Finite Group Databasebeta

← #29597370

Subgroups — #29597370

Order 15616.

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

normal subgroups only

This group has 668 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
1C2×#16839944C211yesmaximal, characteristic
2C2×#16839561C211yesmaximal, characteristic, complemented
3#21445777C212yesmaximal
4#21443946C212yesmaximal, complemented
5C61×#2369C211yesmaximal, characteristic, complemented
6C2×#14519825C2211yescharacteristic
7C2×#14519808—21no—
8#16839944—22no—
9#16839944C414yescomplemented
10C2×#14519705—21no—
11C2×#14519705—22no—
12#16839761—22no—
13C2×#14519696C2212yes—
14C2×#14519696C2211yescharacteristic
15#16839561—22no—
16C2×C4⋊Q8×C61C2211yescharacteristic
17C61×#391—21no—
18C61×#391C2212yes—
19C61×#389—22no—
20C4×C4⋊C4×C61—22no—
21C4×C4⋊C4×C61—21no—
22#14519825—21no—
23#14519825—22no—
24#14519825—42no—
25#14519825C2×C412yes—
26C2×#11758226—41no—
27C2×#11758222—22no—
28C2×#11758222—21no—
29C2×Q8×D61—41no—
30#14519808—24no—
31#14519808—42no—
32C2×#11758202—41no—
33C2×#11758201—41no—
34C2×#11758194—21no—
35C2×#11758194—22no—
36#14519705—24no—
37#14519705—44no—
38#14519700—42no—
39#14519697—42no—
40#14519696—21no—
41#14519696—22no—
42#14519696—22no—
43#14519696C2×C412yes—
44C22×Q8×C61D411yescharacteristic
45C22×Q8×C61C2311yescharacteristic
46C4⋊Q8×C61—22no—
47C4⋊Q8×C61C2×C412yes—
48C4⋊Q8×C61D412yes—
49C2×C4⋊C4×C61—21no—
50C2×C4⋊C4×C61—22no—
51C2×C4⋊C4×C61—21no—
52C2×C4⋊C4×C61—21no—
53C2×C4⋊C4×C61D412yes—
54C2×C42×C61—22no—
55C2×C42×C61—21no—
56C2×C42×C61D411yescharacteristic
57C2×C42×C61D412yes—
58#95×C61—22no—
59#95×C61—22no—
60#95×C61—22no—
61C2×Q8×C61—21no—
62C2×Q8×C61—41no—
63C2×Q8×C61—41no—
64C2×Q8×C61—41no—
65C2×Q8×C61—41no—
66C2×Q8×C61C22⋊C412yes—
67C2×Q8×C61C22×C411yescharacteristic
68C2×Q8×C61C2×D411yescharacteristic
69C2×Q8×C61#4112yes—
70C22×C4×C61—21no—
71C22×C4×C61—22no—
72C22×C4×C61—21no—
73C22×C4×C61—22no—
74C22×C4×C61C2×D411yescharacteristic
75C22×C4×C61C2×D412yes—
76C2×C22⋊D61—41no—
77#11758226—42no—
78#11758226—42no—
79#11758226—44no—
80Q8×D61—42no—
81Q8×D61—42no—
82Q8×D61—44no—
83#11758222—22no—
84#11758222—24no—
85#11758222—22no—
86#11758222—24no—
87#11758222—42no—
88#11758222—41no—
89#11758222—42no—
90#11758222—41no—
91C2×D244—22no—
92C2×D244—21no—
93C2×C4×D61—42no—
94C2×C4×D61—41no—
95C2×Dic122—22no—
96C2×Dic122—21no—
97C4⋊C4×C61—24no—
98C4⋊C4×C61—24no—
99C4⋊C4×C61—24no—
100C4⋊C4×C61—42no—
101C4⋊C4×C61—42no—
102C4⋊C4×C61—81no—
103C42×C61—21no—
104C42×C61—44no—
105C42×C61—42no—
106C42×C61—81no—
107C42×C61C22⋊C412yes—
108#11758202—44no—
109#11758201—44no—
110#11758194—22no—
111#11758194—41no—
112#11758194—42no—
113#11758194—42no—
114C2×#92418—41no—
115C2×#92418—42no—
116C23×C61#12011yescharacteristic
117C22×D61—41no—
118Q8×C61—41no—
119Q8×C61—41no—
120Q8×C61—42no—
121Q8×C61—42no—
122Q8×C61—81no—
123C2×C4×C61—22no—
124C2×C4×C61—22no—
125C2×C4×C61—22no—
126C2×C4×C61—22no—
127C2×C4×C61—21no—
128C2×C4×C61—22no—
129C2×C4×C61—21no—
130C2×C4×C61—24no—
131C2×C4×C61—22no—
132C2×C4×C61—42no—
133C2×C4×C61—42no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—42no—
137C2×C4×C61—42no—
138C2×C4×C61—42no—
139C2×C4×C61C2×C22⋊C411yescharacteristic
140C2×C4×C61C4×D412yes—
141C2×C4×C61#12112yes—
142C2×C4×C61#12311yescharacteristic
143C22⋊D61—42no—
144C22⋊D61—42no—
145C2×Dic61—41no—
146D244—22no—
147D244—24no—
148D244—42no—
149D244—41no—
150C4×D61—42no—
151C4×D61—44no—
152C4×D61—44no—
153C4×D61—42no—
154Dic122—22no—
155Dic122—24no—
156Dic122—42no—
157Dic122—41no—
158#92418—42no—
159#92418—82no—
160#13058—611nomaximal, Sylow
161C22×C61—21no—
162C22×C61—21no—
163C22×C61#38611yescharacteristic
164C22×C61B(4,𝔽2)11yescharacteristic
165C22×C61B(4,𝔽2)11yescharacteristic
166C2×D61—42no—
167C2×D61—81no—
168C2×D61—81no—
169C4×C61—24no—
170C4×C61—21no—
171C4×C61—22no—
172C4×C61—22no—
173C4×C61—21no—
174C4×C61—44no—
175C4×C61—42no—
176C4×C61—82no—
177Dic61—42no—
