Finite Group Databasebeta

← #10695881

Subgroups — #10695881

Order 768.

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

normal subgroups only

This group has 698 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×#451C211yesmaximal, characteristic
2C2×#5228C211yesmaximal, characteristic
3#82742C211yesmaximal, characteristic
4#72133C211yesmaximal, characteristic
5#71691C211yesmaximal, characteristic
6#69923C211yesmaximal, characteristic
7#67680C211yesmaximal, characteristic
8#9991—31nomaximal, Sylow
9C23×C3×Q8C2211yescharacteristic
10C22×C3×Q16—21no—
11C2×C3×C22⋊Q8C2211yescharacteristic
12C22×#216—21no—
13C2×#924C2211yescharacteristic
14C3×#451—22no—
15C3×#451—22no—
16C2×C3×#103—21no—
17C2×C3×C22⋊C8C2211yescharacteristic
18#5228—21no—
19#5228—22no—
20#5228—22no—
21C2×#836—22no—
22C2×#835—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×#451—31no—
34C2×#451D311yescharacteristic
35#1784—31no—
36#1472—31no—
37#1472—31no—
38#1298—31no—
39C22×C3×Q8—21no—
40C22×C3×Q8—21no—
41C22×C3×Q8—21no—
42C22×C3×Q8—21no—
43C22×C3×Q8—21no—
44C22×C3×Q8—44no—
45C22×C3×Q8D411yescharacteristic
46C22×C3×Q8D412yes—
47C23×C3×C4—21no—
48C23×C3×C4C2311yescharacteristic
49C22×Dic6D411yescharacteristic
50C2×C3×Q16—24no—
51C2×C3×Q16—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—24no—
62C2×#216—22no—
63C2×#216—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×#103—24no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8D412yes—
76#836—44no—
77#835—24no—
78#835—22no—
79#830—21no—
80#830—21no—
81#830—21no—
82#830—21no—
83#830—21no—
84#830D412yes—
85#810—22no—
86#808—22no—
87C23×Q8C2×D311yescharacteristic
88C22×Q16—21no—
89C22×Q16—61no—
90C2×C22⋊Q8—31no—
91C2×C22⋊Q8C2×D311yescharacteristic
92#483—61no—
93#483—61no—
94#470—62no—
95#451—22no—
96#451—22no—
97#451—62no—
98#451—61no—
99#451—61no—
100#451—62no—
101C2×#103—21no—
102C2×#103—61no—
103C2×#101—62no—
104#407—31no—
105#407—31no—
106C2×C22⋊C8—31no—
107C2×C22⋊C8C2×D311yescharacteristic
108#324—62no—
109#324—62no—
110C24×C3#4111yescharacteristic
111C2×C3×Q8—24no—
112C2×C3×Q8—24no—
113C2×C3×Q8—24no—
114C2×C3×Q8—22no—
115C2×C3×Q8—22no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—44no—
119C2×C3×Q8—42no—
120C2×C3×Q8—42no—
121C2×C3×Q8—42no—
122C2×C3×Q8—42no—
123C2×C3×Q8—44no—
124C2×C3×Q8—44no—
125C2×C3×Q8—41no—
126C2×C3×Q8—41no—
127C22×C3×C4—22no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—42no—
131C22×C3×C4—44no—
132C22×C3×C4—41no—
133C22×C3×C4—41no—
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×Dic3—21no—
141C22×Dic3—21no—
142C2×Dic6—21no—
143C2×Dic6—21no—
144C2×Dic6—42no—
145C3×Q16—48no—
146C2×C3×C8—22no—
147C2×C3×C8—42no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—41no—
151C3×C4⋊C4—41no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4—41no—
154#217—41no—
155#217—41no—
156#216—48no—
157#216—82no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160Dic3⋊C4—41no—
161Dic3⋊C4—41no—
162#208—42no—
163#208—42no—
164C2×C3⋊C8—22no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—42no—
167C2×C3⋊C8—41no—
168C22×Q8—21no—
169C22×Q8—21no—
170C22×Q8—21no—
171C22×Q8—21no—
172C22×Q8—21no—
173C22×Q8—31no—
174C22×Q8—44no—
175C22×Q8#6711yescharacteristic
176C22×Q8#6712yes—
177C23×C4—21no—
178C23×C4C22×D311yescharacteristic
179C2×Q16—24no—
180C2×Q16—44no—
181C2×Q16—64no—
182C2×Q16—62no—
183C2×Q16—124no—
184C2×M16—32no—
185C2×M16—61no—
186C22×C8—21no—
187C22×C8—61no—
188C22⋊Q8—21no—
189C22⋊Q8—21no—
190C22⋊Q8—21no—
191C22⋊Q8—21no—
192C22⋊Q8—61no—
193C22⋊Q8—61no—
194C22⋊Q8—61no—
195C22⋊Q8—61no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—31no—
198C2×C4⋊C4—61no—
199C2×C4⋊C4#6711yescharacteristic
200C2×C22⋊C4—21no—
201C2×C22⋊C4—61no—
202#106—62no—
203#103—24no—
204#103—62no—
205#103—64no—
206#103—62no—
207#101—124no—
208C22⋊C8—21no—
