Finite Group Databasebeta

← #10698115

Subgroups — #10698115

Order 768.

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

normal subgroups only

This group has 600 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×#2503C211yesmaximal, characteristic, complemented
2#82738C211yesmaximal, characteristic
3#82180C211yesmaximal, characteristic
4#71688C211yesmaximal, characteristic, complemented
5#71562C211yesmaximal, characteristic, complemented
6#69444C211yesmaximal, characteristic
7#67618C211yesmaximal, characteristic, complemented
8#10332—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×C3×C4⋊C4C2211yescharacteristic
11C2×#930C2211yescharacteristic
12C3×#400—22no—
13C3×#391—21no—
14C3×#382—21no—
15C3×#381—22no—
16C3×C4×C22⋊C4C2211yescharacteristic
17#5227—21no—
18#5213—22no—
19C4×#217C2211yescharacteristic
20#5207C2211yescharacteristic
21#5207C2211yescharacteristic
22#5049—22no—
23#5046—21no—
24#5044—21no—
25#4992—22no—
26#4988—21no—
27#4977—21no—
28#4967—21no—
29#4538—22no—
30#4530—22no—
31#4451—22no—
32#3193—31no—
33#2504—31no—
34#2503D311yescharacteristic, complemented
35#1821—31no—
36#1508—31no—
37#1507—31no—
38#1297—31no—
39C22×C3×Q8D411yescharacteristic
40C23×C3×C4D412yes—
41C23×C3×C4C2311yescharacteristic
42C22×#67—21no—
43C22×D12D411yescharacteristic
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
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⋊C4D411yescharacteristic
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C42—21no—
59C2×#217—21no—
60C2×#215—22no—
61#930—21no—
62#930—21no—
63#930—21no—
64#930—21no—
65C2×(C2×D3)⋊C4—21no—
66C2×(C2×D3)⋊C4—24no—
67C2×C4⋊Dic3D411yescharacteristic
68C2×C4×Dic3—21no—
69C2×#208—21no—
70C2×#208D412yes—
71C3×#95—24no—
72C3×#95—24no—
73C3×#95D412yes—
74#835—22no—
75C3⋊#95—24no—
76C3⋊#95D412yes—
77#830—21no—
78#830—21no—
79#830—21no—
80#830—21no—
81#809—22no—
82#807—22no—
83C2×C22⋊Q8C2×D311yescharacteristic
84C2×#121—31no—
85C22×C4⋊C4C2×D311yescharacteristic
86#484—61no—
87#480—61no—
88#468—62no—
89#450—61no—
90#448—61no—
91#407—31no—
92#407—31no—
93#400—22no—
94#397—62no—
95#391—21no—
96#390—61no—
97#386—62no—
98#382—21no—
99#382—61no—
100#381—22no—
101C4×C22⋊C4—31no—
102C4×C22⋊C4C2×D311yescharacteristic
103#343—62no—
104#328—62no—
105#327—62no—
106C24×C3C2×D411yescharacteristic
107C23×D3—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—42no—
111C22×C3×C4—22no—
112C22×C3×C4—22no—
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4—22no—
116C22×C3×C4—22no—
117C22×C3×C4—42no—
118C22×C3×C4—44no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C2×#67—44no—
126C2×#67—41no—
127C2×#67—41no—
128C22×Dic3—21no—
129C22×Dic3—22no—
130C2×D12—21no—
131C2×D12—21no—
132C2×D12—42no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C22⋊C4—42no—
142C3×C22⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C42—42no—
145#217—42no—
146#215—82no—
147(C2×D3)⋊C4—41no—
148(C2×D3)⋊C4—41no—
149(C2×D3)⋊C4—84no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4×Dic3—42no—
153#208—42no—
154#208—42no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—41no—
157C22×Q8#6711yescharacteristic
158C22×D4—31no—
159C22×D4—61no—
160C23×C4#6712yes—
161C23×C4C22×D311yescharacteristic
162C2×SD16—62no—
163C2×M16—32no—
164C2×M16—61no—
165C22⋊Q8—21no—
166C22⋊Q8—21no—
167C22⋊Q8—21no—
168C22⋊Q8—21no—
169#121—61no—
170#121—61no—
171#121—61no—
172#121—61no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—31no—
181C2×C4⋊C4#6711yescharacteristic
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—61no—
186C2×C22⋊C4—64no—
187C2×C42—21no—
188C2×C42—61no—
189#107—62no—
190#103—62no—
191#102—62no—
192C22⋊C8—61no—
193C22⋊C8—61no—
194C22⋊C8—61no—
195C22⋊C8—61no—
196#95—24no—
