Finite Group Databasebeta

← #10690267

Subgroups — #10690267

Order 768.

670 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 111 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×#2397C211yesmaximal, characteristic
2#82302C211yesmaximal, characteristic, complemented
3C4×#833C211yesmaximal, characteristic, complemented
4#71599C212yesmaximal, complemented
5#67615C212yesmaximal, complemented
6#9031—31nomaximal, Sylow
7C2×C3×C4×D4—21no—
8C2×C3×C4×D4—21no—
9C2×C3×C4×D4C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C3×#399—21no—
12C3×#399C411yescharacteristic
13C3×#399C411yescharacteristic
14C3×#392—21no—
15C3×#392C411yescharacteristic
16C3×#392C411yescharacteristic
17C3×#376C2211yescharacteristic
18C2×#833—21no—
19#5207—21no—
20C2×#830—21no—
21#5011—24no—
22#4926C2212yes—
23C4×#208C2212yes—
24D4⋊(C3⋊C8)—24no—
25#4530—21no—
26#2973—31no—
27#2397D311yescharacteristic
28C4×#100—31no—
29#1490—32no—
30#1300—32no—
31C22×C3×D4—21no—
32C22×C3×D4—21no—
33C22×C3×D4C2×C411yescharacteristic
34C22×C3×D4D411yescharacteristic
35C23×C3×C4—21no—
36C23×C3×C4D411yescharacteristic
37C23×C3×C4D411yescharacteristic
38C3×C4×D4—22no—
39C3×C4×D4—22no—
40C3×C4×D4—22no—
41C3×C4×D4—22no—
42C3×C4×D4—44no—
43C3×C4×D4—44no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4C2×C411yescharacteristic
47C2×C3×C4⋊C4D411yescharacteristic
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4—21no—
53C2×C3×C22⋊C4C2×C411yescharacteristic
54C2×C3×C22⋊C4C2×C411yescharacteristic
55C2×C3×C42—21no—
56C2×C3×C42C2311yescharacteristic
57C2×#208—22no—
58C2×#208—22no—
59C22×C3⋊C8—22no—
60C3×#95—21no—
61C3×#95C2×C411yescharacteristic
62C3×#95C2×C411yescharacteristic
63#833—24no—
64#830—22no—
65#830—22no—
66#830—22no—
67C4⋊(C3⋊C8)D414yescomplemented
68#803—22no—
69#803—22no—
70C4×C3⋊C8—22no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4C2×D311yescharacteristic
74C2×C4×D4C2×D311yescharacteristic
75#435—64no—
76#432—32no—
77C2×#100—61no—
78#407—61no—
79C2×C22⋊C8—61no—
80C4×M16—32no—
81#399—21no—
82#399Dic311yescharacteristic
83#399Dic311yescharacteristic
84#392—21no—
85#392Dic311yescharacteristic
86#392Dic311yescharacteristic
87#376C2×D311yescharacteristic
88#343—61no—
89D4⋊C8—64no—
90C24×C3—21no—
91C24×C3C22⋊C411yescharacteristic
92C24×C3C22⋊C411yescharacteristic
93C2×C3×D4—22no—
94C2×C3×D4—22no—
95C2×C3×D4—22no—
96C2×C3×D4—41no—
97C2×C3×D4—42no—
98C2×C3×D4—42no—
99C2×C3×D4—42no—
100C2×C3×D4—42no—
101C2×C3×D4—42no—
102C2×C3×D4—42no—
103C2×C3×D4—42no—
104C22×C3×C4—21no—
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4—21no—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—42no—
112C22×C3×C4—42no—
113C22×C3×C4—42no—
114C22×C3×C4—42no—
115C22×C3×C4—41no—
116C22×C3×C4—41no—
117C22×C3×C4C22⋊C411yescharacteristic
118C22×C3×C4C22⋊C411yescharacteristic
119C22×C3×C4C22×C411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C3×C4⋊C4—22no—
123C3×C4⋊C4—22no—
124C3×C4⋊C4—42no—
125C3×C4⋊C4—42no—
126C3×C22⋊C4—22no—
127C3×C22⋊C4—22no—
128C3×C22⋊C4—42no—
129C3×C22⋊C4—42no—
130C3×C22⋊C4—42no—
131C3×C22⋊C4—42no—
132C3×C22⋊C4—42no—
133C3×C42—24no—
134C3×C42C2×D412yes—
135C3×C42#4112yes—
136#208—24no—
137#208—42no—
138#208—82no—
139C2×C3⋊C8—24no—
140C2×C3⋊C8—42no—
141C2×C3⋊C8—82no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4C2×Dic311yescharacteristic
145C22×D4#6711yescharacteristic
146C23×C4—21no—
147C23×C4#6711yescharacteristic
148C23×C4#6711yescharacteristic
149C2×M16—62no—
150C2×M16—62no—
151C22×C8—62no—
152C4×D4—22no—
153C4×D4—22no—
154C4×D4—22no—
155C4×D4—22no—
156C4×D4—44no—
157C4×D4—44no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4C2×Dic311yescharacteristic
161C2×C4⋊C4#6711yescharacteristic
162C2×C22⋊C4—21no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4C2×Dic311yescharacteristic
168C2×C22⋊C4C2×Dic311yescharacteristic
169C2×C42—21no—
