Finite Group Databasebeta

← #10767972

Subgroups — #10767972

Order 768.

1106 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1106 conjugacy classes of subgroups — too many to draw. Only its 167 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#85766C211yesmaximal, characteristic, complemented
2C3×#2397C211yesmaximal, characteristic
3#82315C212yesmaximal, complemented
4#71603C211yesmaximal, characteristic, complemented
5#69716C214yesmaximal, complemented
6#69698C214yesmaximal, complemented
7#69433C212yesmaximal, complemented
8#18180—31nomaximal, Sylow
9C2×C3×C4×D4—22no—
10C2×C3×C4×D4C2211yescharacteristic
11C2×C3×C4×D4C2211yescharacteristic
12#5798—21no—
13C2×#924—22no—
14#5543—22no—
15#5539—24no—
16D4×Dic6—22no—
17#5500—21no—
18#5483C2214yes—
19#5470C2212yes—
20C3×#399—21no—
21C3×#399C2212yes—
22C3×#392—21no—
23C3×#392C2212yes—
24C3×#376C2211yescharacteristic
25#5218C2214yes—
26#5019—22no—
27#5018—24no—
28#5007—22no—
29#4970C2214yes—
30#4926C2212yes—
31#4525C2214yes—
32#4452C2218yes—
33#3501—31no—
34#3110—32no—
35#2397D311yescharacteristic
36#1614—34no—
37#1596—34no—
38#1506—32no—
39#1466—31no—
40C22×C3×D4—22no—
41C22×C3×D4D411yescharacteristic
42C22×C3×D4C2311yescharacteristic
43C23×C3×C4—21no—
44C23×C3×C4D412yes—
45C22×Dic6—22no—
46C3×C4×D4—22no—
47C3×C4×D4—22no—
48C3×C4×D4—22no—
49C3×C4×D4—22no—
50C3×C4×D4—48no—
51C2×C3×C4⋊C4—22no—
52C2×C3×C4⋊C4D411yescharacteristic
53C2×C3×C4⋊C4C2311yescharacteristic
54C2×C3×C22⋊C4—22no—
55C2×C3×C22⋊C4—21no—
56C2×C3×C22⋊C4—22no—
57C2×C3×C22⋊C4C2312yes—
58C2×C3×C42—21no—
59C2×C3×C42C2311yescharacteristic
60C2×#217—24no—
61D4×Dic3—24no—
62D4×Dic3—24no—
63C2×#214—24no—
64#926—22no—
65C2×C4⋊Dic3—22no—
66#924—24no—
67#924—24no—
68#924—22no—
69#924—24no—
70#924C2314yes—
71C2×Dic3⋊C4—24no—
72#890—24no—
73#888—24no—
74#878—28no—
75#878—28no—
76#870—22no—
77#869D412yescomplemented
78#869D412yescomplemented
79C4×Dic6—22no—
80C4×Dic6D414yescomplemented
81C3×#95—21no—
82C3×#95C2312yes—
83#832—24no—
84#830C2314yes—
85#810—24no—
86#807—24no—
87C4⋊(C3⋊C8)—22no—
88#803—22no—
89#554—64no—
90#554—62no—
91D4×Q8—62no—
92#544—61no—
93#543—34no—
94#533—32no—
95C2×C22⋊Q8—62no—
96#518—61no—
97C2×C4×D4—22no—
98C2×C4×D4C2×D311yescharacteristic
99C2×C4×D4C2×D311yescharacteristic
100#484—34no—
101#474—62no—
102#461—62no—
103#450—34no—
104#440—64no—
105#432—32no—
106#399—21no—
107#399C2×D312yes—
108#392—21no—
109#392C2×D312yes—
110#376C2×D311yescharacteristic
111#328—38no—
112#323—34no—
113C24×C3—21no—
114C24×C3C2×D412yes—
115C2×C3×D4—24no—
116C2×C3×D4—22no—
117C2×C3×D4—22no—
118C2×C3×D4—44no—
119C2×C3×D4—42no—
120C2×C3×D4—44no—
121C2×C3×D4—44no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—21no—
126C22×C3×C4—44no—
127C22×C3×C4—44no—
128C22×C3×C4—41no—
129C22×C3×C4—41no—
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D412yes—
133C22×C3×C4C2411yescharacteristic
134C22×Dic3—24no—
135C22×Dic3—24no—
136C2×Dic6—22no—
137C2×Dic6—24no—
138C2×Dic6—84no—
139C2×Dic6C2×D414yes—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—44no—
143C3×C22⋊C4—24no—
144C3×C22⋊C4—44no—
145C3×C22⋊C4—44no—
146C3×C22⋊C4—42no—
147C3×C42—24no—
148C3×C42C2×D412yes—
149C3×C42C2×D412yes—
150#217—28no—
151#217—24no—
152#214—48no—
153C4⋊Dic3—24no—
154C4⋊Dic3—22no—
155C4⋊Dic3—22no—
156C4⋊Dic3—22no—
157C4⋊Dic3C2×D414yes—
158Dic3⋊C4—28no—
159Dic3⋊C4—24no—
160Dic3⋊C4—24no—
161C4×Dic3—24no—
162C2×C3⋊C8—24no—
163C22×Q8—62no—
164C22×D4—22no—
165C22×D4#6711yescharacteristic
166C22×D4C22×D311yescharacteristic
167C23×C4—21no—
168C23×C4#6712yes—
169C2×SD16—64no—
170C4⋊Q8—32no—
171C4⋊Q8—32no—
172C4⋊Q8—64no—
173#125—64no—
174#125—62no—
175C22⋊Q8—34no—
176C22⋊Q8—68no—
177C22⋊Q8—68no—
178C22⋊Q8—64no—
179C22⋊Q8—64no—
180C22⋊Q8—62no—
181C22⋊Q8—64no—
182C4×Q8—34no—
183C4×Q8—62no—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—22no—
187C4×D4—22no—
188C4×D4—48no—
189C4×D4—64no—
190C4×D4—64no—
191#117—62no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—64no—
194C2×C4⋊C4—62no—
195C2×C4⋊C4#6711yescharacteristic
