Finite Group Databasebeta

← #10691168

Subgroups — #10691168

Order 768.

560 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 99 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#82421C211yesmaximal, characteristic
2#82302C211yesmaximal, characteristic, complemented
3C3×C4×#100C211yesmaximal, characteristic, complemented
4#69860C212yesmaximal, complemented
5#67666C212yesmaximal, complemented
6#9187—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#875C2211yescharacteristic
9C2×C3×#100—21no—
10C3×C4×M16C2212yes—
11#5223—21no—
12#5213—21no—
13#5213—21no—
14#5207—21no—
15C2×#830—21no—
16#5011—24no—
17#4992—21no—
18#4960C411yescharacteristic
19#4960C411yescharacteristic
20#4947C411yescharacteristic
21#4947C411yescharacteristic
22#4937C2211yescharacteristic
23#4926C2212yes—
24C3×#343—21no—
25#4464—24no—
26#2973—31no—
27#2631—31no—
28C4×#100D311yescharacteristic, complemented
29#1527—32no—
30#1288—32no—
31C22×C3×D4D411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C23×C3×C4D411yescharacteristic
34C22×#67—21no—
35C22×#67—21no—
36C22×D12C2×C411yescharacteristic
37C22×Dic6C2×C411yescharacteristic
38C2×C3×M16—22no—
39C3×C4×D4—22no—
40C3×C4×D4—22no—
41C2×C3×C4⋊C4D411yescharacteristic
42C2×C3×C22⋊C4C2×C411yescharacteristic
43C2×C3×C22⋊C4C2×C411yescharacteristic
44C2×C3×C42C2311yescharacteristic
45C2×#217—21no—
46C2×#217—21no—
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—21no—
49C2×(C2×D3)⋊C4—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×#208—22no—
52C22×C3⋊C8—22no—
53#875—22no—
54#875—22no—
55C3×#100—24no—
56C3×#97—22no—
57C3×C4×C8—22no—
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95C2×C411yescharacteristic
63C3⋊#95C2×C411yescharacteristic
64#830—22no—
65#830—22no—
66#830—22no—
67C4⋊(C3⋊C8)—22no—
68#803D414yescomplemented
69C2×#124—31no—
70C2×C4×D4C2×D311yescharacteristic
71#435—64no—
72#432—32no—
73C2×#100—21no—
74#407—61no—
75C2×C22⋊C8—61no—
76C4×M16C2×D312yes—
77#397—61no—
78#393—31no—
79#393—31no—
80#392—31no—
81#392—31no—
82#392—61no—
83#386—61no—
84#386—61no—
85#383—31no—
86#343—21no—
87#329—64no—
88C24×C3C22⋊C411yescharacteristic
89C24×C3C22⋊C411yescharacteristic
90C23×D3—21no—
91C2×C3×D4—22no—
92C2×C3×D4—22no—
93C2×C3×D4—41no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4—81no—
99C22×C3×C4—81no—
100C22×C3×C4C22⋊C411yescharacteristic
101C22×C3×C4C22⋊C411yescharacteristic
102C22×C3×C4C22×C411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C2×#67—42no—
106C2×#67—42no—
107C2×#67—81no—
108C2×#67—81no—
109C2×#67—81no—
110C2×#67—81no—
111C22×Dic3—21no—
112C22×Dic3—21no—
113C22×Dic3—21no—
114C2×D12—22no—
115C2×D12—81no—
116C2×Dic6—22no—
117C2×Dic6—81no—
118C3×M16—24no—
119C3×M16—42no—
120C2×C3×C8—42no—
121C3×C4⋊C4—22no—
122C3×C22⋊C4—22no—
123C3×C22⋊C4—22no—
124C3×C42C2×D412yes—
125C3×C42#4112yes—
126#217—81no—
127#217—81no—
128(C2×D3)⋊C4—41no—
129(C2×D3)⋊C4—42no—
130(C2×D3)⋊C4—41no—
131(C2×D3)⋊C4—81no—
132(C2×D3)⋊C4—81no—
133#208—82no—
134C2×C3⋊C8—24no—
135C2×C3⋊C8—82no—
136C22×Q8—31no—
137C22×D4—31no—
138C22×D4—61no—
139C22×D4—61no—
140C22×D4#6711yescharacteristic
141C23×C4D1211yescharacteristic
142C23×C4D1211yescharacteristic
143C2×M16—22no—
144C2×M16—62no—
145C22×C8—62no—
146#124—62no—
147#124—62no—
148C4×D4—22no—
149C4×D4—22no—
150C2×C4⋊C4#6711yescharacteristic
151C2×C22⋊C4—61no—
152C2×C22⋊C4—61no—
153C2×C22⋊C4—61no—
154C2×C22⋊C4—61no—
155C2×C22⋊C4—61no—
156C2×C22⋊C4—61no—
157C2×C22⋊C4C4×D311yescharacteristic
158C2×C22⋊C4C4×D311yescharacteristic
159C2×C42C22×D311yescharacteristic
160C4⋊C8—62no—
161#100—24no—
162C22⋊C8—62no—
163C22⋊C8—62no—
164C22⋊C8—62no—
165#97—22no—
166#97—34no—
167C4×C8—22no—
168#95—31no—
169#95—31no—
170#95—61no—
171#95—61no—
172#95—61no—
173#95—61no—
174C23×C3—21no—
175C23×C3—21no—
176C23×C3—21no—
177C23×C3—21no—
178C23×C3—21no—
