Finite Group Databasebeta

← #10696067

Subgroups — #10696067

Order 768.

540 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 81 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×#3109C211yesmaximal, characteristic
2#83284C211yesmaximal, characteristic, complemented
3#82391C211yesmaximal, characteristic, complemented
4#71813C211yesmaximal, characteristic, complemented
5#71599C211yesmaximal, characteristic, complemented
6#70814C211yesmaximal, characteristic, complemented
7#67752C211yesmaximal, characteristic, complemented
8#10019—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C3×#481—21no—
13C3×#476—21no—
14C3×#459—21no—
15C3×#447—21no—
16C3×#442—22no—
17C3×#432C2211yescharacteristic
18#5232—21no—
19#5232—21no—
20#5222—21no—
21#5210—22no—
22#5082—21no—
23#5071—21no—
24#5056—22no—
25#4983—21no—
26#4953—21no—
27#4949—21no—
28#4937C2211yescharacteristic
29#4926C2211yescharacteristic
30C4×#208C2211yescharacteristic
31#4537—22no—
32D4⋊(C3⋊C8)—22no—
33#4465—22no—
34#3142—31no—
35#3109D311yescharacteristic
36#2633—31no—
37#1534—31no—
38#1514—31no—
39#1490—31no—
40#1288—31no—
41C2×C3×#41—21no—
42C22×C3×D4D411yescharacteristic
43C23×C3×C4—21no—
44C2×C3×D8—22no—
45C3×C4⋊Q8D411yescharacteristic
46C3×#127D411yescharacteristic
47C3×#124—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C22⋊C4—21no—
58C2×C3×C42C2311yescharacteristic
59C2×#217—22no—
60C2×#217—22no—
61C2×C4⋊Dic3D411yescharacteristic
62C2×C4⋊Dic3D411yescharacteristic
63C2×Dic3⋊C4—21no—
64C2×Dic3⋊C4—21no—
65C2×Dic3⋊C4—22no—
66C2×#208—21no—
67#870—21no—
68#870—21no—
69#870—21no—
70#870—21no—
71C3×#106—22no—
72C3×C4⋊C8—21no—
73C3×#102—22no—
74C3×#102—22no—
75C3×C22⋊C8—21no—
76C3×#97D412yes—
77#835—22no—
78#835—22no—
79#832—22no—
80#832—22no—
81C3⋊#95—22no—
82C3⋊#95—21no—
83C3⋊#95—21no—
84C3⋊#95D412yes—
85#830—21no—
86C4⋊(C3⋊C8)—21no—
87#803—21no—
88#803—21no—
89C4×C3⋊C8D412yes—
90#532C2×D311yescharacteristic
91C2×#125—31no—
92C2×C4×D4C2×D311yescharacteristic
93#491—61no—
94#490—61no—
95#488—62no—
96#481—21no—
97#476—21no—
98#459—21no—
99#447—21no—
100#442—22no—
101#432—31no—
102#432C2×D311yescharacteristic
103#417—61no—
104#417—61no—
105C4×M16—31no—
106#399—61no—
107#398—61no—
108#397—61no—
109#394—61no—
110#385—62no—
111#383—31no—
112#330—62no—
113#327—62no—
114D4⋊C8—62no—
115C24×C3—21no—
116C3×#41—44no—
117C2×C3×Q8—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—42no—
124C2×C3×D4—41no—
125C22×C3×C4—21no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4—41no—
129C22×C3×C4—82no—
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×Dic3—22no—
134C22×Dic3—21no—
135C22×Dic3—21no—
136C3×D8—44no—
137C2×C3×C8—22no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C22⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C22⋊C4—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—
156Dic3⋊C4—41no—
157Dic3⋊C4—41no—
158Dic3⋊C4—41no—
159Dic3⋊C4—41no—
160Dic3⋊C4—82no—
161#208—42no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—41no—
164C2×#41—21no—
165C22×D4#6711yescharacteristic
166C23×C4—21no—
167C2×D8—22no—
168C2×M16—61no—
169C4⋊Q8#6711yescharacteristic
170#127#6711yescharacteristic
171#125—61no—
172#125—61no—
173#125—61no—
174#125—61no—
175#124—21no—
176#121—21no—
177#121—21no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4—21no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—31no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4—61no—
188C2×C4⋊C4—62no—
189C2×C4⋊C4#6711yescharacteristic
190C2×C22⋊C4—21no—
191C2×C22⋊C4—62no—
192C2×C22⋊C4—62no—
193C2×C42C22×D311yescharacteristic
194#106—22no—
195C4⋊C8—21no—
196C4⋊C8—61no—
197#103—62no—
198#103—62no—
199#102—22no—
200#102—22no—
201#102—62no—
202#102—62no—
203C22⋊C8—21no—
204C22⋊C8—61no—
205#97—61no—
206#97—61no—
207#97#6712yes—
