Finite Group Databasebeta

← #10698110

Subgroups — #10698110

Order 768.

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

normal subgroups only

This group has 672 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×#2501C211yesmaximal, characteristic
2#82631C211yesmaximal, characteristic
3#82178C211yesmaximal, characteristic, complemented
4#71673C211yesmaximal, characteristic
5#71562C211yesmaximal, characteristic, complemented
6#69445C211yesmaximal, characteristic, complemented
7#67618C211yesmaximal, characteristic, complemented
8#10331—31nomaximal, Sylow
9C2×C3×#123—22no—
10C2×C3×#121C2211yescharacteristic
11C22×C3×C4⋊C4C2211yescharacteristic
12C2×#924C2211yescharacteristic
13C3×#399—22no—
14C3×#390—21no—
15C3×#384—21no—
16C3×C4×C22⋊C4C2211yescharacteristic
17#5218—21no—
18#5210—22no—
19C4×#217C2211yescharacteristic
20#5207C2211yescharacteristic
21#5207C2211yescharacteristic
22#5042—22no—
23#5040—21no—
24#5033—21no—
25#4983—22no—
26#4975—21no—
27#4973—21no—
28#4970—21no—
29#4537—22no—
30#4530—22no—
31#4452—22no—
32#3193—31no—
33#2502—31no—
34#2501D311yescharacteristic
35#1821—31no—
36#1508—31no—
37#1507—31no—
38#1297—31no—
39C22×C3×D4—21no—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4D412yes—
42C23×C3×C4C2311yescharacteristic
43C22×Dic6D411yescharacteristic
44C3×#123—48no—
45C3×#121—21no—
46C3×#121—21no—
47C3×#121—21no—
48C3×#121—21no—
49C2×C3×C4⋊C4—24no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—24no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C42—21no—
61C2×#217—21no—
62C2×#217—21no—
63C2×#214—22no—
64C2×C4⋊Dic3D411yescharacteristic
65#924—21no—
66#924—21no—
67#924—21no—
68#924—21no—
69C2×Dic3⋊C4—21no—
70C2×Dic3⋊C4—24no—
71C2×C4×Dic3—21no—
72C2×#208—21no—
73C2×#208D412yes—
74C3×#95D412yes—
75#832—22no—
76C3⋊#95—24no—
77C3⋊#95D412yes—
78#830—21no—
79#830—21no—
80#830—21no—
81#830—21no—
82#810—22no—
83#807—22no—
84C2×#123—22no—
85C2×C22⋊Q8—31no—
86C2×#121C2×D311yescharacteristic
87C22×C4⋊C4C2×D311yescharacteristic
88#484—61no—
89#480—61no—
90#468—62no—
91#450—61no—
92#448—61no—
93#407—31no—
94#407—31no—
95#399—22no—
96#398—62no—
97#391—61no—
98#390—21no—
99#385—62no—
100#384—21no—
101#384—61no—
102C4×C22⋊C4—31no—
103C4×C22⋊C4C2×D311yescharacteristic
104#343—62no—
105#328—62no—
106#327—62no—
107C24×C3—21no—
108C24×C3C2×D411yescharacteristic
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—42no—
112C2×C3×D4—44no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C22×C3×C4—22no—
116C22×C3×C4—22no—
117C22×C3×C4—22no—
118C22×C3×C4—21no—
119C22×C3×C4—22no—
120C22×C3×C4—42no—
121C22×C3×C4—44no—
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4#4111yescharacteristic
128C22×Dic3—22no—
129C22×Dic3—21no—
130C22×Dic3—21no—
131C2×Dic6—21no—
132C2×Dic6—21no—
133C2×Dic6—42no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—48no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C22⋊C4—24no—
142C3×C22⋊C4—48no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C42—42no—
146#217—24no—
147#217—41no—
148#217—41no—
149#214—82no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152Dic3⋊C4—41no—
153Dic3⋊C4—41no—
154Dic3⋊C4—84no—
155C4×Dic3—42no—
156#208—42no—
157#208—42no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—41no—
160C22×Q8—31no—
161C22×D4—21no—
162C22×D4#6711yescharacteristic
163C23×C4#6712yes—
164C23×C4C22×D311yescharacteristic
165C2×SD16—62no—
166C2×M16—32no—
167C2×M16—61no—
168#123—48no—
169C22⋊Q8—61no—
170C22⋊Q8—61no—
171C22⋊Q8—61no—
172C22⋊Q8—61no—
173#121—21no—
174#121—21no—
175#121—21no—
176#121—21no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4—64no—
187C2×C4⋊C4#6711yescharacteristic
188C2×C22⋊C4—21no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4—61no—
193C2×C42—21no—
194C2×C42—61no—
195#107—62no—
196#103—62no—
197#102—62no—
198C22⋊C8—61no—
199C22⋊C8—61no—
200C22⋊C8—61no—
201C22⋊C8—61no—
202#95—32no—
203#95—64no—
