Finite Group Databasebeta

← #10696005

Subgroups — #10696005

Order 768.

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

normal subgroups only

This group has 528 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×#3110C211yesmaximal, characteristic
2#83326C211yesmaximal, characteristic, complemented
3#82383C211yesmaximal, characteristic
4#71841C211yesmaximal, characteristic, complemented
5#71599C211yesmaximal, characteristic
6#70812C211yesmaximal, characteristic, complemented
7#67749C211yesmaximal, characteristic, complemented
8#10009—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C3×#484—21no—
13C3×#474—21no—
14C3×#461—21no—
15C3×#450—21no—
16C3×#440—22no—
17C3×#432C2211yescharacteristic
18#5222—21no—
19#5212—21no—
20#5212—21no—
21#5082—21no—
22#5081—21no—
23#5044—21no—
24#5043—21no—
25#5022—22no—
26#4985—21no—
27#4945—22no—
28C4×C4⋊Dic3C2211yescharacteristic
29#4926C2211yescharacteristic
30C4×#208C2211yescharacteristic
31D4⋊(C3⋊C8)—22no—
32#4532—22no—
33#4465—22no—
34#3236—31no—
35#3110D311yescharacteristic
36#2562—31no—
37#1986—31no—
38#1528—31no—
39#1490—31no—
40#1285—31no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C2×C3×SD16—22no—
44C3×C4⋊Q8D411yescharacteristic
45C3×C4⋊Q8D411yescharacteristic
46C3×#125—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×#117—21no—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C22⋊C4—21no—
58C2×C3×C42C2311yescharacteristic
59C2×#217—22no—
60C2×#217—22no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3D411yescharacteristic
63C2×C4⋊Dic3D411yescharacteristic
64C2×Dic3⋊C4—21no—
65C2×Dic3⋊C4—21no—
66C2×C4×Dic3D412yes—
67C2×#208—21no—
68#870—21no—
69#870—21no—
70#870—21no—
71#870—21no—
72C3×#107—22no—
73C3×C4⋊C8—21no—
74C3×#103—22no—
75C3×#102—22no—
76C3×C22⋊C8—21no—
77C3×#97D412yes—
78#835—22no—
79#835—22no—
80C3⋊#95—22no—
81C3⋊#95—22no—
82C3⋊#95—21no—
83#830—21no—
84#808—22no—
85#807—22no—
86C4⋊(C3⋊C8)D412yes—
87#803—21no—
88#803—21no—
89C4×C3⋊C8—21no—
90#533C2×D311yescharacteristic
91C2×#125—31no—
92C2×C4×D4C2×D311yescharacteristic
93#501—61no—
94#491—61no—
95#484—21no—
96#484—61no—
97#483—61no—
98#479—62no—
99#474—21no—
100#461—21no—
101#450—21no—
102#440—22no—
103#432—31no—
104#432C2×D311yescharacteristic
105C4×M16—31no—
106#400—61no—
107#394—61no—
108#388—62no—
109#380—61no—
110#380—61no—
111C4×C4⋊C4—31no—
112#339—62no—
113#330—62no—
114D4⋊C8—62no—
115C24×C3—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—42no—
121C22×C3×C4—21no—
122C22×C3×C4—42no—
123C22×C3×C4—42no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×Dic3—22no—
130C22×Dic3—21no—
131C22×Dic3—21no—
132C3×SD16—44no—
133C2×C3×C8—22no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C42—41no—
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42#4111yescharacteristic
150#217—44no—
151#217—82no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—42no—
157Dic3⋊C4—41no—
158Dic3⋊C4—41no—
159Dic3⋊C4—41no—
160Dic3⋊C4—41no—
161C4×Dic3—42no—
162#208—22no—
163#208—41no—
164C2×C3⋊C8—22no—
165C2×C3⋊C8—41no—
166C22×D4#6711yescharacteristic
167C23×C4—21no—
168C2×SD16—22no—
169C2×M16—61no—
170C4⋊Q8#6711yescharacteristic
171C4⋊Q8#6711yescharacteristic
172#125—21no—
173#125—61no—
174#125—61no—
175#125—61no—
176#125—61no—
177C22⋊Q8—21no—
178C22⋊Q8—21no—
179C4×Q8—21no—
180C4×Q8—21no—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—21no—
185#117—21no—
186C2×C4⋊C4—31no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4#6711yescharacteristic
192C2×C22⋊C4—21no—
193C2×C22⋊C4—62no—
194C2×C22⋊C4—62no—
195C2×C42—32no—
196C2×C42C22×D311yescharacteristic
197#107—22no—
198#107—62no—
199#106—62no—
200C4⋊C8—21no—
201C4⋊C8—32no—
202#103—22no—
203#103—62no—
204#103—62no—
205#102—22no—
206C22⋊C8—21no—
207C22⋊C8—61no—
