Finite Group Databasebeta

← #10695876

Subgroups — #10695876

Order 768.

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

normal subgroups only

This group has 826 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×#450C211yesmaximal, characteristic
2C2×#5218C211yesmaximal, characteristic
3#82742C211yesmaximal, characteristic, complemented
4#72129C211yesmaximal, characteristic
5#71691C211yesmaximal, characteristic, complemented
6#69923C211yesmaximal, characteristic, complemented
7#67680C211yesmaximal, characteristic, complemented
8#9990—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C22×C3×SD16—21no—
11C2×C3×C22⋊Q8C2211yescharacteristic
12C22×#214—21no—
13C2×#924C2211yescharacteristic
14C3×#450—22no—
15C3×#450—22no—
16C2×C3×#102—21no—
17C2×C3×C22⋊C8C2211yescharacteristic
18#5228—21no—
19#5218—22no—
20#5218—22no—
21C2×#833—22no—
22C2×#832—21no—
23#5207C2211yescharacteristic
24#5207C2211yescharacteristic
25C2×#830C2211yescharacteristic
26#5052—22no—
27#5050—21no—
28#5043—21no—
29#4971—21no—
30#4528—22no—
31#4469—22no—
32#3195—31no—
33C2×#450—31no—
34C2×#450D311yescharacteristic
35#1783—31no—
36#1472—31no—
37#1472—31no—
38#1298—31no—
39C25×C3—21no—
40C22×C3×Q8D411yescharacteristic
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×D4D412yes—
48C23×C3×C4C2311yescharacteristic
49C22×Dic6D411yescharacteristic
50C2×C3×SD16—24no—
51C2×C3×SD16—44no—
52C22×C3×C8—21no—
53C3×C22⋊Q8—21no—
54C3×C22⋊Q8—21no—
55C3×C22⋊Q8—21no—
56C3×C22⋊Q8—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C22⋊C4—21no—
60C2×#217—21no—
61C2×#216—22no—
62C2×#214—24no—
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#835—22no—
77#833—44no—
78#832—24no—
79#830—21no—
80#830—21no—
81#830—21no—
82#830—21no—
83#830—21no—
84#830D412yes—
85#810—22no—
86#808—22no—
87C23×D4C2×D311yescharacteristic
88C22×SD16—21no—
89C22×SD16—61no—
90C2×C22⋊Q8—31no—
91C2×C22⋊Q8C2×D311yescharacteristic
92#483—61no—
93#483—61no—
94#470—62no—
95#451—61no—
96#451—61no—
97#450—22no—
98#450—22no—
99#450—62no—
100#450—62no—
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#324—62no—
110C24×C3—22no—
111C24×C3—44no—
112C24×C3—41no—
113C24×C3—41no—
114C24×C3—82no—
115C24×C3—82no—
116C24×C3#4111yescharacteristic
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—42no—
120C2×C3×D4—24no—
121C2×C3×D4—24no—
122C2×C3×D4—24no—
123C2×C3×D4—22no—
124C2×C3×D4—22no—
125C2×C3×D4—42no—
126C2×C3×D4—42no—
127C2×C3×D4—42no—
128C2×C3×D4—44no—
129C2×C3×D4—44no—
130C2×C3×D4—44no—
131C2×C3×D4—41no—
132C2×C3×D4—41no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—42no—
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×Dic3—21no—
143C22×Dic3—21no—
144C2×Dic6—21no—
145C2×Dic6—21no—
146C2×Dic6—42no—
147C3×SD16—48no—
148C2×C3×C8—22no—
149C2×C3×C8—42no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—41no—
153C3×C4⋊C4—41no—
154C3×C22⋊C4—41no—
155C3×C22⋊C4—41no—
156#217—41no—
157#217—41no—
158#216—82no—
159#214—48no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162Dic3⋊C4—41no—
163Dic3⋊C4—41no—
164#208—42no—
165#208—42no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—42no—
169C2×C3⋊C8—41no—
170C25—21no—
171C22×Q8—31no—
172C22×Q8#6711yescharacteristic
173C22×D4—21no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—44no—
179C22×D4#6712yes—
180C23×C4C22×D311yescharacteristic
181C2×Q16—62no—
182C2×SD16—24no—
183C2×SD16—44no—
184C2×SD16—64no—
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—
194C22⋊Q8—61no—
195C22⋊Q8—61no—
196C22⋊Q8—61no—
197C22⋊Q8—61no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—31no—
200C2×C4⋊C4—61no—
201C2×C4⋊C4#6711yescharacteristic
202C2×C22⋊C4—21no—
203C2×C22⋊C4—61no—
204#106—62no—
205#103—62no—
206#103—62no—
207#102—24no—
208#102—64no—
209#100—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—24no—
221C23×C3—41no—
222C23×C3—44no—
223C23×C3—44no—
224C23×C3—42no—
225C23×C3—42no—
226C23×C3—82no—
227C23×C3—82no—
