — #10695870

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
1C3×#3193C211yesmaximal, characteristic
2C2×#5050C211yesmaximal, characteristic, complemented
3C2×#5040C211yesmaximal, characteristic
4#71691C211yesmaximal, characteristic, complemented
5#71676C211yesmaximal, characteristic
6#69990C211yesmaximal, characteristic, complemented
7#67688C211yesmaximal, characteristic, complemented
8#9989—31nomaximal, Sylow
9C23×Dic6C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C2×C3×#121C2211yescharacteristic
12C22×#216—21no—
13C22×#214—21no—
14C3×#484—21no—
15C3×#480—21no—
16C3×#468—22no—
17C3×#450—21no—
18C3×#448—21no—
19C3×#407C2211yescharacteristic
20#5207C2211yescharacteristic
21C2×#830C2211yescharacteristic
22C2×#830C2211yescharacteristic
23C2×#824—22no—
24#5050—22no—
25#5050—22no—
26#5040—22no—
27#5040—22no—
28C2×#810—21no—
29C2×#810—21no—
30#4528—22no—
31#4525—22no—
32#3193D311yescharacteristic
33C2×#451—31no—
34C2×#448—31no—
35#1784—31no—
36#1472—31no—
37#1471—31no—
38#1298—31no—
39C22×C3×Q8D411yescharacteristic
40C22×C3×D4—21no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C23×Dic3—21no—
44C22×Dic6—21no—
45C22×Dic6—21no—
46C22×Dic6—21no—
47C22×Dic6—21no—
48C22×Dic6—21no—
49C22×Dic6—44no—
50C22×Dic6D412yes—
51C2×C3×SD16—22no—
52C2×C3×M16D412yes—
53C3×C22⋊Q8—21no—
54C3×C22⋊Q8—21no—
55C3×C22⋊Q8—21no—
56C3×C22⋊Q8—21no—
57C3×#121—21no—
58C3×#121—21no—
59C3×#121—21no—
60C3×#121—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4D411yescharacteristic
63C2×C3×C4⋊C4D411yescharacteristic
64C2×C3×C22⋊C4—21no—
65C2×C3×C22⋊C4—21no—
66C2×#216—24no—
67C2×#216—44no—
68C2×#214—24no—
69C2×#214—44no—
70C2×#208—21no—
71C22×C3⋊C8—21no—
72C22×C3⋊C8—21no—
73C3×#107—22no—
74C3×#103—22no—
75C3×#102—22no—
76C3×C22⋊C8—21no—
77C3×C22⋊C8—21no—
78#830—21no—
79#830—21no—
80#830—21no—
81#830—21no—
82#830D412yes—
83#830D412yes—
84#824—44no—
85#810—24no—
86#810—24no—
87C23×Q8—31no—
88C22×Q16—61no—
89C22×SD16—61no—
90C2×C22⋊Q8C2×D311yescharacteristic
91C2×#121C2×D311yescharacteristic
92#484—21no—
93#480—21no—
94#468—22no—
95#451—62no—
96#451—62no—
97#450—21no—
98#448—21no—
99#448—62no—
100#448—62no—
101C2×#103—61no—
102C2×#103—61no—
103C2×#101—62no—
104#407—31no—
105#407C2×D311yescharacteristic
106C2×C22⋊C8—31no—
107C2×C22⋊C8—31no—
108#324—62no—
109#323—62no—
110C24×C3—21no—
111C24×C3#4111yescharacteristic
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×Q8—42no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—42no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—82no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—42no—
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×Dic3—22no—
132C22×Dic3—44no—
133C22×Dic3—41no—
134C22×Dic3—41no—
135C2×Dic6—24no—
136C2×Dic6—24no—
137C2×Dic6—22no—
138C2×Dic6—22no—
139C2×Dic6—24no—
140C2×Dic6—44no—
141C2×Dic6—44no—
142C2×Dic6—44no—
143C2×Dic6—42no—
144C2×Dic6—42no—
145C2×Dic6—42no—
146C2×Dic6—41no—
147C2×Dic6—41no—
148C3×SD16—82no—
149C3×M16—42no—
150C2×C3×C8—22no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—41no—
156C3×C4⋊C4—41no—
157C3×C22⋊C4—41no—
158C3×C22⋊C4—41no—
159C3×C22⋊C4—41no—
160C3×C22⋊C4—41no—
161#216—48no—
162#214—48no—
163#208—42no—
164C2×C3⋊C8—22no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—42no—
167C2×C3⋊C8—42no—
168C2×C3⋊C8—41no—
169C22×Q8—32no—
170C22×Q8—61no—
171C22×Q8—61no—
172C22×Q8—61no—
173C22×Q8—61no—
174C22×Q8—61no—
175C22×Q8—124no—
176C22×Q8#6711yescharacteristic
177C22×D4—21no—
178C22×D4#6711yescharacteristic
179C23×C4—61no—
180C23×C4C22×D311yescharacteristic
181C2×Q16—64no—
182C2×Q16—124no—
183C2×SD16—22no—
184C2×SD16—64no—
185C2×SD16—124no—
186C2×M16—61no—
187C2×M16#6712yes—
188C22×C8—61no—
189C22×C8—61no—
190C22⋊Q8—21no—
191C22⋊Q8—21no—
192C22⋊Q8—21no—
193C22⋊Q8—21no—
194#121—21no—
195#121—21no—
196#121—21no—
197#121—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4#6711yescharacteristic
200C2×C4⋊C4#6711yescharacteristic
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203#107—22no—
