Finite Group Databasebeta

← #10697293

Subgroups — #10697293

Order 768.

524 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 107 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#83320C211yesmaximal, characteristic
2#83317C211yesmaximal, characteristic
3#78506C211yesmaximal, characteristic, complemented
4C3×#1980C211yesmaximal, characteristic
5#69489C211yesmaximal, characteristic, complemented
6#69471C211yesmaximal, characteristic, complemented
7#67668C211yesmaximal, characteristic, complemented
8#10197—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C2×C3×#104—22no—
13C3×C4×M16C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15#5121—22no—
16#5094—21no—
17#5093—21no—
18#5076—21no—
19#5075—21no—
20#5041—22no—
21#5040—22no—
22#5019—22no—
23#5019—21no—
24#5008—21no—
25#4925C2211yescharacteristic
26C2×C4⋊(C3⋊C8)C2211yescharacteristic
27#4702—22no—
28#4682—21no—
29#4680—21no—
30#4529—22no—
31#4452—22no—
32#3246—31no—
33#3206—31no—
34#3165—31no—
35#1987—31no—
36#1980D311yescharacteristic
37#1503—31no—
38#1283—31no—
39C2×C3×#41—21no—
40C22×Dic6—21no—
41C2×C3×M16D412yes—
42C3×C4⋊Q8D411yescharacteristic
43C3×#127D411yescharacteristic
44C3×#124—21no—
45C3×#121—21no—
46C3×#121—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C42—21no—
51C2×C3×C42C2311yescharacteristic
52C2×#214—24no—
53C2×#214—22no—
54#926—21no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3—21no—
57#924—21no—
58#924—21no—
59C22×C3⋊C8D412yes—
60#870—21no—
61#869—21no—
62#869—21no—
63#869—21no—
64#869D411yescharacteristic
65#869D411yescharacteristic
66#869D412yes—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C3×#104—44no—
70C3×#97—21no—
71C3×C4×C8—21no—
72C3×#95—21no—
73#832—22no—
74#830D412yes—
75#818—24no—
76#817—24no—
77#810—24no—
78#810—22no—
79#808—22no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4×C3⋊C8—21no—
84#533—31no—
85#532C2×D311yescharacteristic
86C2×C4⋊Q8—31no—
87#501—62no—
88#500—61no—
89#498—61no—
90#492—61no—
91#487—61no—
92#476—61no—
93#475—61no—
94#465—62no—
95#461—62no—
96#461—61no—
97#460—61no—
98#448—62no—
99#431—31no—
100#428—62no—
101C2×C4⋊C8—31no—
102C2×#104—22no—
103C4×M16C2×D311yescharacteristic
104C4×C4⋊C4C2×D311yescharacteristic
105#340—62no—
106#328—62no—
107C3×#41—42no—
108C3×#41—42no—
109C2×C3×Q8—21no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—41no—
114C22×C3×C4—41no—
115C22×C3×C4—41no—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×Dic3—41no—
120C2×Dic6—21no—
121C2×Dic6—22no—
122C2×Dic6—21no—
123C2×Dic6—41no—
124C2×Dic6—82no—
125C3×M16—22no—
126C3×M16SD1614yescomplemented
127C2×C3×C8—22no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—41no—
132C3×C22⋊C4—41no—
133C3×C22⋊C4—41no—
134C3×C42—42no—
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42#4111yescharacteristic
139#217—41no—
140#214—84no—
141#214—82no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—22no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—22no—
148C4⋊Dic3—41no—
149C4⋊Dic3—41no—
150Dic3⋊C4—41no—
151Dic3⋊C4—41no—
152C4×Dic3—41no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—42no—
156C2×#41—21no—
157C22×Q8—61no—
158C2×SD16—64no—
159C2×SD16—62no—
160C2×M16#6712yes—
161C22×C8—32no—
162C4⋊Q8—32no—
163C4⋊Q8—31no—
164C4⋊Q8—31no—
165C4⋊Q8—61no—
166C4⋊Q8—61no—
167C4⋊Q8—61no—
168C4⋊Q8D1211yescharacteristic
169#127D1211yescharacteristic
170#125—61no—
171#124—21no—
172C22⋊Q8—61no—
173C22⋊Q8—61no—
174#121—21no—
175#121—21no—
176C4×Q8—61no—
177C4×Q8—61no—
178C4×D4—21no—
179C4×D4—21no—
180#117—61no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—61no—
184C2×C42—21no—
185C2×C42C22×D311yescharacteristic
186#107—64no—
187#106—62no—
188#106—64no—
189C4⋊C8—61no—
190C4⋊C8—61no—
191C4⋊C8—61no—
192#104—44no—
193#103—64no—
194#103—62no—
195#102—62no—
196C22⋊C8—32no—
197#97—21no—