197#95—24no—
198#95—32no—
199#95—64no—
200#95#6712yes—
201C23×C3—22no—
202C23×C3—22no—
203C23×C3—41no—
204C23×C3#12011yescharacteristic
205C23×C3#12011yescharacteristic
206C23×C3#12111yescharacteristic
207C22×D3—42no—
208C22×D3—42no—
209C22×D3—41no—
210C22×D3—42no—
211C3×Q8—42no—
212C3×Q8—81no—
213C2×C3×C4—22no—
214C2×C3×C4—22no—
215C2×C3×C4—22no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—44no—
222C2×C3×C4—42no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4—44no—
229C2×C3×C4—82no—
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12711yescharacteristic
234#67—82no—
235#67—82no—
236C2×Dic3—42no—
237C2×Dic3—42no—
238C2×Dic3—42no—
239C2×Dic3—41no—
240C2×Dic3—42no—
241D12—42no—
242D12—81no—
243C3⋊C8—42no—
244C3⋊C8—81no—
245C24—61no—
246C24C2×#6711yescharacteristic
247C2×Q8—21no—
248C2×Q8—21no—
249C2×Q8—42no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4—122no—
253C2×D4—124no—
254C2×D4—121no—
255C2×D4—121no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—42no—
263C22×C4—44no—
264C22×C4—62no—
265C22×C4—61no—
266C22×C4D3×D411yescharacteristic
267C22×C4D3×D411yescharacteristic
268C22×C4D3×D411yescharacteristic
269C22×C4#23711yescharacteristic
270C22×C4C2×#6711yescharacteristic
271C22×C4C2×#6711yescharacteristic
272SD16—242no—
273M16—122no—
274M16—122no—
275C2×C8—62no—
276C2×C8—121no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—61no—
286C4⋊C4—61no—
287C22⋊C4—42no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—122no—
291C22⋊C4—121no—
292C22⋊C4—121no—
293C22⋊C4—244no—
294C42—42no—
295C42—122no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—44no—
304C22×C3#39211yescharacteristic
305C22×C3#45412yes—
306C22×C3#45512yes—
307C2×D3—84no—
308C2×D3—81no—
309C2×D3—81no—
310C2×D3—81no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—82no—
315C3×C4—81no—
316C3×C4—82no—
317C3×C4—81no—
318C3×C4#45812yes—
319Dic3—82no—
320Dic3—81no—
321C23—22no—
322C23—22no—
323C23—41no—
324C23—122no—
325C23—122no—
326C23—121no—
327C23—122no—
328C23#93711yescharacteristic
329C23#93911yescharacteristic
330C23#95311yescharacteristic
331Q8—42no—
332Q8—81no—
333D4—122no—
334D4—241no—
335D4—242no—
336D4—242no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—82no—
354C2×C4—122no—
355C2×C4—122no—
356C2×C4—122no—
357C2×C4—121no—
358C2×C4—122no—
359C2×C4#93711yescharacteristic
360C2×C4#93811yescharacteristic
361C2×C4#93911yescharacteristic
362C2×C4#94011yescharacteristic
363C8—122no—
364C8—241no—
365C2×C3—22no—
366C2×C3—22no—
367C2×C3—41no—
368C2×C3#200311yescharacteristic
369C2×C3#200411yescharacteristic
370C2×C3#202211yescharacteristic
371D3—161no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—44no—
380C22—244no—
381C22—241no—
382C22—241no—
383C22—241no—
384C22#506112yes—
385C22#522311yescharacteristic
386C22#524512yes—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—82no—
391C4—81no—
392C4—82no—
393C4—81no—
394C4—242no—
395C4—241no—
396C4#505312yes—
397C3#1033211yescharacteristic, complemented, Sylow
398C2—22no—
399C2—22no—
400C2—41no—
401C2—481no—
402C2#7170611yescharacteristic
403C2#7173111yescharacteristic
404C2#7186711yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2503C2
socleC22×C3#392
prime radicalC2×#930C22
hypercenter#2503D3
hypocenterC3#10332
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2503

Complements

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

NH ≅ G/Ntype
1. C3×#2503C2semidirect
4. #71688C2semidirect
5. #71562C2semidirect
7. #67618C2semidirect
34. #2503D3semidirect
397. C3#10332semidirect

Maximal subgroups

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

subgroupindex
C3×#25032
#827382
#821802
#716882
#715622
#694442
#676182
#103323