Finite Group Databasebeta

← #10695601

Subgroups — #10695601

Order 768.

574 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 83 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
1#82254C211yesmaximal, characteristic
2#82220C211yesmaximal, characteristic
3#73547C211yesmaximal, characteristic
4C3×#1826C211yesmaximal, characteristic
5#69981C211yesmaximal, characteristic
6#69971C211yesmaximal, characteristic
7#67616C211yesmaximal, characteristic
8#9945—31nomaximal, Sylow
9C2×C3×#117C2211yescharacteristic
10C2×#924C2211yescharacteristic
11C22×Dic3⋊C4C2211yescharacteristic
12C3×#407C2211yescharacteristic
13C3×C4×C22⋊C4C2211yescharacteristic
14#5211C2211yescharacteristic
15#5207C2211yescharacteristic
16#5004—21no—
17#4996—22no—
18#4995—21no—
19#4985—21no—
20#4983—21no—
21#4971—21no—
22#4970—21no—
23#4950—21no—
24#4950—21no—
25#4946—21no—
26#4943—21no—
27#4933—22no—
28#4928—22no—
29#4661—21no—
30#4656—21no—
31C3×#339—22no—
32#4469—22no—
33#4467—22no—
34#3095—31no—
35#2583—31no—
36#2484—31no—
37#1826D311yescharacteristic
38#1511—31no—
39#1463—31no—
40#1284—31no—
41C23×C3×C4C2311yescharacteristic
42C23×Dic3D411yescharacteristic
43C23×Dic3D411yescharacteristic
44C22×Dic6D411yescharacteristic
45C2×C3×M16—21no—
46C3×#117—21no—
47C3×#117—21no—
48C3×#117—21no—
49C3×#117—21no—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C22⋊C4—21no—
53C2×C3×C22⋊C4—21no—
54C2×C3×C42—21no—
55C2×C3×C42D412yes—
56C2×#217—21no—
57C2×#217—21no—
58C2×C4⋊Dic3—22no—
59C2×C4⋊Dic3—22no—
60C2×C4⋊Dic3D411yescharacteristic
61#924—21no—
62#924—21no—
63#924—21no—
64#924—21no—
65C2×Dic3⋊C4—22no—
66C2×Dic3⋊C4—21no—
67C2×Dic3⋊C4—21no—
68C2×Dic3⋊C4—21no—
69C2×Dic3⋊C4—21no—
70C2×Dic3⋊C4—21no—
71C2×Dic3⋊C4—21no—
72C2×Dic3⋊C4—21no—
73C2×#208—21no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×#95—21no—
77C3⋊#95—22no—
78C3⋊#95—22no—
79C3⋊#95—22no—
80C3⋊#95—22no—
81C3⋊#95—22no—
82C3⋊#95—21no—
83C3⋊#95D412yes—
84#830—21no—
85#830D412yes—
86#810—22no—
87#810—22no—
88#808—22no—
89#807—22no—
90C2×C22⋊Q8—31no—
91C2×#117C2×D311yescharacteristic
92C22×C4⋊C4—31no—
93#485—62no—
94#484—61no—
95#483—61no—
96#428—61no—
97#419—61no—
98#407—31no—
99#407C2×D311yescharacteristic
100#400—61no—
101#398—61no—
102#398—61no—
103#395—61no—
104#389—62no—
105#387—62no—
106#385—61no—
107#385—61no—
108#385—61no—
109#381—31no—
110#381—61no—
111C4×C22⋊C4C2×D311yescharacteristic
112#339—22no—
113#328—62no—
114#324—62no—
115C24×C3C2×D411yescharacteristic
116C22×C3×C4—22no—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4—42no—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C22×Dic3—21no—
127C22×Dic3—21no—
128C22×Dic3—21no—
129C22×Dic3—22no—
130C22×Dic3—21no—
131C22×Dic3—21no—
132C22×Dic3—21no—
133C22×Dic3—21no—
134C22×Dic3—21no—
135C22×Dic3—42no—
136C22×Dic3—42no—
137C2×Dic6—21no—
138C2×Dic6—21no—
139C2×Dic6—42no—
140C3×M16—42no—
141C2×C3×C8—41no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C22⋊C4—24no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C42—42no—
150C3×C42—41no—
151C3×C42—41no—
152#217—42no—
153#217—41no—
154#217—41no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—44no—
158C4⋊Dic3—82no—
159Dic3⋊C4—44no—
160Dic3⋊C4—41no—
161Dic3⋊C4—41no—
162Dic3⋊C4—41no—
163Dic3⋊C4—41no—
164Dic3⋊C4—41no—
165Dic3⋊C4—41no—
166#208—42no—
167C2×C3⋊C8—22no—
168C22×Q8—31no—
169C23×C4—31no—
170C23×C4—31no—
171C23×C4C22×D311yescharacteristic
172C2×M16—21no—
173C2×M16—61no—
174C22⋊Q8—61no—
175C22⋊Q8—61no—
176C22⋊Q8—61no—
177C22⋊Q8—61no—
178#117—21no—
179#117—21no—
180#117—21no—
181#117—21no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—62no—
184C2×C4⋊C4—62no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4—61no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4—61no—
192C2×C4⋊C4—62no—
193C2×C4⋊C4D1211yescharacteristic
194C2×C4⋊C4D1211yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—61no—
198C2×C22⋊C4—61no—