204#103—22no—
205#103—64no—
206#103—64no—
207#102—22no—
208#101—124no—
209C22⋊C8—21no—
210C22⋊C8—21no—
211C22⋊C8—32no—
212C22⋊C8—32no—
213C22⋊C8—61no—
214C22⋊C8—61no—
215C22⋊C8—61no—
216C22⋊C8—61no—
217C23×C3—22no—
218C23×C3—22no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3—42no—
222C23×C3—81no—
223C23×C3—81no—
224C23×C3#12111yescharacteristic
225C23×C3#12111yescharacteristic
226C23×C3#12111yescharacteristic
227C3×Q8—42no—
228C3×Q8—42no—
229C3×D4—42no—
230C3×D4—42no—
231C3×D4—82no—
232C3×D4—82no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—42no—
243C2×C3×C4—81no—
244C2×C3×C4—81no—
245C2×C3×C4#12011yescharacteristic
246C2×C3×C4#12011yescharacteristic
247C2×C3×C4#12011yescharacteristic
248C2×C3×C4#12711yescharacteristic
249C2×Dic3—24no—
250C2×Dic3—44no—
251C2×Dic3—44no—
252C2×Dic3—42no—
253C2×Dic3—42no—
254Dic6—24no—
255Dic6—24no—
256Dic6—42no—
257Dic6—42no—
258Dic6—44no—
259Dic6—44no—
260C3×C8—42no—
261C3⋊C8—42no—
262C3⋊C8—42no—
263C3⋊C8—81no—
264C24—21no—
265C24#23711yescharacteristic
266C2×Q8—21no—
267C2×Q8—21no—
268C2×Q8—42no—
269C2×Q8—64no—
270C2×Q8—64no—
271C2×Q8—64no—
272C2×Q8—62no—
273C2×Q8—62no—
274C2×Q8—122no—
275C2×Q8—122no—
276C2×Q8—122no—
277C2×Q8—124no—
278C2×Q8—124no—
279C2×Q8—124no—
280C2×Q8—121no—
281C2×Q8—121no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—42no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—82no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—42no—
292C22×C4—62no—
293C22×C4—124no—
294C22×C4—121no—
295C22×C4—121no—
296C22×C4D3×D411yescharacteristic
297C22×C4D3×D411yescharacteristic
298C22×C4D3×D411yescharacteristic
299C22×C4C2×#6711yescharacteristic
300C22×C4C2×#6711yescharacteristic
301C22×C4C2×#6711yescharacteristic
302Q16—128no—
303SD16—82no—
304SD16—128no—
305M16—42no—
306M16—122no—
307C2×C8—22no—
308C2×C8—62no—
309C2×C8—62no—
310C2×C8—122no—
311C2×C8—122no—
312C2×C8—121no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—41no—
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—81no—
332C22×C3—81no—
333C22×C3—81no—
334C22×C3—162no—
335C22×C3#39211yescharacteristic
336C22×C3#47112yes—
337C22×C3#47312yes—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—81no—
342C3×C4—81no—
343C3×C4—81no—
344C3×C4B(4,𝔽2)12yes—
345Dic3—44no—
346C23—22no—
347C23—22no—
348C23—41no—
349C23—41no—
350C23—42no—
351C23—81no—
352C23—81no—
353C23#93811yescharacteristic
354C23#93911yescharacteristic
355C23#93911yescharacteristic
356Q8—42no—
357Q8—42no—
358Q8—64no—
359Q8—64no—
360Q8—122no—
361Q8—122no—
362Q8—124no—
363Q8—124no—
364D4—42no—
365D4—42no—
366D4—82no—
367D4—82no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—64no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—122no—
382C2×C4—124no—
383C2×C4—124no—
384C2×C4—122no—
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—161no—
397C2×C3#201911yescharacteristic
398C2×C3#202011yescharacteristic
399C2×C3#202111yescharacteristic
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—44no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—162no—
412C22#520012yes—
413C22#520412yes—
414C22#522311yescharacteristic
415C4—21no—
416C4—21no—
417C4—21no—
418C4—81no—
419C4—81no—
420C4—81no—
421C4—124no—
422C4#524412yes—
423C3#998911yescharacteristic, complemented, Sylow
424C2—22no—
425C2—22no—
426C2—41no—
427C2—161no—
428C2#7185111yescharacteristic
429C2#7209711yescharacteristic
430C2#7210911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#3193C2
socleC22×C3#392
prime radicalC2×C3×#121C22
hypercenter#3193D3
hypocenterC3#9989
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3193

Complements

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

NH ≅ G/Ntype
2. C2×#5050C2semidirect
4. #71691C2semidirect
6. #69990C2semidirect
7. #67688C2semidirect
423. C3#9989semidirect

Maximal subgroups

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

subgroupindex
C3×#31932
C2×#50502
C2×#50402
#716912
#716762
#699902
#676882
#99893