Finite Group Databasebeta

← #10698108

Subgroups — #10698108

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×#2502C211yesmaximal, characteristic, complemented
2#82738C211yesmaximal, characteristic
3#82177C211yesmaximal, characteristic
4#71688C211yesmaximal, characteristic, complemented
5#71562C211yesmaximal, characteristic, complemented
6#69444C211yesmaximal, characteristic
7#67618C211yesmaximal, characteristic, complemented
8#10331—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×C3×C4⋊C4C2211yescharacteristic
11C2×#930C2211yescharacteristic
12C2×#929—22no—
13C3×#398—22no—
14C3×#391—21no—
15C3×#385—22no—
16C3×#384—21no—
17C3×C4×C22⋊C4C2211yescharacteristic
18#5227—21no—
19C4×#217C2211yescharacteristic
20#5207C2211yescharacteristic
21#5207C2211yescharacteristic
22#5049—22no—
23#5046—21no—
24#5044—21no—
25#4993—22no—
26#4988—21no—
27#4975—21no—
28#4967—21no—
29#4538—22no—
30#4530—22no—
31#4451—22no—
32#3193—31no—
33#2502D311yescharacteristic, complemented
34#2501—31no—
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⋊C4—24no—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C42—21no—
60C2×#217—21no—
61C2×#215—22no—
62#930—21no—
63#930—21no—
64#930—21no—
65#930—21no—
66#929—48no—
67C2×(C2×D3)⋊C4—24no—
68C2×(C2×D3)⋊C4—21no—
69C2×C4⋊Dic3D411yescharacteristic
70C2×Dic3⋊C4—24no—
71C2×C4×Dic3—21no—
72C2×#208—21no—
73C2×#208D412yes—
74C3×#95—24no—
75C3×#95D412yes—
76#835—22no—
77C3⋊#95D412yes—
78#830—21no—
79#830—21no—
80#830—21no—
81#830—21no—
82#809—22no—
83#807—22no—
84C2×#123—62no—
85C2×C22⋊Q8C2×D311yescharacteristic
86C2×#121—31no—
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—62no—
96#398—22no—
97#391—21no—
98#390—61no—
99#385—22no—
100#384—21no—
101#384—61no—
102C4×C22⋊C4—31no—
103C4×C22⋊C4C2×D311yescharacteristic
104#343—62no—
105#328—62no—
106#327—62no—
107C24×C3C2×D411yescharacteristic
108C23×D3—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—42no—
112C22×C3×C4—22no—
113C22×C3×C4—21no—
114C22×C3×C4—22no—
115C22×C3×C4—22no—
116C22×C3×C4—21no—
117C22×C3×C4—22no—
118C22×C3×C4—42no—
119C22×C3×C4—44no—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C2×#67—44no—
127C2×#67—41no—
128C2×#67—41no—
129C22×Dic3—22no—
130C22×Dic3—21no—
131C2×D12—21no—
132C2×D12—21no—
133C2×D12—42no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—84no—
143C3×C22⋊C4—24no—
144C3×C22⋊C4—41no—
145C3×C22⋊C4—41no—
146C3×C42—42no—
147#217—24no—
148#215—82no—
149(C2×D3)⋊C4—48no—
150(C2×D3)⋊C4—41no—
151(C2×D3)⋊C4—41no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154Dic3⋊C4—48no—
155C4×Dic3—42no—
156#208—42no—
157#208—42no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—41no—
160C22×Q8#6711yescharacteristic
161C22×D4—31no—
162C22×D4—61no—
163C23×C4#6712yes—
164C23×C4C22×D311yescharacteristic
165C2×SD16—62no—
166C2×M16—32no—
167C2×M16—61no—
168#123—128no—
169C22⋊Q8—21no—
170C22⋊Q8—21no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173#121—61no—
174#121—61no—
175#121—61no—
176#121—61no—
177C2×C4⋊C4—21no—
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—24no—
185C2×C4⋊C4—31no—
186C2×C4⋊C4—64no—
187C2×C4⋊C4#6711yescharacteristic
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—61no—
191C2×C22⋊C4—64no—
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—24no—
203#95—32no—
204#95#6712yes—
205C23×C3—22no—
206C23×C3—22no—
207C23×C3—41no—
208C23×C3#12011yescharacteristic
209C23×C3#12011yescharacteristic
210C23×C3#12111yescharacteristic
211C22×D3—42no—
212C22×D3—42no—
213C22×D3—41no—
214C22×D3—42no—
215C3×Q8—42no—
216C3×Q8—81no—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—22no—
220C2×C3×C4—24no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—42no—
225C2×C3×C4—42no—
226C2×C3×C4—44no—
227C2×C3×C4—42no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—42no—
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
238#67—82no—
239#67—82no—
240C2×Dic3—24no—
241C2×Dic3—42no—
242C2×Dic3—42no—
243C2×Dic3—42no—
244C2×Dic3—41no—
245D12—42no—
246D12—81no—
247C3⋊C8—42no—
248C3⋊C8—81no—
249C24—61no—
250C24C2×#6711yescharacteristic
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—42no—
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—122no—
257C2×D4—124no—
258C2×D4—121no—
259C2×D4—121no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—42no—
267C22×C4—44no—
268C22×C4—62no—
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—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4—84no—
292C4⋊C4—128no—
293C22⋊C4—24no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—64no—
297C22⋊C4—128no—
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#39211yescharacteristic
311C22×C3#45312yes—
312C22×C3#45612yes—
313C2×D3—84no—
314C2×D3—81no—
315C2×D3—81no—
316C2×D3—81no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—44no—
321C3×C4—81no—
322C3×C4—82no—
323C3×C4—81no—
324C3×C4#45812yes—
325Dic3—44no—
326Dic3—81no—
327C23—22no—
328C23—22no—
329C23—41no—
330C23—122no—
331C23—122no—
332C23—121no—
333C23—122no—
334C23#93711yescharacteristic
335C23#93911yescharacteristic
336C23#95311yescharacteristic
337Q8—42no—
338Q8—81no—
339D4—122no—
340D4—241no—
341D4—242no—
342D4—242no—
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—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—64no—
360C2×C4—82no—
361C2×C4—122no—
362C2×C4—122no—
363C2×C4—122no—
364C2×C4—121no—
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#200211yescharacteristic
375C2×C3#200511yescharacteristic
376C2×C3#202211yescharacteristic
377D3—161no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—44no—
386C22—244no—
387C22—241no—
388C22—241no—
389C22—241no—
390C22#509512yes—
391C22#522311yescharacteristic
392C22#524112yes—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—44no—
397C4—81no—
398C4—82no—
399C4—81no—
400C4—124no—
401C4—241no—
402C4#505312yes—
403C3#1033111yescharacteristic, complemented, Sylow
404C2—22no—
405C2—22no—
406C2—41no—
407C2—481no—
408C2#7170611yescharacteristic
409C2#7179511yescharacteristic
410C2#7185611yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2502C2
socleC22×C3#392
prime radicalC2×#930C22
hypercenter#2502D3
hypocenterC3#10331
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2502

Complements

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

NH ≅ G/Ntype
1. C3×#2502C2semidirect
4. #71688C2semidirect
5. #71562C2semidirect
7. #67618C2semidirect
33. #2502D3semidirect
403. C3#10331semidirect

Maximal subgroups

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

subgroupindex
C3×#25022
#827382
#821772
#716882
#715622
#694442
#676182
#103313