204#95#6712yes—
205C23×C3—22no—
206C23×C3—22no—
207C23×C3—41no—
208C23×C3—42no—
209C23×C3—42no—
210C23×C3—41no—
211C23×C3—42no—
212C23×C3#12011yescharacteristic
213C23×C3#12011yescharacteristic
214C23×C3#12111yescharacteristic
215C3×D4—42no—
216C3×D4—81no—
217C3×D4—82no—
218C3×D4—82no—
219C2×C3×C4—22no—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—24no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4—44no—
229C2×C3×C4—42no—
230C2×C3×C4—41no—
231C2×C3×C4—42no—
232C2×C3×C4—44no—
233C2×C3×C4—82no—
234C2×C3×C4#12011yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12711yescharacteristic
238C2×Dic3—24no—
239C2×Dic3—42no—
240C2×Dic3—42no—
241C2×Dic3—42no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—42no—
245Dic6—42no—
246Dic6—81no—
247C3⋊C8—42no—
248C3⋊C8—81no—
249C24—21no—
250C24C2×#6711yescharacteristic
251C2×Q8—61no—
252C2×Q8—61no—
253C2×Q8—122no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—42no—
257C2×D4—44no—
258C2×D4—41no—
259C2×D4—41no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—42no—
266C22×C4—44no—
267C22×C4—62no—
268C22×C4—61no—
269C22×C4—61no—
270C22×C4D3×D411yescharacteristic
271C22×C4D3×D411yescharacteristic
272C22×C4D3×D411yescharacteristic
273C22×C4#23711yescharacteristic
274C22×C4C2×#6711yescharacteristic
275C22×C4C2×#6711yescharacteristic
276SD16—242no—
277M16—122no—
278M16—122no—
279C2×C8—62no—
280C2×C8—121no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—48no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—61no—
289C4⋊C4—61no—
290C4⋊C4—121no—
291C4⋊C4—121no—
292C4⋊C4—244no—
293C22⋊C4—24no—
294C22⋊C4—48no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—64no—
298C22⋊C4—121no—
299C22⋊C4—121no—
300C42—42no—
301C42—122no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—44no—
310C22×C3—84no—
311C22×C3—81no—
312C22×C3—81no—
313C22×C3—81no—
314C22×C3#39211yescharacteristic
315C22×C3#45312yes—
316C22×C3#45612yes—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—44no—
321C3×C4—82no—
322C3×C4—81no—
323C3×C4#45812yes—
324Dic3—44no—
325Dic3—81no—
326Dic3—81no—
327C23—22no—
328C23—22no—
329C23—41no—
330C23—42no—
331C23—42no—
332C23—41no—
333C23—42no—
334C23#93711yescharacteristic
335C23#93911yescharacteristic
336C23#95311yescharacteristic
337Q8—122no—
338Q8—241no—
339D4—42no—
340D4—81no—
341D4—82no—
342D4—82no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—64no—
358C2×C4—82no—
359C2×C4—122no—
360C2×C4—122no—
361C2×C4—122no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4—122no—
365C2×C4#93711yescharacteristic
366C2×C4#93811yescharacteristic
367C2×C4#93911yescharacteristic
368C2×C4#94011yescharacteristic
369C8—122no—
370C8—241no—
371C2×C3—22no—
372C2×C3—22no—
373C2×C3—41no—
374C2×C3—161no—
375C2×C3#200211yescharacteristic
376C2×C3#200511yescharacteristic
377C2×C3#202211yescharacteristic
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—44no—
386C22—84no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22#507712yes—
391C22#522311yescharacteristic
392C22#524612yes—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—44no—
397C4—82no—
398C4—81no—
399C4—124no—
400C4—241no—
401C4—241no—
402C4#505312yes—
403C3#1033111yescharacteristic, complemented, Sylow
404C2—22no—
405C2—22no—
406C2—41no—
407C2—161no—
408C2#7170611yescharacteristic
409C2#7176911yescharacteristic
410C2#7186811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2501C2
socleC22×C3#392
prime radicalC2×C3×#121C22
hypercenter#2501D3
hypocenterC3#10331
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2501

Complements

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

NH ≅ G/Ntype
3. #82178C2semidirect
5. #71562C2semidirect
6. #69445C2semidirect
7. #67618C2semidirect
403. C3#10331semidirect

Maximal subgroups

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

subgroupindex
C3×#25012
#826312
#821782
#716732
#715622
#694452
#676182
#103313