170C2×C42C22×D311yescharacteristic
171C4⋊C8—34no—
172#100—64no—
173C22⋊C8—62no—
174C22⋊C8—62no—
175C22⋊C8—62no—
176#97—62no—
177#97—62no—
178C4×C8—62no—
179#95—21no—
180#95C2×Dic311yescharacteristic
181#95C2×Dic311yescharacteristic
182C23×C3—21no—
183C23×C3—21no—
184C23×C3—22no—
185C23×C3—21no—
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—41no—
189C23×C3—41no—
190C23×C3—41no—
191C23×C3—41no—
192C23×C3—41no—
193C23×C3—41no—
194C23×C3—41no—
195C23×C3—41no—
196C23×C3—41no—
197C23×C3—41no—
198C23×C3#10412yes—
199C23×C3C2×C22⋊C411yescharacteristic
200C3×D4—24no—
201C3×D4—44no—
202C3×D4—42no—
203C3×D4—48no—
204C3×D4—48no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—22no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—81no—
225C2×C3×C4—81no—
226C2×C3×C4—82no—
227C2×C3×C4—82no—
228C2×C3×C4C4×D412yes—
229C2×C3×C4#12011yescharacteristic
230C2×C3×C4#12112yes—
231C2×C3×C4#12311yescharacteristic
232C3⋊C8—44no—
233C3⋊C8—44no—
234C24—21no—
235C24#21711yescharacteristic
236C24#21711yescharacteristic
237C2×D4—22no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—41no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4#21711yescharacteristic
262C22×C4#21711yescharacteristic
263C22×C4C22×Dic311yescharacteristic
264C22×C4C2×#6711yescharacteristic
265C22×C4C2×#6711yescharacteristic
266M16—64no—
267M16—122no—
268M16—242no—
269C2×C8—64no—
270C2×C8—122no—
271C2×C8—242no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C22⋊C4—22no—
277C22⋊C4—22no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C42—24no—
284C42D3×D412yes—
285C42#23712yes—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—22no—
289C22×C3—42no—
290C22×C3—42no—
291C22×C3—42no—
292C22×C3—42no—
293C22×C3—41no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3—42no—
298C22×C3—41no—
299C22×C3—81no—
300C22×C3—81no—
301C22×C3—81no—
302C22×C3—81no—
303C22×C3—81no—
304C22×C3—81no—
305C22×C3#38611yescharacteristic
306C22×C3C2×#10411yescharacteristic
307C22×C3#42111yescharacteristic
308C3×C4—22no—
309C3×C4—21no—
310C3×C4—22no—
311C3×C4—42no—
312C3×C4—42no—
313C3×C4—41no—
314C3×C4—41no—
315C3×C4—81no—
316C3×C4#45312yes—
317C23—21no—
318C23—21no—
319C23—22no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23#83712yes—
334C23C2×#21711yescharacteristic
335D4—24no—
336D4—44no—
337D4—42no—
338D4—48no—
339D4—48no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4D4×Dic312yes—
364C2×C4#93511yescharacteristic
365C2×C4#93711yescharacteristic
366C2×C4#93912yes—
367C8—124no—
368C8—124no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—42no—
372C2×C3—81no—
373C2×C3—82no—
374C2×C3#187111yescharacteristic
375C2×C3#188711yescharacteristic
376C2×C3#189111yescharacteristic
377C22—21no—
378C22—21no—
379C22—22no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—42no—
388C22—41no—
389C22—41no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22#522011yescharacteristic
397C22C2×#83711yescharacteristic
398C22#523611yescharacteristic
399C4—22no—
400C4—22no—
401C4—21no—
402C4—42no—
403C4—42no—
404C4—41no—
405C4—41no—
406C4—81no—
407C4#507712yes—
408C3#903111yescharacteristic, complemented, Sylow
409C2—21no—
410C2—21no—
411C2—42no—
412C2—81no—
413C2—82no—
414C2#7171311yescharacteristic
415C2#7171411yescharacteristic
416C2#7176311yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#2397C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#2397D3
hypocenterC3#9031
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2397

Complements

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

NH ≅ G/Ntype
2. #82302C2semidirect
3. C4×#833C2semidirect
4. #71599C2semidirect
5. #67615C2semidirect
67. C4⋊(C3⋊C8)D4semidirect
408. C3#9031semidirect

Maximal subgroups

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

subgroupindex
C3×#23972
#823022
C4×#8332
#715992
#676152
#90313