198C4×C8—21no—
199C4×C8—61no—
200#95—21no—
201C23×C3—41no—
202C23×C3#12711yescharacteristic
203C3×Q8—42no—
204C3×D4—42no—
205C3×D4—41no—
206C3×D4—41no—
207C3×D4—42no—
208C3×D4—81no—
209C3×D4—81no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—41no—
213C2×C3×C4—42no—
214C2×C3×C4—41no—
215C2×C3×C4—81no—
216C2×C3×C4—81no—
217C2×C3×C4—82no—
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4C2×SD1612yes—
225C2×C3×C4#13712yes—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—42no—
229C2×Dic3—82no—
230Dic6—44no—
231Dic6—41no—
232Dic6—41no—
233Dic6—82no—
234Dic6—81no—
235C3×C8—24no—
236C3⋊C8—42no—
237C3⋊C8—42no—
238#41—42no—
239#41—42no—
240C2×Q8—21no—
241C2×Q8—61no—
242C2×Q8—62no—
243C2×Q8—61no—
244C2×Q8—121no—
245C2×Q8—242no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—121no—
253C22×C4D3×D411yescharacteristic
254C22×C4D3×D411yescharacteristic
255C22×C4C2×#6711yescharacteristic
256SD16—244no—
257SD16—242no—
258M16—22no—
259M16#21414yescomplemented
260C2×C8—22no—
261C2×C8—62no—
262C2×C8—62no—
263C2×C8—122no—
264C4⋊C4—22no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—41no—
268C4⋊C4—61no—
269C4⋊C4—62no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—62no—
273C4⋊C4—61no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C4⋊C4—121no—
277C4⋊C4—121no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—121no—
281C42—42no—
282C42—121no—
283C42C2×D1211yescharacteristic
284C42#23511yescharacteristic
285C42D3×D411yescharacteristic
286C42D3×D411yescharacteristic
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—81no—
290C22×C3—81no—
291C22×C3—81no—
292C22×C3#39211yescharacteristic
293C22×C3#49211yescharacteristic
294C22×C3#49711yescharacteristic
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—82no—
299C3×C4—81no—
300C3×C4#44812yes—
301C3×C4#45511yescharacteristic
302C3×C4#45611yescharacteristic
303C3×C4#46112yes—
304Dic3—81no—
305Dic3—82no—
306Dic3—81no—
307C23—41no—
308C23#94011yescharacteristic
309Q8—42no—
310Q8—124no—
311Q8—121no—
312Q8—121no—
313Q8—241no—
314Q8—242no—
315D4—42no—
316D4—42no—
317D4—41no—
318D4—41no—
319D4—81no—
320D4—81no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—82no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4—122no—
332C2×C4—242no—
333C2×C4#88211yescharacteristic
334C2×C4#88211yescharacteristic
335C2×C4#88411yescharacteristic
336C2×C4#88411yescharacteristic
337C2×C4#93011yescharacteristic
338C2×C4C2×#21412yes—
339C2×C4#93711yescharacteristic
340C2×C4#95112yes—
341C8—24no—
342C8—122no—
343C8—122no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—161no—
347C2×C3#200711yescharacteristic
348C2×C3#200811yescharacteristic
349C2×C3#201311yescharacteristic
350C22—21no—
351C22—21no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22#496011yescharacteristic
356C22#506011yescharacteristic
357C22#507611yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—82no—
362C4—81no—
363C4—242no—
364C4—241no—
365C4—241no—
366C4#482511yescharacteristic
367C4#482611yescharacteristic
368C4#501912yes—
369C4#504012yes—
370C3#1019711yescharacteristic, complemented, Sylow
371C2—21no—
372C2—21no—
373C2—161no—
374C2#6966111yescharacteristic
375C2#7039311yescharacteristic
376C2#7040911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1980C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#1980D3
hypocenterC3#10197
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1980

Complements

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

NH ≅ G/Ntype
3. #78506C2semidirect
5. #69489C2semidirect
6. #69471C2semidirect
7. #67668C2semidirect
126. C3×M16SD16semidirect
259. M16#214semidirect
370. C3#10197semidirect

Maximal subgroups

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

subgroupindex
#833202
#833172
#785062
C3×#19802
#694892
#694712
#676682
#101973