Finite Group Databasebeta

← #10698114

Subgroups — #10698114

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×#3193C211yesmaximal, characteristic
2#82776C211yesmaximal, characteristic, complemented
3#82665C211yesmaximal, characteristic
4#71688C211yesmaximal, characteristic, complemented
5#71673C211yesmaximal, characteristic
6#69973C211yesmaximal, characteristic, complemented
7#67683C211yesmaximal, characteristic, complemented
8#10332—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C2×C3×#121C2211yescharacteristic
11C22×C4⋊Dic3C2211yescharacteristic
12C3×#484—21no—
13C3×#480—21no—
14C3×#468—22no—
15C3×#450—21no—
16C3×#448—21no—
17C3×#407C2211yescharacteristic
18#5229—21no—
19#5221—21no—
20#5220—22no—
21C4×#217C2211yescharacteristic
22C4×#217C2211yescharacteristic
23#5207C2211yescharacteristic
24#4986—22no—
25#4977—21no—
26#4977—21no—
27#4953—22no—
28#4943—22no—
29#4554—22no—
30#4538—22no—
31#4537—22no—
32#3193D311yescharacteristic
33#2504—31no—
34#2503—31no—
35#1821—31no—
36#1508—31no—
37#1507—31no—
38#1297—31no—
39C22×C3×Q8D411yescharacteristic
40C22×C3×D4—21no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C23×Dic3D412yes—
44C2×C3×SD16—22no—
45C2×C3×M16D412yes—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×#121—21no—
51C3×#121—21no—
52C3×#121—21no—
53C3×#121—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×#217—21no—
60C2×#217—21no—
61C2×#217—24no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3—21no—
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3—21no—
67C2×C4⋊Dic3—21no—
68C2×C4×Dic3—21no—
69C2×C4×Dic3—21no—
70C2×#208—21no—
71C3×#107—22no—
72C3×#103—22no—
73C3×#102—22no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8—21no—
76C3⋊#95—24no—
77C3⋊#95—24no—
78C3⋊#95—24no—
79C3⋊#95D412yes—
80C3⋊#95D412yes—
81#830—21no—
82#830—21no—
83C2×C22⋊Q8C2×D311yescharacteristic
84C2×#121C2×D311yescharacteristic
85C22×C4⋊C4—31no—
86#484—21no—
87#480—21no—
88#468—22no—
89#450—21no—
90#448—21no—
91#407—31no—
92#407C2×D311yescharacteristic
93#400—62no—
94#397—62no—
95#391—61no—
96#390—61no—
97#386—62no—
98#382—61no—
99#382—61no—
100#381—62no—
101C4×C22⋊C4—31no—
102C4×C22⋊C4—31no—
103#343—62no—
104#328—62no—
105#327—62no—
106C24×C3—21no—
107C24×C3C2×D411yescharacteristic
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—42no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—42no—
114C2×C3×D4—44no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4—21no—
120C22×C3×C4—42no—
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4#4111yescharacteristic
127C22×Dic3—22no—
128C22×Dic3—22no—
129C22×Dic3—22no—
130C22×Dic3—22no—
131C22×Dic3—22no—
132C22×Dic3—44no—
133C3×SD16—82no—
134C3×M16—42no—
135C2×C3×C8—22no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C22⋊C4—41no—
146#217—42no—
147#217—42no—
148#217—84no—
149C4⋊Dic3—41no—
150C4⋊Dic3—41no—
151C4⋊Dic3—41no—
152C4⋊Dic3—41no—
153C4×Dic3—42no—
154C4×Dic3—42no—
155#208—42no—
156C2×C3⋊C8—41no—
157C22×Q8#6711yescharacteristic
158C22×D4—21no—
159C22×D4#6711yescharacteristic
160C23×C4—32no—
161C23×C4C22×D311yescharacteristic
162C2×SD16—22no—
163C2×M16—61no—
164C2×M16#6712yes—
165C22⋊Q8—21no—
166C22⋊Q8—21no—
167C22⋊Q8—21no—
168C22⋊Q8—21no—
169#121—21no—
170#121—21no—
171#121—21no—
172#121—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4#6711yescharacteristic
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—61no—
188C2×C42—61no—
189#107—22no—
190#103—22no—
191#102—22no—
192C22⋊C8—21no—
193C22⋊C8—21no—
194C22⋊C8—61no—
195C22⋊C8—61no—
196#95—32no—
197#95—32no—
198#95—64no—
199#95—64no—
200#95—64no—
201C23×C3—22no—
202C23×C3—22no—
203C23×C3—41no—
204C23×C3—42no—
205C23×C3—42no—