208C4×C8—32no—
209#95—32no—
210#95—62no—
211#95—61no—
212#95—61no—
213C23×C3—22no—
214C23×C3—42no—
215C23×C3—41no—
216C23×C3—41no—
217C23×C3—42no—
218C23×C3—41no—
219C23×C3#12011yescharacteristic
220C3×Q8—42no—
221C3×D4—22no—
222C3×D4—22no—
223C3×D4—41no—
224C3×D4—41no—
225C3×D4—42no—
226C3×D4—42no—
227C3×D4—81no—
228C3×D4—82no—
229C2×C3×C4—22no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—42no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—82no—
236C2×C3×C4—82no—
237C2×C3×C4—82no—
238C2×C3×C4#12011yescharacteristic
239C2×C3×C4#12011yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12711yescharacteristic
244C2×Dic3—41no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3—42no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251C2×Dic3—42no—
252C3×C8—42no—
253C3⋊C8—42no—
254C3⋊C8—81no—
255C24—21no—
256#41—44no—
257C2×Q8—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—42no—
264C2×D4—41no—
265C22×C4—21no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—62no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4—82no—
273C22×C4D3×D411yescharacteristic
274C22×C4D3×D411yescharacteristic
275C22×C4C2×#6711yescharacteristic
276D8—44no—
277M16—122no—
278C2×C8—22no—
279C2×C8—62no—
280C2×C8—121no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—61no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C4⋊C4—61no—
289C4⋊C4—121no—
290C4⋊C4—121no—
291C4⋊C4—121no—
292C4⋊C4—121no—
293C4⋊C4—242no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—124no—
299C22⋊C4—242no—
300C42D3×D411yescharacteristic
301C42#23711yescharacteristic
302C42C2×#6711yescharacteristic
303C42C2×#6711yescharacteristic
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—44no—
307C22×C3—42no—
308C22×C3—82no—
309C22×C3—81no—
310C22×C3—81no—
311C22×C3—81no—
312C22×C3—81no—
313C22×C3—81no—
314C22×C3#39211yescharacteristic
315C22×C3#45412yes—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—81no—
323C3×C4—81no—
324Dic3—82no—
325Dic3—81no—
326Dic3—81no—
327C23—22no—
328C23—42no—
329C23—41no—
330C23—42no—
331C23—41no—
332C23—41no—
333C23#93711yescharacteristic
334Q8—42no—
335D4—22no—
336D4—22no—
337D4—41no—
338D4—41no—
339D4—42no—
340D4—42no—
341D4—81no—
342D4—82no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—122no—
353C2×C4—122no—
354C2×C4—121no—
355C2×C4—122no—
356C2×C4—121no—
357C2×C4—122no—
358C2×C4—121no—
359C2×C4—121no—
360C2×C4#93711yescharacteristic
361C2×C4#93811yescharacteristic
362C2×C4#93911yescharacteristic
363C2×C4#93911yescharacteristic
364C2×C4#94011yescharacteristic
365C2×C4#95311yescharacteristic
366C8—42no—
367C8—122no—
368C8—241no—
369C2×C3—22no—
370C2×C3—82no—
371C2×C3—161no—
372C2×C3#200311yescharacteristic
373C2×C3#200811yescharacteristic
374C2×C3#201011yescharacteristic
375C22—21no—
376C22—21no—
377C22—44no—
378C22—42no—
379C22—82no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22#520112yes—
386C22#522311yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—81no—
394C4—81no—
395C4—242no—
396C4—241no—
397C4—241no—
398C3#1001911yescharacteristic, complemented, Sylow
399C2—22no—
400C2—82no—
401C2—161no—
402C2#7171511yescharacteristic
403C2#7209611yescharacteristic
404C2#7209911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3109C2
socleC22×C3#392
hypercenter#3109D3
hypocenterC3#10019
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3109

Complements

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

NH ≅ G/Ntype
2. #83284C2semidirect
3. #82391C2semidirect
4. #71813C2semidirect
5. #71599C2semidirect
6. #70814C2semidirect
7. #67752C2semidirect
398. C3#10019semidirect

Maximal subgroups

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

subgroupindex
C3×#31092
#832842
#823912
#718132
#715992
#708142
#677522
#100193