208#97—61no—
209#97—61no—
210#97#6712yes—
211C4×C8—61no—
212#95—62no—
213#95—62no—
214#95—61no—
215C23×C3—22no—
216C23×C3—41no—
217C23×C3—41no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3#12011yescharacteristic
223C3×Q8—42no—
224C3×Q8—42no—
225C3×D4—22no—
226C3×D4—22no—
227C3×D4—41no—
228C3×D4—41no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—22no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—82no—
239C2×C3×C4—82no—
240C2×C3×C4#12011yescharacteristic
241C2×C3×C4#12011yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12111yescharacteristic
244C2×C3×C4#12111yescharacteristic
245C2×C3×C4#12711yescharacteristic
246C2×Dic3—42no—
247C2×Dic3—42no—
248C2×Dic3—42no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251C2×Dic3—42no—
252C2×Dic3—41no—
253C2×Dic3—41no—
254C3×C8—42no—
255C3⋊C8—42no—
256C3⋊C8—42no—
257C24—21no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—42no—
263C22×C4—21no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—62no—
269C22×C4—61no—
270C22×C4—61no—
271C22×C4D3×D411yescharacteristic
272C22×C4D3×D411yescharacteristic
273C22×C4C2×#6711yescharacteristic
274SD16—44no—
275M16—62no—
276M16—121no—
277C2×C8—22no—
278C2×C8—62no—
279C2×C8—121no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—61no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—122no—
293C4⋊C4—121no—
294C4⋊C4—121no—
295C4⋊C4—121no—
296C4⋊C4—121no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—124no—
301C22⋊C4—242no—
302C42—41no—
303C42—122no—
304C42D3×D411yescharacteristic
305C42#23711yescharacteristic
306C42C2×#6711yescharacteristic
307C42C2×#6711yescharacteristic
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—42no—
311C22×C3—42no—
312C22×C3—42no—
313C22×C3—81no—
314C22×C3—81no—
315C22×C3—81no—
316C22×C3—81no—
317C22×C3#39211yescharacteristic
318C22×C3#45511yescharacteristic
319C22×C3#45611yescharacteristic
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—81no—
326C3×C4—81no—
327C3×C4—81no—
328C3×C4#45412yes—
329Dic3—82no—
330Dic3—81no—
331Dic3—81no—
332C23—22no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23#93711yescharacteristic
340Q8—42no—
341Q8—42no—
342D4—22no—
343D4—22no—
344D4—41no—
345D4—41no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—122no—
358C2×C4—122no—
359C2×C4—122no—
360C2×C4—122no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4#93711yescharacteristic
366C2×C4#93811yescharacteristic
367C2×C4#93911yescharacteristic
368C2×C4#93911yescharacteristic
369C2×C4#94011yescharacteristic
370C2×C4#95311yescharacteristic
371C8—42no—
372C8—122no—
373C8—122no—
374C2×C3—21no—
375C2×C3—21no—
376C2×C3—82no—
377C2×C3#200311yescharacteristic
378C2×C3#200711yescharacteristic
379C2×C3#201011yescharacteristic
380C22—21no—
381C22—21no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22#520211yescharacteristic
390C22#520511yescharacteristic
391C22#522311yescharacteristic
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—81no—
398C4—81no—
399C4—81no—
400C4—242no—
401C4—241no—
402C4—241no—
403C4#520112yes—
404C3#1000911yescharacteristic, complemented, Sylow
405C2—21no—
406C2—21no—
407C2—82no—
408C2#7171511yescharacteristic
409C2#7209611yescharacteristic
410C2#7211011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3110C2
socleC22×C3#392
prime radicalC22×C3×D4D4
hypercenter#3110D3
hypocenterC3#10009
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3110

Complements

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

NH ≅ G/Ntype
2. #83326C2semidirect
4. #71841C2semidirect
6. #70812C2semidirect
7. #67749C2semidirect
404. C3#10009semidirect

Maximal subgroups

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

subgroupindex
C3×#31102
#833262
#823832
#718412
#715992
#708122
#677492
#100093