179C23×C3—41no—
180C23×C3—41no—
181C23×C3—41no—
182C23×C3—41no—
183C23×C3#10412yes—
184C23×C3C2×C22⋊C411yescharacteristic
185C22×D3—41no—
186C22×D3—41no—
187C22×D3—41no—
188C22×D3—41no—
189C22×D3—41no—
190C22×D3—41no—
191C22×D3—41no—
192C3×D4—44no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—81no—
204C2×C3×C4—81no—
205C2×C3×C4C4×D412yes—
206C2×C3×C4#12011yescharacteristic
207C2×C3×C4#12112yes—
208C2×C3×C4#12311yescharacteristic
209#67—84no—
210#67—84no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220D12—82no—
221Dic6—82no—
222C3×C8—44no—
223C3⋊C8—44no—
224C24—61no—
225C24(C2×D3)⋊C411yescharacteristic
226C24(C2×D3)⋊C411yescharacteristic
227C2×Q8—62no—
228C2×Q8—241no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—41no—
232C2×D4—62no—
233C2×D4—122no—
234C2×D4—122no—
235C2×D4—241no—
236C2×D4—241no—
237C2×D4—241no—
238C2×D4—241no—
239C2×D4—241no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—61no—
245C22×C4—61no—
246C22×C4—61no—
247C22×C4—81no—
248C22×C4—81no—
249C22×C4(C2×D3)⋊C411yescharacteristic
250C22×C4(C2×D3)⋊C411yescharacteristic
251C22×C4C2×C4×D311yescharacteristic
252C22×C4C2×D1211yescharacteristic
253C22×C4C2×#6711yescharacteristic
254M16—24no—
255M16—42no—
256M16—242no—
257C2×C8—42no—
258C2×C8—64no—
259C2×C8—242no—
260C4⋊C4—22no—
261C22⋊C4—22no—
262C22⋊C4—22no—
263C22⋊C4—121no—
264C22⋊C4—121no—
265C22⋊C4—122no—
266C22⋊C4—241no—
267C22⋊C4—241no—
268C22⋊C4—241no—
269C22⋊C4—241no—
270C42D3×D412yes—
271C42#23712yes—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—22no—
275C22×C3—42no—
276C22×C3—42no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—81no—
283C22×C3—81no—
284C22×C3#38611yescharacteristic
285C22×C3C2×#10411yescharacteristic
286C22×C3#42111yescharacteristic
287C2×D3—81no—
288C2×D3—81no—
289C2×D3—81no—
290C2×D3—81no—
291C2×D3—81no—
292C2×D3—81no—
293C2×D3—81no—
294C3×C4—22no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—22no—
298C3×C4—41no—
299C3×C4—41no—
300Dic3—81no—
301Dic3—81no—
302Dic3—81no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—121no—
313C23—121no—
314C23—121no—
315C23—121no—
316C23—121no—
317C23—121no—
318C23—121no—
319C23#80512yes—
320C23C2×(C2×D3)⋊C411yescharacteristic
321Q8—242no—
322D4—44no—
323D4—242no—
324D4—244no—
325D4—244no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4—121no—
347C2×C4#67⋊C412yes—
348C2×C4#88211yescharacteristic
349C2×C4#88611yescharacteristic
350C2×C4#93912yes—
351C8—44no—
352C8—124no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—42no—
356C2×C3—81no—
357C2×C3#187111yescharacteristic
358C2×C3#191111yescharacteristic
359C2×C3#191411yescharacteristic
360D3—161no—
361C22—21no—
362C22—21no—
363C22—22no—
364C22—42no—
365C22—42no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—81no—
372C22—81no—
373C22—241no—
374C22—241no—
375C22—241no—
376C22—241no—
377C22—241no—
378C22—241no—
379C22—241no—
380C22C2×#80511yescharacteristic
381C22#495811yescharacteristic
382C22#500511yescharacteristic
383C4—22no—
384C4—21no—
385C4—21no—
386C4—22no—
387C4—41no—
388C4—41no—
389C4—241no—
390C4—241no—
391C4—241no—
392C3#918711yescharacteristic, complemented, Sylow
393C2—21no—
394C2—21no—
395C2—42no—
396C2—81no—
397C2—481no—
398C2#6945311yescharacteristic
399C2#7007111yescharacteristic
400C2#7013711yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×C4×#100C2
socleC22×C3#386
prime radical#82421C2
cosocleC2×C3×C42C23
hypercenterC4×#100D3
hypocenterC3#9187
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
2. #82302C2semidirect
3. C3×C4×#100C2semidirect
4. #69860C2semidirect
5. #67666C2semidirect
28. C4×#100D3semidirect
68. #803D4semidirect
392. C3#9187semidirect

Maximal subgroups

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

subgroupindex
#824212
#823022
C3×C4×#1002
#698602
#676662
#91873