206C23×C3—41no—
207C23×C3—42no—
208C23×C3#12011yescharacteristic
209C23×C3#12011yescharacteristic
210C23×C3#12111yescharacteristic
211C3×Q8—42no—
212C3×Q8—81no—
213C3×D4—42no—
214C3×D4—81no—
215C3×D4—82no—
216C3×D4—82no—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—22no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—42no—
224C2×C3×C4—42no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—42no—
229C2×C3×C4#12011yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12711yescharacteristic
233C2×Dic3—42no—
234C2×Dic3—42no—
235C2×Dic3—42no—
236C2×Dic3—42no—
237C2×Dic3—44no—
238C2×Dic3—42no—
239C2×Dic3—42no—
240C2×Dic3—42no—
241C2×Dic3—44no—
242C2×Dic3—82no—
243C3×C8—42no—
244C3⋊C8—81no—
245C24—21no—
246C24D3×D411yescharacteristic
247C2×Q8—21no—
248C2×Q8—21no—
249C2×Q8—42no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—42no—
253C2×D4—44no—
254C2×D4—41no—
255C2×D4—41no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—42no—
260C22×C4—62no—
261C22×C4—62no—
262C22×C4—62no—
263C22×C4—62no—
264C22×C4—62no—
265C22×C4—124no—
266C22×C4D3×D411yescharacteristic
267C22×C4D3×D411yescharacteristic
268C22×C4#23711yescharacteristic
269C22×C4C2×#6711yescharacteristic
270C22×C4C2×#6711yescharacteristic
271C22×C4C2×#6711yescharacteristic
272SD16—82no—
273M16—42no—
274M16—122no—
275C2×C8—22no—
276C2×C8—121no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C4⋊C4—121no—
286C4⋊C4—121no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—122no—
292C22⋊C4—122no—
293C22⋊C4—244no—
294C42—122no—
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—84no—
305C22×C3—81no—
306C22×C3—81no—
307C22×C3—81no—
308C22×C3#39211yescharacteristic
309C22×C3#45412yes—
310C22×C3#45512yes—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—81no—
315C3×C4—81no—
316C3×C4—81no—
317C3×C4#45812yes—
318Dic3—82no—
319Dic3—82no—
320Dic3—82no—
321C23—22no—
322C23—22no—
323C23—41no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—41no—
328C23#93711yescharacteristic
329C23#93711yescharacteristic
330C23#93811yescharacteristic
331Q8—42no—
332Q8—81no—
333D4—42no—
334D4—81no—
335D4—82no—
336D4—82no—
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—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—122no—
350C2×C4—122no—
351C2×C4—122no—
352C2×C4—122no—
353C2×C4—124no—
354C2×C4—122no—
355C2×C4—122no—
356C2×C4—124no—
357C2×C4—122no—
358C2×C4—242no—
359C2×C4#93911yescharacteristic
360C2×C4#93911yescharacteristic
361C2×C4#94011yescharacteristic
362C2×C4#95311yescharacteristic
363C8—42no—
364C8—241no—
365C2×C3—22no—
366C2×C3—22no—
367C2×C3—41no—
368C2×C3—161no—
369C2×C3#200311yescharacteristic
370C2×C3#200411yescharacteristic
371C2×C3#202211yescharacteristic
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—44no—
380C22—84no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22#520112yes—
385C22#520212yes—
386C22#522311yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—81no—
391C4—81no—
392C4—81no—
393C4—242no—
394C4—242no—
395C4—242no—
396C4#524312yes—
397C3#1033211yescharacteristic, complemented, Sylow
398C2—22no—
399C2—22no—
400C2—41no—
401C2—161no—
402C2#7184911yescharacteristic
403C2#7209611yescharacteristic
404C2#7210611yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#3193C2
socleC22×C3#392
prime radicalC2×C3×#121C22
hypercenter#3193D3
hypocenterC3#10332
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. #82776C2semidirect
4. #71688C2semidirect
6. #69973C2semidirect
7. #67683C2semidirect
397. C3#10332semidirect

Maximal subgroups

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

subgroupindex
C3×#31932
#827762
#826652
#716882
#716732
#699732
#676832
#103323