228C23×C3—82no—
229C23×C3—82no—
230C23×C3—816no—
231C23×C3—82no—
232C23×C3—82no—
233C23×C3#12111yescharacteristic
234C23×C3#12111yescharacteristic
235C23×C3#12111yescharacteristic
236C3×Q8—42no—
237C3×Q8—42no—
238C3×D4—24no—
239C3×D4—24no—
240C3×D4—42no—
241C3×D4—44no—
242C3×D4—42no—
243C3×D4—44no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—42no—
253C2×C3×C4—81no—
254C2×C3×C4#12011yescharacteristic
255C2×C3×C4#12011yescharacteristic
256C2×C3×C4#12011yescharacteristic
257C2×C3×C4#12711yescharacteristic
258C2×Dic3—41no—
259C2×Dic3—41no—
260C2×Dic3—42no—
261C2×Dic3—81no—
262Dic6—42no—
263Dic6—42no—
264C3×C8—42no—
265C3⋊C8—42no—
266C3⋊C8—42no—
267C3⋊C8—81no—
268C24—22no—
269C24—44no—
270C24—41no—
271C24—41no—
272C24—82no—
273C24—82no—
274C24#23711yescharacteristic
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—42no—
278C2×Q8—61no—
279C2×Q8—61no—
280C2×Q8—122no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—44no—
290C2×D4—44no—
291C2×D4—44no—
292C2×D4—41no—
293C2×D4—41no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—42no—
297C22×C4—61no—
298C22×C4—61no—
299C22×C4D3×D411yescharacteristic
300C22×C4D3×D411yescharacteristic
301C22×C4D3×D411yescharacteristic
302C22×C4C2×#6711yescharacteristic
303C22×C4C2×#6711yescharacteristic
304C22×C4C2×#6711yescharacteristic
305Q16—242no—
306SD16—48no—
307SD16—128no—
308M16—122no—
309M16—122no—
310C2×C8—22no—
311C2×C8—42no—
312C2×C8—62no—
313C2×C8—62no—
314C2×C8—122no—
315C2×C8—121no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—61no—
321C4⋊C4—61no—
322C4⋊C4—121no—
323C4⋊C4—121no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—121no—
327C22⋊C4—121no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—44no—
336C22×C3—44no—
337C22×C3—88no—
338C22×C3—88no—
339C22×C3—84no—
340C22×C3—84no—
341C22×C3—82no—
342C22×C3—82no—
343C22×C3#39211yescharacteristic
344C22×C3#47112yes—
345C22×C3#47112yes—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—81no—
350C3×C4—81no—
351C3×C4B(4,𝔽2)12yes—
352Dic3—81no—
353Dic3—81no—
354C23—22no—
355C23—22no—
356C23—24no—
357C23—41no—
358C23—44no—
359C23—44no—
360C23—42no—
361C23—42no—
362C23—82no—
363C23—82no—
364C23—82no—
365C23—82no—
366C23—816no—
367C23—82no—
368C23—82no—
369C23#93811yescharacteristic
370C23#93911yescharacteristic
371C23#93911yescharacteristic
372Q8—42no—
373Q8—42no—
374Q8—122no—
375Q8—122no—
376D4—24no—
377D4—24no—
378D4—42no—
379D4—42no—
380D4—44no—
381D4—44no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—81no—
392C2×C4—122no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—241no—
396C2×C4#93711yescharacteristic
397C2×C4#93711yescharacteristic
398C2×C4#94011yescharacteristic
399C2×C4#95311yescharacteristic
400C8—42no—
401C8—122no—
402C8—122no—
403C8—241no—
404C2×C3—22no—
405C2×C3—22no—
406C2×C3—41no—
407C2×C3—84no—
408C2×C3#201911yescharacteristic
409C2×C3#201911yescharacteristic
410C2×C3#202111yescharacteristic
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—44no—
419C22—44no—
420C22—88no—
421C22—88no—
422C22—84no—
423C22—84no—
424C22—82no—
425C22—82no—
426C22#517712yes—
427C22#520012yes—
428C22#522311yescharacteristic
429C4—21no—
430C4—21no—
431C4—21no—
432C4—81no—
433C4—81no—
434C4—241no—
435C4—241no—
436C4#503212yes—
437C3#999011yescharacteristic, complemented, Sylow
438C2—22no—
439C2—22no—
440C2—41no—
441C2—84no—
442C2#7167111yescharacteristic
443C2#7205911yescharacteristic
444C2#7209711yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#450C2
socleC22×C3#392
prime radicalC23×C3×D4C22
hypercenterC2×#450D3
hypocenterC3#9990
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#450

Complements

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

NH ≅ G/Ntype
3. #82742C2semidirect
5. #71691C2semidirect
6. #69923C2semidirect
7. #67680C2semidirect
437. C3#9990semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4502
C2×#52182
#827422
#721292
#716912
#699232
#676802
#99903