Finite Group Databasebeta

← #10695543

Subgroups — #10695543

Order 768.

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

normal subgroups only

This group has 526 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#82742C211yesmaximal, characteristic
2#82727C211yesmaximal, characteristic
3#73951C211yesmaximal, characteristic
4C3×#1996C211yesmaximal, characteristic
5#69445C211yesmaximal, characteristic
6#69442C211yesmaximal, characteristic
7#67626C211yesmaximal, characteristic
8#9937—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C2×#926C2211yescharacteristic
11C2×#924C2211yescharacteristic
12C3×#380C2211yescharacteristic
13C3×C4×C22⋊C4C2211yescharacteristic
14#5228—21no—
15#5225—21no—
16#5207C2211yescharacteristic
17#5207C2211yescharacteristic
18#5052—22no—
19#5050—21no—
20#5045—22no—
21#5044—21no—
22#5043—21no—
23#5043—21no—
24#4984—21no—
25#4973—21no—
26#4971—21no—
27#4964—21no—
28#4953—22no—
29#4661—21no—
30#4659—21no—
31C3×#342—22no—
32#4523—22no—
33#4452—22no—
34#3195—31no—
35#3095—31no—
36#2625—31no—
37#1996D311yescharacteristic
38#1826—31no—
39#1508—31no—
40#1305—31no—
41C22×C3×Q8D411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C22×Dic6D411yescharacteristic
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⋊C4D411yescharacteristic
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4—21no—
53C2×C3×C22⋊C4D412yes—
54C2×C3×C42—21no—
55C2×#217—21no—
56C2×#217—21no—
57C2×#216—22no—
58#926—21no—
59#926—21no—
60#926—21no—
61#926—21no—
62C2×C4⋊Dic3—22no—
63C2×C4⋊Dic3D411yescharacteristic
64C2×C4⋊Dic3D411yescharacteristic
65C2×C4⋊Dic3D411yescharacteristic
66#924—21no—
67#924—21no—
68#924—21no—
69#924—21no—
70C2×Dic3⋊C4—21no—
71C2×C4×Dic3—21no—
72C2×#208—21no—
73C2×#208D412yes—
74C3×#95—21no—
75C3×#95—21no—
76#835—22no—
77#835—22no—
78#835—22no—
79C3⋊#95—22no—
80C3⋊#95—22no—
81C3⋊#95—22no—
82#830—21no—
83#830—21no—
84#830—21no—
85#830D412yes—
86#810—22no—
87#808—22no—
88#808—22no—
89#807—22no—
90C2×C22⋊Q8—31no—
91C2×C22⋊Q8C2×D311yescharacteristic
92C2×#117—31no—
93#485—62no—
94#484—61no—
95#483—61no—
96#483—61no—
97#483—61no—
98#470—62no—
99#451—61no—
100#451—61no—
101#428—61no—
102#419—61no—
103#407—31no—
104#407—31no—
105#400—61no—
106#398—61no—
107#397—62no—
108#391—61no—
109#389—61no—
110#380C2×D311yescharacteristic
111C4×C22⋊C4C2×D311yescharacteristic
112#342—22no—
113#339—62no—
114#328—62no—
115C24×C3C2×D411yescharacteristic
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—42no—
119C22×C3×C4—22no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—42no—
123C22×C3×C4—41no—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4#4111yescharacteristic
130C22×Dic3—21no—
131C22×Dic3—21no—
132C22×Dic3—21no—
133C22×Dic3—21no—
134C2×Dic6—21no—
135C2×Dic6—21no—
136C2×Dic6—42no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—42no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C22⋊C4—22no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C22⋊C4—41no—
146C3×C22⋊C4—42no—
147C3×C42—42no—
148#217—41no—
149#217—41no—
150#217—41no—
151#217—41no—
152#216—82no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—82no—
160Dic3⋊C4—41no—
161Dic3⋊C4—41no—
162C4×Dic3—41no—
163C4×Dic3—41no—
164#208—42no—
165#208—42no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—22no—
168C22×Q8—31no—
169C22×Q8D1211yescharacteristic
170C23×C4C22×D311yescharacteristic
171C2×Q16—62no—
172C2×M16—32no—
173C2×M16—61no—
174C22⋊Q8—21no—
175C22⋊Q8—21no—
176C22⋊Q8—21no—
177C22⋊Q8—21no—
178C22⋊Q8—61no—
179C22⋊Q8—61no—
180C22⋊Q8—61no—
