Finite Group Databasebeta

← #10695542

Subgroups — #10695542

Order 768.

526 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 85 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×#3195C211yesmaximal, characteristic
2#82787C211yesmaximal, characteristic
3#82727C211yesmaximal, characteristic
4#71701C211yesmaximal, characteristic
5#71688C211yesmaximal, characteristic
6#69975C211yesmaximal, characteristic
7#67684C211yesmaximal, characteristic
8#9937—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C2×#926C2211yescharacteristic
12C3×#483—21no—
13C3×#483—21no—
14C3×#470—22no—
15C3×#451—21no—
16C3×#451—21no—
17C3×#407C2211yescharacteristic
18#5229—21no—
19#5225—21no—
20#5212C2211yescharacteristic
21C4×#217C2211yescharacteristic
22#5207C2211yescharacteristic
23#5045—22no—
24#5044—21no—
25#5043—21no—
26#4986—21no—
27#4981—21no—
28#4972—21no—
29#4964—21no—
30#4952—22no—
31#4555—22no—
32#4538—22no—
33#4526—22no—
34#3195D311yescharacteristic
35#3095—31no—
36#2625—31no—
37#1996—31no—
38#1826—31no—
39#1508—31no—
40#1305—31no—
41C22×C3×Q8D411yescharacteristic
42C22×C3×Q8D411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C2×C3×Q16—22no—
45C2×C3×M16D412yes—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8—21no—
51C3×C22⋊Q8—21no—
52C3×C22⋊Q8—21no—
53C3×C22⋊Q8—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—21no—
60C2×#217—21no—
61C2×#217—21no—
62C2×#217D412yes—
63#926—21no—
64#926—21no—
65#926—21no—
66#926—21no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3D411yescharacteristic
69C2×C4⋊Dic3D411yescharacteristic
70C2×Dic3⋊C4—22no—
71C2×C4×Dic3—21no—
72C2×C4×Dic3—21no—
73C2×#208—21no—
74C3×#106—22no—
75C3×#103—22no—
76C3×#103—22no—
77C3×C22⋊C8—21no—
78C3×C22⋊C8—21no—
79#835—22no—
80#835—22no—
81C3⋊#95—22no—
82C3⋊#95—22no—
83C3⋊#95—22no—
84C3⋊#95—21no—
85C3⋊#95—21no—
86#830—21no—
87#830D412yes—
88#808—22no—
89#807—22no—
90C2×C22⋊Q8C2×D311yescharacteristic
91C2×C22⋊Q8C2×D311yescharacteristic
92C2×#117—31no—
93#485—62no—
94#484—61no—
95#483—21no—
96#483—21no—
97#483—61no—
98#470—22no—
99#451—21no—
100#451—21no—
101#428—61no—
102#419—61no—
103#407—31no—
104#407C2×D311yescharacteristic
105#400—61no—
106#398—61no—
107#397—62no—
108#391—61no—
109#389—61no—
110#380—31no—
111C4×C22⋊C4—31no—
112#342—62no—
113#339—62no—
114#328—62no—
115C24×C3C2×D411yescharacteristic
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—42no—
121C2×C3×Q8—42no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—42no—
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4#4111yescharacteristic
133C22×Dic3—21no—
134C22×Dic3—22no—
135C22×Dic3—21no—
136C22×Dic3—41no—
137C3×Q16—82no—
138C3×M16—42no—
139C2×C3×C8—22no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—41no—
146C3×C4⋊C4—41no—
147C3×C4⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—41no—
152#217—22no—
153#217—41no—
154#217—41no—
155#217—41no—
156#217—42no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3—42no—
162Dic3⋊C4—82no—
163C4×Dic3—41no—
164C4×Dic3—41no—
165C4×Dic3—42no—
166#208—42no—
167C2×C3⋊C8—22no—
168C22×Q8#6711yescharacteristic
169C22×Q8#6711yescharacteristic
170C23×C4C22×D311yescharacteristic
171C2×Q16—22no—
172C2×M16—61no—
173C2×M16#6712yes—
174C22⋊Q8—21no—
175C22⋊Q8—21no—
176C22⋊Q8—21no—
177C22⋊Q8—21no—
178C22⋊Q8—21no—
179C22⋊Q8—21no—
180C22⋊Q8—21no—
181C22⋊Q8—21no—
182#117—61no—
183#117—61no—
184#117—61no—
185#117—61no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—31no—
189C2×C4⋊C4—31no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4—62no—
192C2×C4⋊C4#6711yescharacteristic
193C2×C4⋊C4#6711yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—32no—
197C2×C22⋊C4—61no—
198C2×C22⋊C4—61no—
199C2×C42—61no—
200C2×C42—61no—
201#107—62no—