178C2×#456—611no—
179#2369D6111yescharacteristic, complemented
180C2×#356—611no—
181#1793—612no—
182#1394—612no—
183C2×C61—21no—
184C2×C61—21no—
185C2×C61#188911yescharacteristic
186C2×C61#220411yescharacteristic
187C2×C61#220511yescharacteristic
188D61—82no—
189C2×2−1+4—611no—
190C2×#137—1221no—
191C2×C4⋊Q8C2×D6111yescharacteristic
192#456—614no—
193#456—1222no—
194C2×#104—1221no—
195C2×#104—1222no—
196#417—1222no—
197C2×#101—611no—
198C2×#101—612no—
199#391—21no—
200#391C2×D6112yes—
201#389—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204#356—1222no—
205C61#1305811yescharacteristic, complemented, Sylow
2062−1+4—612no—
2072−1+4—1221no—
2082−1+4—1222no—
2092−1+4—2442no—
210C2×#41—1221no—
211C2×#41—1222no—
212C2×#41—2441no—
213C22×Q8—2441no—
214C22×Q8C22⋊D6111yescharacteristic
215C22×Q8C22×D6111yescharacteristic
216#137—1224no—
217#137—2442no—
218C2×Q16—2441no—
219C2×SD16—2441no—
220C2×M16—1221no—
221C2×M16—1222no—
222C4⋊Q8—22no—
223C4⋊Q8C4×D6112yes—
224C4⋊Q8D24412yes—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4C22⋊D6112yes—
230C2×C42—22no—
231C2×C42—21no—
232C2×C42C22⋊D6111yescharacteristic
233C2×C42C22⋊D6112yes—
234#104—1224no—
235#104—2444no—
236#103—2442no—
237#102—2442no—
238#101—612no—
239#101—1221no—
240#101—1222no—
241#101—1222no—
242#95—22no—
243#95—22no—
244#95—22no—
245#41—1224no—
246#41—1222no—
247#41—1222no—
248#41—1224no—
249#41—2442no—
250#41—2441no—
251#41—2442no—
252#41—2442no—
253#41—2444no—
254#41—2442no—
255#41—2441no—
256C2×Q8—21no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—1222no—
262C2×Q8—1221no—
263C2×Q8—2442no—
264C2×Q8—2442no—
265C2×Q8—2444no—
266C2×Q8#1175819812yes—
267C2×Q8C2×C4×D6111yescharacteristic
268C2×Q8C2×D24411yescharacteristic
269C2×Q8#1175822212yes—
270C2×D4—1222no—
271C2×D4—1221no—
272C2×D4—2441no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—2442no—
278C22×C4—2441no—
279C22×C4C2×C22⋊D6111yescharacteristic
280C22×C4C2×C22⋊D6112yes—
281Q16—2444no—
282SD16—2444no—
283M16—1222no—
284M16—2442no—
285M16—2442no—
286M16—2441no—
287C2×C8—2442no—
288C2×C8—2441no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—81no—
295C42—21no—
296C42—42no—
297C42—44no—
298C42—81no—
299C42#1175819812yes—
300C23—2441no—
301C23#1451974611yescharacteristic
302Q8—41no—
303Q8—41no—
304Q8—42no—
305Q8—42no—
306Q8—81no—
307Q8—1222no—
308Q8—1224no—
309Q8—2442no—
310Q8—2441no—
311D4—1222no—
312D4—1224no—
313D4—2442no—
314D4—2441no—
315D4—2442no—
316D4—2442no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—22no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—2444no—
334C2×C4—2442no—
335C2×C4—2444no—
336C2×C4—2442no—
337C2×C4—2441no—
338C2×C4C2×#1175819811yescharacteristic
339C2×C4C4×C22⋊D6112yes—
340C2×C4#1451975112yes—
341C2×C4#1451976011yescharacteristic
342C8—4882no—
343C8—2442no—
344C22—21no—
345C22—21no—
346C22—4881no—
347C22—4881no—
348C22—2442no—
349C22#1683964111yescharacteristic
350C22#1683994911yescharacteristic
351C22#1683994911yescharacteristic
352C4—24no—
353C4—22no—
354C4—21no—
355C4—22no—
356C4—21no—
357C4—44no—
358C4—42no—
359C4—82no—
360C4—2442no—
361C2—21no—
362C2—21no—
363C2—4882no—
364C2#2144622711yescharacteristic
365C2#2144784511yescharacteristic
366C2#2144784911yescharacteristic
subgroupG/H
centerC22#16839949
commutatorC2×Q8×C61C22×C4
FrattiniC22×Q8C22×D61
FittingC61×#2369C2
socleC22×C61B(4,𝔽2)
prime radicalC2×#14519825C22
cosocleC22×Q8×C61C23
hypercenter#2369D61
hypocenterC61#13058
LCSG ▷ C2×Q8×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #2369

Complements

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

NH ≅ G/Ntype
2. C2×#16839561C2semidirect
4. #21443946C2semidirect
5. C61×#2369C2semidirect
9. #16839944C4semidirect
179. #2369D61semidirect
205. C61#13058semidirect

Maximal subgroups

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

subgroupindex
C2×#168399442
C2×#168395612
#214457772
#214439462
C61×#23692
#1305861