Finite Group Databasebeta

← #10696049

Subgroups — #10696049

Order 768.

560 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 97 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×#3101C211yesmaximal, characteristic
2#83368C211yesmaximal, characteristic, complemented
3#82348C211yesmaximal, characteristic
4#71840C211yesmaximal, characteristic, complemented
5#71598C211yesmaximal, characteristic
6#70770C211yesmaximal, characteristic, complemented
7#67737C211yesmaximal, characteristic, complemented
8#10016—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#483—21no—
13C3×#476—21no—
14C3×#461—21no—
15C3×#450—21no—
16C3×C4×SD16—22no—
17C3×#431C2211yescharacteristic
18#5222—21no—
19C4×#217—22no—
20#5091—21no—
21#5090—21no—
22#5050—21no—
23#5044—21no—
24C4×#216—22no—
25#4985—21no—
26#4953—21no—
27#4947—21no—
28#4932C2211yescharacteristic
29#4925C2211yescharacteristic
30#4924C2211yescharacteristic
31D4⋊(C3⋊C8)—22no—
32#4532—22no—
33#4456—22no—
34#3104—31no—
35#3101D311yescharacteristic
36#2398—31no—
37#1995—31no—
38#1487—31no—
39#1486—31no—
40#1274—31no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C22×Dic6D411yescharacteristic
44C2×C3×SD16—22no—
45C3×C4⋊Q8D411yescharacteristic
46C3×C4⋊Q8D411yescharacteristic
47C3×#125—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C3×#117—21no—
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C22⋊C4—21no—
59C2×C3×C42C2311yescharacteristic
60C2×#217—22no—
61C2×#217—22no—
62C2×#216—22no—
63C2×C4⋊Dic3D411yescharacteristic
64C2×Dic3⋊C4—21no—
65C2×C4×Dic3—21no—
66C2×C4×Dic3D412yes—
67C2×#208—21no—
68C4×Dic6—21no—
69C4×Dic6—21no—
70C4×Dic6—21no—
71C4×Dic6—21no—
72C3×#106—22no—
73C3×C4⋊C8—21no—
74C3×#103—22no—
75C3×#102—22no—
76C3×C22⋊C8—21no—
77C3×C4×C8D412yes—
78#835—22no—
79C3⋊#95—22no—
80C3⋊#95—22no—
81C3⋊#95—21no—
82C3⋊#95—21no—
83#830—21no—
84#810—22no—
85#807—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)—21no—
89C4×C3⋊C8D412yes—
90#533C2×D311yescharacteristic
91C2×C4×Q8—31no—
92C2×C4×D4C2×D311yescharacteristic
93#484—61no—
94#483—21no—
95#477—61no—
96#476—21no—
97#462—61no—
98#461—21no—
99#451—61no—
100#450—21no—
101C4×Q16—62no—
102C4×SD16—22no—
103#431—31no—
104#431C2×D311yescharacteristic
105#423—31no—
106#400—61no—
107#397—61no—
108#394—61no—
109#393—61no—
110C4×C22⋊C4—62no—
111#376—31no—
112#339—62no—
113Q8⋊C8—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—41no—
123C22×C3×C4—41no—
124C22×C3×C4—42no—
125C22×C3×C4—42no—
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×Dic3—21no—
130C22×Dic3—22no—
131C22×Dic3—21no—
132C2×Dic6—21no—
133C2×Dic6—21no—
134C2×Dic6—42no—
135C3×SD16—44no—
136C2×C3×C8—22no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4—41no—
145C3×C22⋊C4—41no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C42—41no—
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42#4111yescharacteristic
153#217—44no—
154#217—44no—
155#216—44no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158Dic3⋊C4—41no—
159Dic3⋊C4—41no—
160C4×Dic3—22no—
161C4×Dic3—22no—
162C4×Dic3—41no—
163C4×Dic3—41no—
164#208—42no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—41no—
167C22×Q8—31no—
168C22×D4#6711yescharacteristic
169C23×C4—21no—
170C2×Q16—62no—
171C2×SD16—22no—
172C2×M16—61no—
173C4⋊Q8#6711yescharacteristic
174C4⋊Q8#6711yescharacteristic
175#125—21no—
176C22⋊Q8—21no—
177C22⋊Q8—21no—
178C4×Q8—21no—
179C4×Q8—21no—
180C4×Q8—61no—
181C4×Q8—61no—
182C4×Q8—61no—
183C4×Q8—61no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—21no—
188#117—21no—
189C2×C4⋊C4—31no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4#6711yescharacteristic
192C2×C22⋊C4—21no—
193C2×C22⋊C4—62no—
194C2×C22⋊C4—62no—
195C2×C42—32no—
196C2×C42—61no—
197C2×C42C22×D311yescharacteristic
198#107—62no—
199#106—22no—
200C4⋊C8—21no—
201C4⋊C8—61no—
202C4⋊C8—61no—
203C4⋊C8—61no—
204#103—22no—
205#103—62no—
206#103—62no—
207#102—22no—
208C22⋊C8—21no—
209C22⋊C8—61no—
210C4×C8—32no—