199C2×C42—21no—
200C2×C42#6712yes—
201#107—62no—
202#106—62no—
203#103—62no—
204#103—62no—
205C22⋊C8—21no—
206C22⋊C8—21no—
207C22⋊C8—32no—
208C22⋊C8—61no—
209#95—21no—
210#95—32no—
211#95—62no—
212#95—62no—
213#95—62no—
214#95—62no—
215#95—62no—
216#95—61no—
217C23×C3—22no—
218C23×C3—22no—
219C23×C3—41no—
220C23×C3#12011yescharacteristic
221C23×C3#12011yescharacteristic
222C23×C3#12111yescharacteristic
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—24no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—42no—
236C2×C3×C4#12011yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12711yescharacteristic
240C2×Dic3—22no—
241C2×Dic3—22no—
242C2×Dic3—22no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—42no—
248C2×Dic3—42no—
249C2×Dic3—42no—
250C2×Dic3—42no—
251C2×Dic3—41no—
252C2×Dic3—41no—
253C2×Dic3—41no—
254C2×Dic3—42no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—81no—
258Dic6—42no—
259Dic6—81no—
260C3×C8—81no—
261C3⋊C8—42no—
262C24D3×D411yescharacteristic
263C2×Q8—61no—
264C2×Q8—61no—
265C2×Q8—122no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—42no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4—62no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4—61no—
277C22×C4—61no—
278C22×C4—61no—
279C22×C4—122no—
280C22×C4—122no—
281C22×C4C2×D1211yescharacteristic
282C22×C4#23511yescharacteristic
283C22×C4D3×D411yescharacteristic
284C22×C4D3×D411yescharacteristic
285C22×C4D3×D411yescharacteristic
286C22×C4C2×#6711yescharacteristic
287M16—42no—
288M16—122no—
289C2×C8—41no—
290C2×C8—62no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—61no—
296C4⋊C4—61no—
297C4⋊C4—124no—
298C4⋊C4—124no—
299C4⋊C4—121no—
300C4⋊C4—121no—
301C4⋊C4—121no—
302C4⋊C4—121no—
303C4⋊C4—121no—
304C4⋊C4—121no—
305C4⋊C4—242no—
306C22⋊C4—24no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—122no—
310C22⋊C4—121no—
311C22⋊C4—121no—
312C42—42no—
313C42—41no—
314C42—41no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—44no—
323C22×C3#39211yescharacteristic
324C22×C3#45612yes—
325C22×C3#45812yes—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—44no—
331C3×C4—81no—
332C3×C4—81no—
333Dic3—42no—
334Dic3—82no—
335Dic3—81no—
336Dic3—81no—
337Dic3—81no—
338C23—22no—
339C23—22no—
340C23—41no—
341C23#88211yescharacteristic
342C23#88411yescharacteristic
343C23#93711yescharacteristic
344Q8—122no—
345Q8—241no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—62no—
360C2×C4—62no—
361C2×C4—62no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4—121no—
366C2×C4—122no—
367C2×C4—122no—
368C2×C4—122no—
369C2×C4—122no—
370C2×C4—121no—
371C2×C4—121no—
372C2×C4—121no—
373C2×C4—122no—
374C2×C4—122no—
375C2×C4—122no—
376C2×C4—241no—
377C2×C4#88211yescharacteristic
378C2×C4#88411yescharacteristic
379C2×C4#93011yescharacteristic
380C2×C4#94011yescharacteristic
381C8—81no—
382C8—122no—
383C2×C3—22no—
384C2×C3—22no—
385C2×C3—41no—
386C2×C3#200511yescharacteristic
387C2×C3#201011yescharacteristic
388C2×C3#202211yescharacteristic
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—44no—
397C22#482612yes—
398C22#496011yescharacteristic
399C22#505312yes—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—44no—
405C4—81no—
406C4—81no—
407C4—122no—
408C4—242no—
409C4—241no—
410C4—241no—
411C4—241no—
412C3#994511yescharacteristic, complemented, Sylow
413C2—22no—
414C2—22no—
415C2—41no—
416C2#6953911yescharacteristic
417C2#7038911yescharacteristic
418C2#7040611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1826C2
socleC22×C3#392
prime radicalC24×C3C2×D4
hypercenter#1826D3
hypocenterC3#9945
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1826

Complements

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

NH ≅ G/Ntype
412. C3#9945semidirect

Maximal subgroups

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

subgroupindex
#822542
#822202
#735472
C3×#18262
#699812
#699712
#676162
#99453