202#106—22no—
203#106—62no—
204#103—22no—
205#103—22no—
206#103—62no—
207#103—62no—
208C22⋊C8—21no—
209C22⋊C8—21no—
210C22⋊C8—32no—
211C22⋊C8—61no—
212#95—62no—
213#95—62no—
214#95—62no—
215#95—61no—
216#95—61no—
217C23×C3—21no—
218C23×C3—21no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3#12011yescharacteristic
222C23×C3#12111yescharacteristic
223C23×C3#12711yescharacteristic
224C3×Q8—42no—
225C3×Q8—42no—
226C3×Q8—42no—
227C3×Q8—81no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—42no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—81no—
242C2×C3×C4—81no—
243C2×C3×C4#12011yescharacteristic
244C2×C3×C4#12011yescharacteristic
245C2×C3×C4#12111yescharacteristic
246C2×C3×C4#12111yescharacteristic
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—41no—
256C2×Dic3—81no—
257C2×Dic3—81no—
258C3×C8—42no—
259C3⋊C8—42no—
260C24D3×D411yescharacteristic
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—42no—
266C2×Q8—42no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—42no—
272C22×C4—61no—
273C22×C4—62no—
274C22×C4—61no—
275C22×C4—121no—
276C22×C4D3×D411yescharacteristic
277C22×C4D3×D411yescharacteristic
278C22×C4#23711yescharacteristic
279C22×C4C2×#6711yescharacteristic
280C22×C4C2×#6711yescharacteristic
281C22×C4C2×#6711yescharacteristic
282Q16—82no—
283M16—42no—
284M16—122no—
285C2×C8—22no—
286C2×C8—62no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—61no—
296C4⋊C4—61no—
297C4⋊C4—61no—
298C4⋊C4—61no—
299C4⋊C4—122no—
300C4⋊C4—242no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—62no—
306C22⋊C4—121no—
307C22⋊C4—121no—
308C22⋊C4—121no—
309C22⋊C4—122no—
310C42—122no—
311C42—121no—
312C42—121no—
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—81no—
322C22×C3—81no—
323C22×C3#39211yescharacteristic
324C22×C3#45511yescharacteristic
325C22×C3#45611yescharacteristic
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—81no—
330C3×C4—81no—
331C3×C4—81no—
332C3×C4—81no—
333C3×C4#45812yes—
334Dic3—82no—
335Dic3—82no—
336Dic3—81no—
337Dic3—81no—
338C23—21no—
339C23—21no—
340C23—41no—
341C23—41no—
342C23#93711yescharacteristic
343C23#93911yescharacteristic
344C23#94011yescharacteristic
345Q8—42no—
346Q8—42no—
347Q8—42no—
348Q8—81no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—122no—
365C2×C4—122no—
366C2×C4—122no—
367C2×C4—122no—
368C2×C4—122no—
369C2×C4—121no—
370C2×C4—121no—
371C2×C4—121no—
372C2×C4—121no—
373C2×C4—241no—
374C2×C4—241no—
375C2×C4#93711yescharacteristic
376C2×C4#93811yescharacteristic
377C2×C4#93911yescharacteristic
378C2×C4#95311yescharacteristic
379C8—42no—
380C8—122no—
381C2×C3—21no—
382C2×C3—21no—
383C2×C3—41no—
384C2×C3—41no—
385C2×C3#200711yescharacteristic
386C2×C3#200911yescharacteristic
387C2×C3#202211yescharacteristic
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—81no—
397C22—81no—
398C22#520211yescharacteristic
399C22#520511yescharacteristic
400C22#522311yescharacteristic
401C4—21no—
402C4—21no—
403C4—21no—
404C4—81no—
405C4—81no—
406C4—81no—
407C4—81no—
408C4—242no—
409C4—242no—
410C4—241no—
411C4—241no—
412C4#524312yes—
413C3#993711yescharacteristic, complemented, Sylow
414C2—21no—
415C2—21no—
416C2—41no—
417C2—41no—
418C2#7184911yescharacteristic
419C2#7211011yescharacteristic
420C2#7211111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#3195C2
socleC22×C3#392
prime radicalC24×C3C2×D4
hypercenter#3195D3
hypocenterC3#9937
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3195

Complements

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

NH ≅ G/Ntype
413. C3#9937semidirect

Maximal subgroups

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

subgroupindex
C3×#31952
#827872
#827272
#717012
#716882
#699752
#676842
#99373