211C4×C8#6712yes—
212#95—62no—
213#95—62no—
214#95—61no—
215#95—61no—
216C23×C3—22no—
217C23×C3—42no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3—42no—
221C23×C3#12011yescharacteristic
222C3×Q8—42no—
223C3×Q8—42no—
224C3×D4—22no—
225C3×D4—22no—
226C3×D4—41no—
227C3×D4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—42no—
234C2×C3×C4—82no—
235C2×C3×C4—81no—
236C2×C3×C4—81no—
237C2×C3×C4#12011yescharacteristic
238C2×C3×C4#12011yescharacteristic
239C2×C3×C4#12111yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12711yescharacteristic
243C2×C3×C4#13512yes—
244C2×C3×C4#13712yes—
245C2×Dic3—22no—
246C2×Dic3—41no—
247C2×Dic3—41no—
248C2×Dic3—42no—
249C2×Dic3—42no—
250C2×Dic3—42no—
251C2×Dic3—41no—
252C2×Dic3—42no—
253Dic6—22no—
254Dic6—22no—
255Dic6—41no—
256Dic6—41no—
257C3×C8—42no—
258C3⋊C8—42no—
259C3⋊C8—81no—
260C24—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—61no—
264C2×Q8—61no—
265C2×Q8—122no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—42no—
269C22×C4—21no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—61no—
275C22×C4—62no—
276C22×C4—61no—
277C22×C4D3×D411yescharacteristic
278C22×C4D3×D411yescharacteristic
279C22×C4C2×#6711yescharacteristic
280Q16—124no—
281SD16—44no—
282M16—122no—
283C2×C8—22no—
284C2×C8—62no—
285C2×C8—121no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—121no—
297C4⋊C4—121no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—124no—
302C22⋊C4—124no—
303C42—41no—
304C42—62no—
305C42—62no—
306C42—121no—
307C42—121no—
308C42D3×D411yescharacteristic
309C42#23711yescharacteristic
310C42C2×#6711yescharacteristic
311C42C2×#6711yescharacteristic
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—44no—
315C22×C3—42no—
316C22×C3—82no—
317C22×C3—81no—
318C22×C3—81no—
319C22×C3#39211yescharacteristic
320C22×C3#45212yes—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—81no—
326C3×C4—81no—
327C3×C4—81no—
328C3×C4#46412yes—
329C3×C4#49511yescharacteristic
330C3×C4#49711yescharacteristic
331Dic3—42no—
332Dic3—82no—
333Dic3—81no—
334C23—22no—
335C23—42no—
336C23—41no—
337C23—42no—
338C23—41no—
339C23#93711yescharacteristic
340Q8—42no—
341Q8—42no—
342Q8—62no—
343Q8—62no—
344Q8—121no—
345Q8—121no—
346D4—22no—
347D4—22no—
348D4—41no—
349D4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—62no—
357C2×C4—82no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4—122no—
363C2×C4—122no—
364C2×C4—122no—
365C2×C4—122no—
366C2×C4—121no—
367C2×C4#91512yes—
368C2×C4#91612yes—
369C2×C4#93711yescharacteristic
370C2×C4#93811yescharacteristic
371C2×C4#93911yescharacteristic
372C2×C4#93911yescharacteristic
373C2×C4#94011yescharacteristic
374C2×C4#95311yescharacteristic
375C8—42no—
376C8—122no—
377C8—241no—
378C2×C3—22no—
379C2×C3—82no—
380C2×C3#201011yescharacteristic
381C2×C3#201011yescharacteristic
382C2×C3#201511yescharacteristic
383C22—21no—
384C22—21no—
385C22—44no—
386C22—42no—
387C22—82no—
388C22—81no—
389C22—81no—
390C22#517312yes—
391C22#522311yescharacteristic
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—81no—
397C4—81no—
398C4—81no—
399C4—122no—
400C4—242no—
401C4—241no—
402C4#516612yes—
403C4#516711yescharacteristic
404C4#516811yescharacteristic
405C3#1001611yescharacteristic, complemented, Sylow
406C2—22no—
407C2—82no—
408C2#7171511yescharacteristic
409C2#7205811yescharacteristic
410C2#7211811yescharacteristic
subgroupG/H
centerC2×C4#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3101C2
socleC22×C3#392
prime radicalC22×C3×D4D4
hypercenter#3101D3
hypocenterC3#10016
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3101

Complements

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

NH ≅ G/Ntype
2. #83368C2semidirect
4. #71840C2semidirect
6. #70770C2semidirect
7. #67737C2semidirect
405. C3#10016semidirect

Maximal subgroups

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

subgroupindex
C3×#31012
#833682
#823482
#718402
#715982
#707702
#677372
#100163