181C22⋊Q8—61no—
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—31no—
191C2×C4⋊C4—61no—
192C2×C4⋊C4—62no—
193C2×C4⋊C4D1211yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—61no—
197C2×C22⋊C4—61no—
198C2×C22⋊C4#6712yes—
199C2×C42—21no—
200C2×C42—61no—
201#107—62no—
202#106—62no—
203#106—62no—
204#103—62no—
205#103—62no—
206#103—62no—
207#103—62no—
208C22⋊C8—32no—
209C22⋊C8—61no—
210C22⋊C8—61no—
211C22⋊C8—61no—
212#95—21no—
213#95—21no—
214#95—62no—
215#95—62no—
216#95—62no—
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—
226C2×C3×C4—22no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—42no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—42no—
238C2×C3×C4—81no—
239C2×C3×C4—81no—
240C2×C3×C4—81no—
241C2×C3×C4—81no—
242C2×C3×C4#12011yescharacteristic
243C2×C3×C4#12011yescharacteristic
244C2×C3×C4#12111yescharacteristic
245C2×C3×C4#12111yescharacteristic
246C2×Dic3—42no—
247C2×Dic3—42no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252C2×Dic3—41no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256Dic6—42no—
257Dic6—81no—
258C3⋊C8—42no—
259C3⋊C8—42no—
260C24C2×#6711yescharacteristic
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—42no—
264C2×Q8—61no—
265C2×Q8—61no—
266C2×Q8—122no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—42no—
271C22×C4—41no—
272C22×C4—61no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4C2×D1211yescharacteristic
277C22×C4#23511yescharacteristic
278C22×C4D3×D411yescharacteristic
279C22×C4D3×D411yescharacteristic
280C22×C4D3×D411yescharacteristic
281C22×C4D3×D411yescharacteristic
282Q16—242no—
283M16—122no—
284M16—122no—
285C2×C8—62no—
286C2×C8—62no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—42no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—61no—
297C4⋊C4—61no—
298C4⋊C4—121no—
299C4⋊C4—121no—
300C4⋊C4—242no—
301C22⋊C4—22no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—42no—
306C22⋊C4—121no—
307C22⋊C4—121no—
308C22⋊C4—121no—
309C22⋊C4—121no—
310C42—42no—
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—82no—
330C3×C4—82no—
331C3×C4—81no—
332C3×C4—81no—
333C3×C4#45812yes—
334Dic3—81no—
335Dic3—81no—
336Dic3—81no—
337Dic3—81no—
338C23—21no—
339C23—21no—
340C23—41no—
341C23—41no—
342C23#93011yescharacteristic
343C23#93711yescharacteristic
344C23#94011yescharacteristic
345Q8—42no—
346Q8—42no—
347Q8—122no—
348Q8—241no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—122no—
366C2×C4—122no—
367C2×C4—121no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4—121no—
371C2×C4—121no—
372C2×C4—121no—
373C2×C4—121no—
374C2×C4—121no—
375C2×C4#88211yescharacteristic
376C2×C4#88211yescharacteristic
377C2×C4#88411yescharacteristic
378C2×C4#88411yescharacteristic
379C8—122no—
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#496011yescharacteristic
399C22#506111yescharacteristic
400C22#509511yescharacteristic
401C4—21no—
402C4—21no—
403C4—21no—
404C4—82no—
405C4—82no—
406C4—81no—
407C4—81no—
408C4—241no—
409C4—241no—
410C4—241no—
411C4—241no—
412C4#474212yes—
413C3#993711yescharacteristic, complemented, Sylow
414C2—21no—
415C2—21no—
416C2—41no—
417C2—41no—
418C2#6969411yescharacteristic
419C2#7003111yescharacteristic
420C2#7005311yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1996C2
socleC22×C3#392
prime radicalC24×C3C2×D4
hypercenter#1996D3
hypocenterC3#9937
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

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
#827422
#827272
#739512
C3×#19962
#694452
#694422
#676262
#99373