196C2×C4⋊C4C22×D311yescharacteristic
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—64no—
201C2×C22⋊C4C22×D312yes—
202C2×C42—21no—
203C2×C42C22×D311yescharacteristic
204#107—64no—
205C4⋊C8—62no—
206#103—64no—
207#102—64no—
208C22⋊C8—34no—
209#97—62no—
210#95—21no—
211#95C22×D312yes—
212C23×C3—22no—
213C23×C3—22no—
214C23×C3—22no—
215C23×C3—42no—
216C23×C3—42no—
217C23×C3—42no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3—42no—
221C23×C3#13714yes—
222C23×C3C22×D411yescharacteristic
223C3×D4—28no—
224C3×D4—24no—
225C3×D4—42no—
226C3×D4—416no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—24no—
230C2×C3×C4—22no—
231C2×C3×C4—41no—
232C2×C3×C4—42no—
233C2×C3×C4—42no—
234C2×C3×C4—44no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—41no—
238C2×C3×C4—82no—
239C2×C3×C4—82no—
240C2×C3×C4—82no—
241C2×C3×C4#12012yes—
242C2×C3×C4#12011yescharacteristic
243C2×C3×C4#12011yescharacteristic
244C2×C3×C4C22×D412yes—
245C2×Dic3—24no—
246C2×Dic3—24no—
247C2×Dic3—84no—
248C2×Dic3—84no—
249Dic6—44no—
250Dic6—44no—
251Dic6—84no—
252C3⋊C8—44no—
253C24—21no—
254C24C2×#6712yes—
255C2×Q8—34no—
256C2×Q8—64no—
257C2×Q8—62no—
258C2×Q8—244no—
259C2×D4—24no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—44no—
263C2×D4—42no—
264C2×D4—44no—
265C2×D4—44no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—64no—
275C22×C4—64no—
276C22×C4C2×#6711yescharacteristic
277C22×C4C2×#6711yescharacteristic
278C22×C4C2×#6712yes—
279C22×C4C23×D311yescharacteristic
280SD16—128no—
281C2×C8—64no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—34no—
285C4⋊C4—44no—
286C4⋊C4—64no—
287C4⋊C4—68no—
288C4⋊C4—64no—
289C4⋊C4—62no—
290C4⋊C4—62no—
291C4⋊C4—62no—
292C4⋊C4—64no—
293C22⋊C4—24no—
294C22⋊C4—44no—
295C22⋊C4—44no—
296C22⋊C4—42no—
297C22⋊C4—68no—
298C22⋊C4—64no—
299C42—24no—
300C42—64no—
301C42D3×D412yes—
302C42D3×D412yes—
303C22×C3—21no—
304C22×C3—21no—
305C22×C3—24no—
306C22×C3—44no—
307C22×C3—44no—
308C22×C3—42no—
309C22×C3—42no—
310C22×C3—42no—
311C22×C3—82no—
312C22×C3—82no—
313C22×C3—81no—
314C22×C3—81no—
315C22×C3C2×#12011yescharacteristic
316C22×C3C2×#13712yes—
317C3×C4—22no—
318C3×C4—21no—
319C3×C4—22no—
320C3×C4—44no—
321C3×C4—42no—
322C3×C4—81no—
323C3×C4#45312yes—
324Dic3—44no—
325Dic3—44no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—41no—
333C23—41no—
334C23—42no—
335C23#95114yes—
336C23C22×#6711yescharacteristic
337Q8—124no—
338Q8—124no—
339Q8—244no—
340D4—28no—
341D4—24no—
342D4—42no—
343D4—416no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—64no—
356C2×C4—64no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—244no—
361C2×C4—244no—
362C2×C4#93712yes—
363C2×C4#93711yescharacteristic
364C2×C4#93711yescharacteristic
365C2×C4C2×D3×D412yes—
366C8—124no—
367C2×C3—22no—
368C2×C3—44no—
369C2×C3—82no—
370C2×C3#300411yescharacteristic
371C2×C3C2×#45311yescharacteristic
372C2×C3#302511yescharacteristic
373C22—21no—
374C22—21no—
375C22—24no—
376C22—44no—
377C22—44no—
378C22—42no—
379C22—42no—
380C22—42no—
381C22—82no—
382C22—82no—
383C22—81no—
384C22—81no—
385C22C2×#93711yescharacteristic
386C22C2×#95112yes—
387C4—21no—
388C4—22no—
389C4—22no—
390C4—44no—
391C4—42no—
392C4—81no—
393C4—124no—
394C4—124no—
395C4#507712yes—
396C3#1818011yescharacteristic, complemented, Sylow
397C2—22no—
398C2—44no—
399C2—82no—
400C2#8293311yescharacteristic
401C2#8293611yescharacteristic
402C2C2×#507711yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#2397C2
socleC22×C3C2×#120
hypercenter#2397D3
hypocenterC3#18180
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2397

Complements

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

NH ≅ G/Ntype
1. #85766C2semidirect
3. #82315C2semidirect
4. #71603C2semidirect
5. #69716C2semidirect
6. #69698C2semidirect
7. #69433C2semidirect
77. #869D4semidirect
78. #869D4semidirect
80. C4×Dic6D4semidirect
396. C3#18180semidirect

Maximal subgroups

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

subgroupindex
#857662
C3×#23972
#823152
#716032
#697162
#696982
#694332
#181803