209C22⋊C8—32no—
210C22⋊C8—61no—
211C22⋊C8—61no—
212C22⋊C8—61no—
213C22⋊C8—61no—
214C22⋊C8—61no—
215C22⋊C8#6712yes—
216C23×C3—22no—
217C23×C3—22no—
218C23×C3—41no—
219C23×C3#12111yescharacteristic
220C23×C3#12111yescharacteristic
221C23×C3#12111yescharacteristic
222C3×Q8—24no—
223C3×Q8—24no—
224C3×Q8—42no—
225C3×Q8—44no—
226C3×Q8—42no—
227C3×Q8—42no—
228C3×Q8—44no—
229C3×Q8—42no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—22no—
233C2×C3×C4—24no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—42no—
238C2×C3×C4—44no—
239C2×C3×C4—44no—
240C2×C3×C4—42no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—42no—
244C2×C3×C4—81no—
245C2×C3×C4#12011yescharacteristic
246C2×C3×C4#12011yescharacteristic
247C2×C3×C4#12011yescharacteristic
248C2×C3×C4#12711yescharacteristic
249C2×Dic3—41no—
250C2×Dic3—41no—
251C2×Dic3—42no—
252C2×Dic3—81no—
253Dic6—42no—
254Dic6—42no—
255C3×C8—42no—
256C3⋊C8—42no—
257C3⋊C8—42no—
258C3⋊C8—81no—
259C24#23711yescharacteristic
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—22no—
264C2×Q8—21no—
265C2×Q8—22no—
266C2×Q8—21no—
267C2×Q8—44no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—44no—
273C2×Q8—44no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—61no—
277C2×Q8—61no—
278C2×Q8—122no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—42no—
283C22×C4—44no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—61no—
287C22×C4—61no—
288C22×C4D3×D411yescharacteristic
289C22×C4D3×D411yescharacteristic
290C22×C4D3×D411yescharacteristic
291C22×C4C2×#6711yescharacteristic
292C22×C4C2×#6711yescharacteristic
293C22×C4C2×#6711yescharacteristic
294Q16—48no—
295Q16—128no—
296Q16—242no—
297M16—122no—
298M16—122no—
299C2×C8—22no—
300C2×C8—42no—
301C2×C8—62no—
302C2×C8—62no—
303C2×C8—122no—
304C2×C8—121no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—61no—
310C4⋊C4—61no—
311C4⋊C4—121no—
312C4⋊C4—121no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—121no—
316C22⋊C4—121no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—44no—
325C22×C3#39211yescharacteristic
326C22×C3#47312yes—
327C22×C3#47312yes—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—44no—
332C3×C4—81no—
333C3×C4—81no—
334C3×C4B(4,𝔽2)12yes—
335Dic3—81no—
336Dic3—81no—
337C23—22no—
338C23—22no—
339C23—41no—
340C23#93811yescharacteristic
341C23#93911yescharacteristic
342C23#93911yescharacteristic
343Q8—24no—
344Q8—24no—
345Q8—42no—
346Q8—42no—
347Q8—44no—
348Q8—42no—
349Q8—44no—
350Q8—42no—
351Q8—122no—
352Q8—122no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—24no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—81no—
368C2×C4—122no—
369C2×C4—121no—
370C2×C4—121no—
371C2×C4—241no—
372C2×C4#93711yescharacteristic
373C2×C4#93711yescharacteristic
374C2×C4#94011yescharacteristic
375C2×C4#95311yescharacteristic
376C8—42no—
377C8—122no—
378C8—122no—
379C8—241no—
380C2×C3—22no—
381C2×C3—22no—
382C2×C3—41no—
383C2×C3#202011yescharacteristic
384C2×C3#202011yescharacteristic
385C2×C3#202111yescharacteristic
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—44no—
394C22#519212yes—
395C22#520412yes—
396C22#522311yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—44no—
401C4—81no—
402C4—81no—
403C4—241no—
404C4—241no—
405C4#503212yes—
406C3#999111yescharacteristic, complemented, Sylow
407C2—22no—
408C2—22no—
409C2—41no—
410C2#7167111yescharacteristic
411C2#7207411yescharacteristic
412C2#7210911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#451C2
socleC22×C3#392
prime radicalC24×C3#41
hypercenterC2×#451D3
hypocenterC3#9991
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#451

Complements

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

NH ≅ G/Ntype
406. C3#9991semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4512
C2×#52282
#827422
#721332
#716912
#699232
#676802
#99913